Finite Group Databasebeta

← #33249461

Subgroups — #33249461

Order 18688.

2286 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989158C211yesmaximal, characteristic
2#22613110C212yesmaximal, complemented
3#22613074C212yesmaximal, complemented
4#22613064C212yesmaximal, complemented
5#22613019C212yesmaximal, complemented
6#22612958C212yesmaximal, complemented
7#22612936C212yesmaximal, complemented
8#22612826C212yesmaximal, complemented
9#22612763C212yesmaximal, complemented
10#22612740C214yesmaximal, complemented
11#22612718C214yesmaximal, complemented
12#22612609C212yesmaximal, complemented
13#22612459C212yesmaximal, complemented
14#22612422C211yesmaximal, characteristic
15C73×#2311C211yesmaximal, characteristic
16C23×Dic146C2211yescharacteristic
17#17989158C4116yescomplemented
18C2×#14556589C2212yes—
19C2×#14556578C2212yes—
20C2×#14556572C2212yes—
21C2×#14556572C2212yes—
22C2×#14556570C2212yes—
23C2×#14556570C2212yes—
24C2×#14556556C2212yes—
25C2×#14556556C2212yes—
26C2×#14556550C2211yescharacteristic
27C2×#14556550C2211yescharacteristic
28C22×C4×Dic73C2211yescharacteristic
29#17988494C2211yescharacteristic
30#17988494C2211yescharacteristic
31#17988492C2212yes—
32#17988492C2212yes—
33#17988491C2214yes—
34#17988491C2214yes—
35#17988490C2211yescharacteristic
36#17988490C2211yescharacteristic
37#17988484C2214yes—
38#17988483C2212yes—
39#17988483C2212yes—
40#17988482C2214yes—
41#17988481C2214yes—
42#17988481C2214yes—
43#17988475C2214yes—
44#17988475C2214yes—
45#17988475C2214yes—
46#17988475C2214yes—
47#17988474C2214yes—
48#17988471C2214yes—
49#17988469C2214yes—
50#17988468C2214yes—
51#17988467C2212yes—
52#17988467C2212yes—
53#17988460C2214yes—
54#17988459C2214yes—
55#17988454C2214yes—
56#17988445C2214yes—
57C4⋊C4×Dic73C2212yes—
58C4⋊C4×Dic73C2212yes—
59C4×#11924007C2214yes—
60#17988438C2214yes—
61#17988438C2214yes—
62C22⋊C4×Dic73C2214yes—
63C22⋊C4×Dic73C2214yes—
64C4×C22⋊Dic73C2214yes—
65C4×C22⋊Dic73C2212yes—
66C4×C22⋊Dic73C2212yes—
67C22×#41×C73C2211yescharacteristic
68C2×#117×C73C2211yescharacteristic
69C22×C4⋊C4×C73C2211yescharacteristic
70#391×C73C2212yes—
71#390×C73C2212yes—
72#386×C73C2212yes—
73#386×C73C2212yes—
74#380×C73C2212yes—
75#380×C73C2212yes—
76C22×Dic146—42no—
77C22×Dic146—42no—
78C22×Dic146C2×C418yescomplemented
79C22×Dic146C2312yes—
80C22×Dic146C2312yes—
81C23×Dic73C2312yes—
82C23×Dic73C2312yes—
83#14556589C2×C4116yescomplemented
84#14556578C2×C4116yes—
85#14556572C2×C4116yescomplemented
86#14556572C2×C4116yescomplemented
87#14556570C2×C4116yescomplemented
88#14556570C2×C4116yescomplemented
89#14556556—44no—
90#14556556—44no—
91#14556550—42no—
92#14556550C2×C418yescomplemented
93C2×C4⋊Dic73C2311yescharacteristic
94C2×C4⋊Dic73C2311yescharacteristic
95C2×C4⋊Dic73C2312yes—
96C2×#11924007C2314yes—
97C2×#11924007C2314yes—
98C2×#11924007C2314yes—
99C2×#11924007C2314yes—
100C2×#11924007C2314yes—
101C2×#11924007C2314yes—
102C2×C22⋊Dic73C2312yes—
103C2×C22⋊Dic73C2312yes—
104C2×C22⋊Dic73C2312yes—
105C2×C22⋊Dic73C2312yes—
106C2×C22⋊Dic73C2312yes—
107C2×C22⋊Dic73C2314yes—
108C2×C22⋊Dic73C2314yes—
109C2×C22⋊Dic73C2312yes—
110C2×C4×Dic73—42no—
111C2×C4×Dic73C2312yes—
112C2×C4×Dic73C2312yes—
113C2×C4×Dic73C2314yes—
114C2×C4×Dic73C2314yes—
115C2×C4×Dic73C2314yes—
116C2×C4×Dic73C2311yescharacteristic
117C2×C4×Dic73C2314yes—
118C2×C4×Dic73C2311yescharacteristic
119C2×C4×Dic73C2312yes—
120#14556498C2314yes—
121#14556498C2312yes—
122#14556498C2312yes—
123#14556498C2312yes—
124#14556498C2314yes—
125#14556498C2312yes—
126#14556498C2314yes—
127#14556498C2314yes—
128#14556498C2314yes—
129#14556498C2314yes—
130#14556498C2314yes—
131#14556498C2314yes—
132C2×#41×C73—42no—
133C2×#41×C73—42no—
134C2×#41×C73C2×C418yes—
135C22×Q8×C73C2311yescharacteristic
136C22×D4×C73C2311yescharacteristic
137C22×D4×C73C2311yescharacteristic
138C22×D4×C73C2311yescharacteristic
139C23×C4×C73C2311yescharacteristic
140C23×C4×C73C2311yescharacteristic
141C23×C4×C73C2311yescharacteristic
142C23×C4×C73C2312yes—
143#117×C73—42no—
144C2×C4⋊C4×C73—42no—
145C2×C4⋊C4×C73C2312yes—
146C2×C4⋊C4×C73C2312yes—
147C2×C4⋊C4×C73C2312yes—
148C2×C22⋊C4×C73C2314yes—
149C2×C22⋊C4×C73C2314yes—
150C2×C22⋊C4×C73C2312yes—
151C2×C42×C73C2312yes—
152#95×C73C2314yes—
153#95×C73C2314yes—
154C2×Dic146—24no—
155C2×Dic146—24no—
156C2×Dic146—416no—
157C2×Dic146—42no—
158C2×Dic146—42no—
159C2×Dic146C22×C418yes—
160C2×Dic146C22×C418yes—
161C22×Dic73—24no—
162C22×Dic73—42no—
163C22×Dic73—42no—
164C22×Dic73—42no—
165C22×Dic73—42no—
166C22×Dic73C22×C418yes—
167C22×Dic73C2412yes—
168C22×Dic73C2412yes—
169C22×Dic73C2412yes—
170C22×Dic73C2414yes—
171C22×Dic73C2414yes—
172C22×Dic73C2412yes—
173C4⋊Dic73—41no—
174C4⋊Dic73—41no—
175C4⋊Dic73C22×C418yes—
176#11924007—28no—
177#11924007—44no—
178#11924007—44no—
179#11924007—44no—
180#11924007C22×C4116yes—
181#11924007C22×C4116yes—
182C22⋊Dic73—24no—
183C22⋊Dic73—24no—
184C22⋊Dic73—42no—
185C22⋊Dic73—42no—
186C22⋊Dic73—42no—
187C22⋊Dic73C22×C418yes—
188C22⋊Dic73C22×C4116yes—
189C22⋊Dic73C22×C4116yes—
190C4×Dic73—24no—
191C4×Dic73—24no—
192C4×Dic73—41no—
193C4×Dic73—41no—
194C4×Dic73—44no—
195C4×Dic73—44no—
196C4×Dic73—44no—
197C4×Dic73—44no—
198C4×Dic73C22×C418yes—
199C24×C73C2411yescharacteristic
200C24×C73C2411yescharacteristic
201C24×C73C2411yescharacteristic
202#41×C73—416no—
203C2×Q8×C73—41no—
204C2×Q8×C73—41no—
205C2×Q8×C73C22×C414yes—
206C2×D4×C73—41no—
207C2×D4×C73—41no—
208C2×D4×C73—41no—
209C2×D4×C73—41no—
210C2×D4×C73—41no—
211C2×D4×C73—41no—
212C2×D4×C73C22×C414yes—
213C2×D4×C73C22×C414yes—
214C2×D4×C73C22×C414yes—
215C22×C4×C73—22no—
216C22×C4×C73—22no—
217C22×C4×C73—22no—
218C22×C4×C73—22no—
219C22×C4×C73—22no—
220C22×C4×C73—22no—
221C22×C4×C73—42no—
222C22×C4×C73—44no—
223C22×C4×C73C22×C414yes—
224C22×C4×C73C22×C414yes—
225C22×C4×C73C22×C414yes—
226C22×C4×C73C2414yes—
227C22×C4×C73C2414yes—
228C22×C4×C73C2411yescharacteristic
229C22×C4×C73C2411yescharacteristic
230C22×C4×C73C2411yescharacteristic
231C22×C4×C73C2411yescharacteristic
232C4⋊C4×C73—24no—
233C4⋊C4×C73—24no—
234C4⋊C4×C73—42no—
235C22⋊C4×C73—44no—
236C22⋊C4×C73—44no—
237C22⋊C4×C73—42no—
238C42×C73—42no—
239C23×C73—21no—
240C23×C73—21no—
241C23×C73—21no—
242C23×C73—21no—
243C23×C73—21no—
244C23×C73—21no—
245C23×C73—22no—
246C23×C73—22no—
247C23×C73—22no—
248C23×C73—22no—
249C23×C73—22no—
250C23×C73—22no—
251C23×C73C23×C412yes—
252C23×C73C23×C412yes—
253C23×C73C23×C412yes—
254C23×C73C2511yescharacteristic
255Q8×C73—44no—
256D4×C73—44no—
257D4×C73—44no—
258D4×C73—44no—
259C2×C4×C73—21no—
260C2×C4×C73—21no—
261C2×C4×C73—21no—
262C2×C4×C73—21no—
263C2×C4×C73—21no—
264C2×C4×C73—21no—
265C2×C4×C73—28no—
266C2×C4×C73—28no—
267C2×C4×C73—28no—
268C2×C4×C73—24no—
269C2×C4×C73—24no—
270C2×C4×C73—24no—
271C2×C4×C73—24no—
272C2×C4×C73—24no—
273C2×C4×C73—24no—
274C2×C4×C73—84no—
275C2×C4×C73C23×C412yes—
276C2×C4×C73C23×C412yes—
277C2×C4×C73C23×C412yes—
278C2×C4×C73C23×C412yes—
279C2×C4×C73H(5,𝔽2)12yes—
280C2×C4×C73H(5,𝔽2)12yes—
281C2×Dic73—22no—
282C2×Dic73—24no—
283C2×Dic73—24no—
284C2×Dic73—24no—
285C2×Dic73—24no—
286C2×Dic73—22no—
287C2×Dic73—22no—
288C2×Dic73—22no—
289C2×Dic73—22no—
290C2×Dic73—22no—
291C2×Dic73—22no—
292C2×Dic73—22no—
293C2×Dic73—22no—
294C2×Dic73—22no—
295C2×Dic73—22no—
296C2×Dic73—48no—
297C2×Dic73—48no—
298C2×Dic73C23×C418yes—
299C2×Dic73C23×C418yes—
300C2×Dic732−1+414yes—
301Dic146—216no—
302Dic146—48no—
303C22×C73—22no—
304C22×C73—22no—
305C22×C73—22no—
306C22×C73—24no—
307C22×C73—24no—
308C22×C73—24no—
309C22×C73—42no—
310C22×C73—42no—
311C22×C73—42no—
312C22×C73—41no—
313C22×C73—41no—
314C22×C73—41no—
315C22×C73—41no—
316C22×C73—41no—
317C22×C73—41no—
318C22×C73C24×C411yescharacteristic
319C22×C73C2×H(5,𝔽2)11yescharacteristic
320C22×C73C2×H(5,𝔽2)11yescharacteristic
321C22×C73C2×2−1+412yes—
322C22×C73#58512yes—
323C4×C73—22no—
324C4×C73—22no—
325C4×C73—22no—
326C4×C73—44no—
327C4×C73—44no—
328C4×C73#51814yes—
329Dic73—28no—
330Dic73—28no—
331Dic73—24no—
332Dic73—42no—
333Dic73—42no—
334Dic73—42no—
335#34413—731nomaximal, Sylow
336C2×C73—42no—
337C2×C73—42no—
338C2×C73—42no—
339C2×C73C2×#51811yescharacteristic
340C2×C73C4×2−1+412yes—
341C2×C73#354112yes—
342C2×C73#355011yescharacteristic
343C2×C73#355111yescharacteristic
344C2×#536—731no—
345#2381—732no—
346#2376—732no—
347#2374—732no—
348#2369—732no—
349#2362—732no—
350#2360—732no—
351#2347—732no—
352Q8×C22⋊C4—732no—
353#2337—734no—
354#2335—734no—
355#2323—732no—
356#2311D7311yescharacteristic
357C4×C22⋊Q8—732no—
358#2297—731no—
359C73#3441311yescharacteristic, complemented, Sylow
360C22×#41C2×D7311yescharacteristic
361C23×Q8—731no—
362#536—7316no—
363C2×C4⋊Q8—732no—
364C2×#124—732no—
365C2×#123—732no—
366C2×#123—732no—
367C2×C22⋊Q8—732no—
368C2×C22⋊Q8—732no—
369C2×C4×Q8—732no—
370C2×C4×Q8—732no—
371C2×#117—731no—
372C2×#117—731no—
373C2×#117C2×D7311yescharacteristic
374C22×C4⋊C4C2×D7311yescharacteristic
375C22×C42—731no—
376#391—731no—
377#391—731no—
378#391—732no—
379#391—732no—
380#391—734no—
381#391—734no—
382#391C2×D7312yes—
383#390—731no—
384#390—731no—
385#390C2×D7312yes—
386#389—732no—
387#389—734no—
388#389—732no—
389#389—734no—
390#388—734no—
391#388—734no—
392#388—734no—
393#388—734no—
394#388—734no—
395#388—734no—
396#387—734no—
397#387—734no—
398#387—734no—
399#387—734no—
400#386—732no—
401#386—732no—
402#386C2×D7312yes—
403#386C2×D7312yes—
404#385—734no—
405#385—734no—
406#383—734no—
407#380—734no—
408#380C2×D7312yes—
409#380C2×D7312yes—
410C4×C4⋊C4—732no—
411C4×C4⋊C4—732no—
412C4×C4⋊C4—734no—
413C4×C4⋊C4—734no—
414C4×C4⋊C4—734no—
415C4×C22⋊C4—734no—
416C4×C22⋊C4—732no—
417C4×C22⋊C4—732no—
418C4×C22⋊C4—734no—
419C4×C22⋊C4—734no—
420C2×#41—42no—
421C2×#41—42no—
422C2×#41C4×D7318yes—
423C22×Q8—2922no—
424C22×Q8—2922no—
425C22×Q8—738no—
426C22×Q8—732no—
427C22×Q8—732no—
428C22×Q8C22×D7311yescharacteristic
429C22×D4C22×D7311yescharacteristic
430C22×D4C22×D7311yescharacteristic
431C22×D4C22×D7311yescharacteristic
432C23×C4—732no—
433C23×C4—732no—
434C23×C4C22×D7311yescharacteristic
435C23×C4C22×D7311yescharacteristic
436C23×C4C22×D7311yescharacteristic
437C23×C4C22×D7312yes—
438C4⋊Q8—7316no—
439#124—7316no—
440#123—7316no—
441#123—7316no—
442C22⋊Q8—7316no—
443C22⋊Q8—7316no—
444C4×Q8—2924no—
445C4×Q8—2924no—
446#117—42no—
447#117—2922no—
448#117—738no—
449C2×C4⋊C4—42no—
450C2×C4⋊C4—731no—
451C2×C4⋊C4—731no—
452C2×C4⋊C4—734no—
453C2×C4⋊C4—734no—
454C2×C4⋊C4—734no—
455C2×C4⋊C4—732no—
456C2×C4⋊C4—734no—
457C2×C4⋊C4—734no—
458C2×C4⋊C4—734no—
459C2×C4⋊C4C22×D7312yes—
460C2×C4⋊C4C22×D7312yes—
461C2×C4⋊C4C22×D7312yes—
462C2×C22⋊C4—732no—
463C2×C22⋊C4—732no—
464C2×C22⋊C4—732no—
465C2×C22⋊C4—732no—
466C2×C22⋊C4—734no—
467C2×C22⋊C4—734no—
468C2×C22⋊C4—732no—
469C2×C22⋊C4—732no—
470C2×C22⋊C4C22×D7312yes—
471C2×C22⋊C4C22×D7314yes—
472C2×C22⋊C4C22×D7314yes—
473C2×C42—2922no—
474C2×C42—734no—
475C2×C42—734no—
476C2×C42—734no—
477C2×C42—731no—
478C2×C42—734no—
479C2×C42—731no—
480C2×C42—732no—
481C2×C42—732no—
482C2×C42—732no—
483C2×C42C22×D7312yes—
484#95—734no—
485#95—732no—
486#95—732no—
487#95—734no—
488#95—732no—
489#95—734no—
490#95—734no—
491#95—732no—
492#95—734no—
493#95—734no—
494#95—734no—
495#95—734no—
496#95C22×D7314yes—
497#95C22×D7314yes—
498C24C23×D7311yescharacteristic
499C24C23×D7311yescharacteristic
500C24C23×D7311yescharacteristic
501#41—416no—
502C2×Q8—41no—
503C2×Q8—41no—
504C2×Q8—29216no—
505C2×Q8—2922no—
506C2×Q8—2922no—
507C2×Q8—738no—
508C2×Q8—738no—
509C2×Q8—1464no—
510C2×Q8—1464no—
511C2×Q8C2×C4×D7314yes—
512C2×D4—41no—
513C2×D4—41no—
514C2×D4—41no—
515C2×D4—41no—
516C2×D4—41no—
517C2×D4—41no—
518C2×D4C2×C4×D7314yes—
519C2×D4C2×C4×D7314yes—
520C2×D4C2×C4×D7314yes—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—42no—
528C22×C4—44no—
529C22×C4—2922no—
530C22×C4—2922no—
531C22×C4—2922no—
532C22×C4—2922no—
533C22×C4—738no—
534C22×C4—732no—
535C22×C4—734no—
536C22×C4—734no—
537C22×C4—732no—
538C22×C4—732no—
539C22×C4—732no—
540C22×C4—1464no—
541C22×C4C2×C4×D7314yes—
542C22×C4C2×C4×D7314yes—
543C22×C4C2×C4×D7314yes—
544C22×C4C23×D7314yes—
545C22×C4C23×D7314yes—
546C22×C4C23×D7311yescharacteristic
547C22×C4C23×D7311yescharacteristic
548C22×C4C23×D7311yescharacteristic
549C22×C4C23×D7311yescharacteristic
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—42no—
553C4⋊C4—2924no—
554C4⋊C4—2924no—
555C4⋊C4—2924no—
556C4⋊C4—2921no—
557C4⋊C4—2921no—
558C4⋊C4—7316no—
559C4⋊C4—7316no—
560C4⋊C4—738no—
561C4⋊C4—1468no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—42no—
565C22⋊C4—2922no—
566C22⋊C4—2922no—
567C22⋊C4—2922no—
568C22⋊C4—738no—
569C22⋊C4—7316no—
570C22⋊C4—7316no—
571C22⋊C4—1464no—
572C22⋊C4—1464no—
573C42—42no—
574C42—2921no—
575C42—2921no—
576C42—2924no—
577C42—2924no—
578C42—2924no—
579C42—2924no—
580C42—738no—
581C42—1464no—
582C42—1464no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23C22×C4×D7312yes—
596C23C22×C4×D7312yes—
597C23C22×C4×D7312yes—
598C23C24×D7311yescharacteristic
599Q8—44no—
600Q8—2928no—
601Q8—14616no—
602D4—44no—
603D4—44no—
604D4—44no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—84no—
621C2×C4—2928no—
622C2×C4—2928no—
623C2×C4—738no—
624C2×C4—738no—
625C2×C4—734no—
626C2×C4—1462no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4—1462no—
630C2×C4—1462no—
631C2×C4—1462no—
632C2×C4—1462no—
633C2×C4—1462no—
634C2×C4—1462no—
635C2×C4—1462no—
636C2×C4—1464no—
637C2×C4—1464no—
638C2×C4—1464no—
639C2×C4—1464no—
640C2×C4—1462no—
641C2×C4C22×C4×D7312yes—
642C2×C4C22×C4×D7312yes—
643C2×C4C22×C4×D7312yes—
644C2×C4C22×C4×D7312yes—
645C2×C4#1455663612yes—
646C2×C4#1455663612yes—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22C23×C4×D7311yescharacteristic
663C22C2×#1455663611yescharacteristic
664C22C2×#1455663611yescharacteristic
665C222−1+4×D7312yes—
666C22#1798942212yes—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—44no—
672C4—2922no—
673C4—2922no—
674C4—2922no—
675C4—1468no—
676C4—1468no—
677C4—1464no—
678C4#1798906014yes—
679C2—42no—
680C2—42no—
681C2—42no—
682C2C2×#1798906011yescharacteristic
683C2#2262314212yes—
684C2#2262421212yes—
685C2#2262434011yescharacteristic
686C2#2262435511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2311C2
socle, cosocleC23×C73C25
prime radicalC22×#41×C73C22
hypercenter#2311D73
hypocenterC73#34413
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2311

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22613110C2semidirect
3. #22613074C2semidirect
4. #22613064C2semidirect
5. #22613019C2semidirect
6. #22612958C2semidirect
7. #22612936C2semidirect
8. #22612826C2semidirect
9. #22612763C2semidirect
10. #22612740C2semidirect
11. #22612718C2semidirect
12. #22612609C2semidirect
13. #22612459C2semidirect
17. #17989158C4semidirect
78. C22×Dic146C2×C4semidirect
83. #14556589C2×C4semidirect
85. #14556572C2×C4semidirect
86. #14556572C2×C4semidirect
87. #14556570C2×C4semidirect
88. #14556570C2×C4semidirect
92. #14556550C2×C4semidirect
359. C73#34413semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179891582
#226131102
#226130742
#226130642
#226130192
#226129582
#226129362
#226128262
#226127632
#226127402
#226127182
#226126092
#226124592
#226124222
C73×#23112
#3441373