Finite Group Databasebeta

← #33103263

Subgroups — #33103263

Order 18688.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988597C211yesmaximal, characteristic
2#22612822C211yesmaximal, characteristic
3C22⋊C4×Dic146C211yesmaximal, characteristic
4#22610621C211yesmaximal, characteristic, complemented
5#22610607C211yesmaximal, characteristic, complemented
6#22610573C211yesmaximal, characteristic, complemented
7#22610566C211yesmaximal, characteristic, complemented
8#22610202C211yesmaximal, characteristic, complemented
9#22610188C211yesmaximal, characteristic, complemented
10#22610027C211yesmaximal, characteristic
11#22609964C211yesmaximal, characteristic
12#22609867C211yesmaximal, characteristic
13#22609762C211yesmaximal, characteristic, complemented
14C4×#14556516C211yesmaximal, characteristic, complemented
15C73×#1753C211yesmaximal, characteristic
16C23×Dic146C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556523C2211yescharacteristic
22#17988597C418yescomplemented
23C2×#14556516C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556516C2211yescharacteristic
29C2×#14556516C2211yescharacteristic
30C2×#14556516C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988493C2211yescharacteristic
33#17988491—21no—
34#17988491—21no—
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988481—21no—
38#17988481—21no—
39#17988454—21no—
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988445—21no—
43C4×C4⋊Dic73C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45#17988438—21no—
46C22⋊C4×Dic73—21no—
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C22×M16×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C4×C22⋊C4×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Dic146—22no—
61C22×Dic146—22no—
62C22×Dic146—21no—
63C22×Dic146—22no—
64C22×Dic146—41no—
65C22×Dic146—41no—
66C22×Dic146C2×C414yes—
67C22×Dic146C2311yescharacteristic
68C22×Dic146C2311yescharacteristic
69C23×Dic73—21no—
70C4×Dic146—22no—
71C4×Dic146—22no—
72C4×Dic146—41no—
73C4×Dic146—41no—
74#14556550—22no—
75#14556550C2×C414yes—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94#14556525—41no—
95#14556523—41no—
96#14556516—22no—
97#14556516—22no—
98#14556516—41no—
99#14556516—41no—
100#14556516C2×C414yes—
101#14556516C2×C414yes—
102#14556516C2×C414yes—
103#14556516C2×C414yes—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113#14556498—21no—
114C23×C4×C73C2311yescharacteristic
115C2×M16×C73—41no—
116C2×M16×C73—41no—
117C2×M16×C73C2×C414yes—
118C22×C8×C73C2311yescharacteristic
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C2×C22⋊C4×C73D411yescharacteristic
123C2×C22⋊C4×C73D411yescharacteristic
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C22⋊C8×C73—22no—
127#97×C73—41no—
128C4×C8×C73—41no—
129#95×C73D411yescharacteristic
130#95×C73D411yescharacteristic
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—28no—
139C2×Dic146—24no—
140C2×Dic146—21no—
141C2×Dic146—44no—
142C2×Dic146—44no—
143C2×Dic146—44no—
144C2×Dic146—42no—
145C2×Dic146—42no—
146C2×Dic146C22×C412yes—
147C2×Dic146C22×C412yes—
148C2×Dic146#4112yes—
149C2×Dic146#4112yes—
150C22×Dic73—21no—
151C22×Dic73—24no—
152C22×Dic73—21no—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C22×Dic73—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73—42no—
160C22×Dic73—42no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73C22×C412yes—
172C4⋊Dic73C22×C412yes—
173#11924007—42no—
174#11924007—42no—
175#11924007—42no—
176#11924007—42no—
177C22⋊Dic73—24no—
178C22⋊Dic73—42no—
179C22⋊Dic73—81no—
180C4×Dic73—24no—
181C4×Dic73—42no—
182C4×Dic73—42no—
183C4×Dic73—81no—
184C4×Dic73—81no—
185C24×C73C2×D411yescharacteristic
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73C22×C412yes—
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73#4112yes—
197C22×C4×C73C2411yescharacteristic
198M16×C73—44no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73C22×C412yes—
210C2×C8×C73C22×C412yes—
211C22⋊C4×C73—22no—
212C22⋊C4×C73—41no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—21no—
216C42×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73C4×D412yes—
224C23×C73C22×D411yescharacteristic
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73C4×D412yes—
243C2×C4×C73#13712yes—
244C2×C4×C73C23×C411yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73H(5,𝔽2)11yescharacteristic
249C2×C4×C73H(5,𝔽2)11yescharacteristic
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—48no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275Dic146—22no—
276Dic146—22no—
277Dic146—22no—
278Dic146—22no—
279Dic146—22no—
280Dic146—48no—
281Dic146—41no—
282Dic146—41no—
283Dic146#11714yes—
284C8×C73—22no—
285C8×C73—22no—
286C8×C73—41no—
287C8×C73—41no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73C2×C4×D411yescharacteristic
296C22×C73#54611yescharacteristic
297C22×C73#54611yescharacteristic
298C22×C73C2×#13711yescharacteristic
299C22×C73#57511yescharacteristic
300C22×C73#57711yescharacteristic
301C22×C73#57811yescharacteristic
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—41no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73C2×#11711yescharacteristic
308C4×C73#51811yescharacteristic
309C4×C73#55111yescharacteristic
310C4×C73#55111yescharacteristic
311Dic73—24no—
312Dic73—42no—
313Dic73—42no—
314Dic73—42no—
315Dic73—81no—
316Dic73—81no—
317Dic73—81no—
318Dic73—81no—
319#22581—731nomaximal, Sylow
320C2×C73—42no—
321C2×C73#234211yescharacteristic
322C2×C73C4×#13711yescharacteristic
323C2×C73#294711yescharacteristic
324C2×C73#331311yescharacteristic
325C2×C73#332011yescharacteristic
326C2×C73#332611yescharacteristic
327C2×C73#332811yescharacteristic
328C2×#419—731no—
329#2347—731no—
330Q8×C22⋊C4—731no—
331#1935—731no—
332#1933—731no—
333#1928—731no—
334#1927—731no—
335#1865—731no—
336#1863—731no—
337#1825—731no—
338#1809—731no—
339#1787—731no—
340#1763—731no—
341C4×#103—731no—
342#1753D7311yescharacteristic
343C73#2258111yescharacteristic, complemented, Sylow
344C23×Q8—731no—
345C22×M16C2×D7311yescharacteristic
346C2×C4×Q8—731no—
347C2×C4×Q8—731no—
348C2×#117—731no—
349C2×#107—731no—
350C2×#106—731no—
351#419—738no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×#103—731no—
357C2×#103—731no—
358C2×#103—731no—
359C2×#103—731no—
360C2×C22⋊C8C2×D7311yescharacteristic
361C2×#97C2×D7311yescharacteristic
362C2×C4×C8C2×D7311yescharacteristic
363#391—731no—
364#391—731no—
365#391—1461no—
366#391—1461no—
367#389—731no—
368#389—731no—
369#388—1461no—
370#388—1461no—
371#383—1461no—
372#380—731no—
373#380—731no—
374#380—1461no—
375C4×C4⋊C4—731no—
376C4×C4⋊C4—731no—
377C4×C4⋊C4—1461no—
378C4×C22⋊C4—1461no—
379C4×C22⋊C4C2×D7311yescharacteristic
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340—731no—
386#336C2×D7311yescharacteristic
387#336C2×D7311yescharacteristic
388C22×Q8—2921no—
389C22×Q8—2921no—
390C22×Q8—734no—
391C22×Q8—731no—
392C22×Q8—731no—
393C22×Q8—1462no—
394C22×Q8—1462no—
395C22×Q8—1461no—
396C22×Q8—1462no—
397C23×C4—1461no—
398C23×C4C22×D7311yescharacteristic
399C2×M16—41no—
400C2×M16—41no—
401C2×M16C4×D7314yes—
402C22×C8C22×D7311yescharacteristic
403C22×C8C22×D7311yescharacteristic
404C22×C8C22×D7311yescharacteristic
405C22×C8C22×D7311yescharacteristic
406C4×Q8—2921no—
407C4×Q8—2921no—
408C4×Q8—1462no—
409C4×Q8—1462no—
410#117—734no—
411#117—1462no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4D29211yescharacteristic
426C2×C22⋊C4D29211yescharacteristic
427C2×C42—1461no—
428C2×C42—1461no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42C22×D7311yescharacteristic
433C2×C42C22×D7311yescharacteristic
434#107—2921no—
435#106—2921no—
436#103—2921no—
437#103—2921no—
438#103—734no—
439#103—734no—
440#103—734no—
441#103—734no—
442#103—1462no—
443#103—1462no—
444C22⋊C8—22no—
445#97—41no—
446C4×C8—41no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95D29211yescharacteristic
458#95D29211yescharacteristic
459C24C2×D29211yescharacteristic
460C2×Q8—2924no—
461C2×Q8—2924no—
462C2×Q8—2924no—
463C2×Q8—2922no—
464C2×Q8—2922no—
465C2×Q8—732no—
466C2×Q8—732no—
467C2×Q8—732no—
468C2×Q8—732no—
469C2×Q8—1468no—
470C2×Q8—1461no—
471C2×Q8—1461no—
472C2×Q8—1461no—
473C2×Q8—1461no—
474C2×Q8—1464no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1464no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4C2×C4×D7312yes—
495C22×C4C2×D29211yescharacteristic
496C22×C4C2×D29211yescharacteristic
497C22×C4C2×D29211yescharacteristic
498C22×C4C2×D29211yescharacteristic
499C22×C4C2×D29211yescharacteristic
500C22×C4#1192402812yes—
501C22×C4C23×D7311yescharacteristic
502M16—44no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8C2×C4×D7312yes—
514C2×C8C2×C4×D7312yes—
515C4⋊C4—2922no—
516C4⋊C4—2922no—
517C4⋊C4—2922no—
518C4⋊C4—2922no—
519C4⋊C4—732no—
520C4⋊C4—732no—
521C4⋊C4—1461no—
522C4⋊C4—1461no—
523C4⋊C4—1461no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C22⋊C4—22no—
532C22⋊C4—41no—
533C22⋊C4—2922no—
534C22⋊C4—5841no—
535C22⋊C4—1464no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—2922no—
541C42—2922no—
542C42—5841no—
543C42—5841no—
544C42—1464no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×D29212yes—
552C23C22×D29211yescharacteristic
553Q8—2928no—
554Q8—2921no—
555Q8—2921no—
556Q8—734no—
557Q8—1462no—
558Q8—1462no—
559Q8—1462no—
560Q8—1462no—
561Q8—1462no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2928no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—1462no—
599C2×C4—1462no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4—1462no—
603C2×C4—1462no—
604C2×C4C4×D29212yes—
605C2×C4#1455662412yes—
606C2×C4C22×C4×D7311yescharacteristic
607C2×C4#41×D7311yescharacteristic
608C2×C4#41×D7311yescharacteristic
609C2×C4C2×#1192402811yescharacteristic
610C2×C4#1455663611yescharacteristic
611C2×C4#1455663611yescharacteristic
612C8—22no—
613C8—22no—
614C8—41no—
615C8—41no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×D29211yescharacteristic
624C22#1798923211yescharacteristic
625C22#1798923211yescharacteristic
626C22C2×#1455662411yescharacteristic
627C22#1798938711yescharacteristic
628C22#1798939911yescharacteristic
629C22#1798940411yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—2922no—
636C4—2922no—
637C4—2922no—
638C4—5841no—
639C4—5841no—
640C4—5841no—
641C4—5841no—
642C4—1464no—
643C4#117×D7311yescharacteristic
644C4#1798906011yescharacteristic
645C4#1798926611yescharacteristic
646C4#1798926611yescharacteristic
647C2—42no—
648C2#2261278311yescharacteristic
649C2C4×#1455662411yescharacteristic
650C2#2261877611yescharacteristic
651C2#2262199911yescharacteristic
652C2#2262210411yescharacteristic
653C2#2262218611yescharacteristic
654C2#2262221511yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1753C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1753D73
hypocenterC73#22581
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #22610621C2semidirect
5. #22610607C2semidirect
6. #22610573C2semidirect
7. #22610566C2semidirect
8. #22610202C2semidirect
9. #22610188C2semidirect
13. #22609762C2semidirect
14. C4×#14556516C2semidirect
22. #17988597C4semidirect
343. C73#22581semidirect

Maximal subgroups

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

subgroupindex
C2×#179885972
#226128222
C22⋊C4×Dic1462
#226106212
#226106072
#226105732
#226105662
#226102022
#226101882
#226100272
#226099642
#226098672
#226097622
C4×#145565162
C73×#17532
#2258173