Finite Group Databasebeta

← #33246448

Subgroups — #33246448

Order 18688.

2422 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556590C211yesmaximal, characteristic
2#22613018C212yesmaximal, complemented
3#22613017C212yesmaximal, complemented
4#22612986C214yesmaximal, complemented
5#22612810C211yesmaximal, characteristic
6#22612808C211yesmaximal, characteristic
7#22612806C211yesmaximal, characteristic
8C22⋊C4×Dic146C212yesmaximal
9#22612763C212yesmaximal
10#22612676C214yesmaximal, complemented
11#22612548C212yesmaximal
12#22612502C212yesmaximal, complemented
13C4×#14556590C212yesmaximal, complemented
14#22612333C211yesmaximal, characteristic
15#22612286C212yesmaximal
16C4×#14556547C211yesmaximal, characteristic
17C73×#2345C211yesmaximal, characteristic
18C23×Dic146C2212yes—
19C2×#14556590C4116yescomplemented
20C2×#14556590C2211yescharacteristic
21C2×#14556590C2211yescharacteristic
22C2×#14556590C2211yescharacteristic
23C2×#14556590C2211yescharacteristic
24C2×#14556590C2212yes—
25C2×#14556590C2212yes—
26C2×C4×Dic146C2214yes—
27C2×#14556556C2214yes—
28C2×#14556547C2211yescharacteristic
29C2×#14556547C2212yes—
30C2×#14556547C2211yescharacteristic
31C22×C4⋊Dic73C2211yescharacteristic
32C22×#11924007C2212yes—
33C22×C4×Dic73C2211yescharacteristic
34#17988493C2214yes—
35#17988492C2214yes—
36#17988491C2214yes—
37#17988491C2214yes—
38#17988491C2214yes—
39#17988491C2214yes—
40#17988486C2212yes—
41#17988485C2212yes—
42#17988484C2214yes—
43#17988484C2214yes—
44#17988483C2212yes—
45#17988482C2214yes—
46#17988482C2214yes—
47#17988473C2214yes—
48#17988472C2214yes—
49#17988470C2214yes—
50#17988456C2212yes—
51#17988451C2212yes—
52#17988447C2212yes—
53#17988447C2211yescharacteristic
54#17988445C2212yes—
55#17988445C2212yes—
56#17988445C2212yes—
57C4⋊C4×Dic73C2212yes—
58C4⋊C4×Dic73C2212yes—
59C4⋊C4×Dic73C2212yes—
60C4×#11924007C2214yes—
61C4×#11924007C2212yes—
62C4×#11924007C2214yes—
63C4×C4⋊Dic73C2212yes—
64#17988438C2214yes—
65#17988438C2214yes—
66#17988438C2212yes—
67#17988438C2212yes—
68C22⋊C4×Dic73C2211yescharacteristic
69C22⋊C4×Dic73C2212yes—
70C22⋊C4×Dic73C2211yescharacteristic
71C22⋊C4×Dic73C2212yes—
72C4×C22⋊Dic73C2212yes—
73C4×C22⋊Dic73C2211yescharacteristic
74#17988432C2212yes—
75C42×Dic73C2212yes—
76C22×C4⋊C4×C73C2211yescharacteristic
77#389×C73C2212yes—
78#387×C73C2214yes—
79#380×C73C2212yes—
80#380×C73C2211yescharacteristic
81C4×C4⋊C4×C73C2212yes—
82C4×C22⋊C4×C73C2212yes—
83C4×C22⋊C4×C73C2211yescharacteristic
84C22×Dic146—48no—
85C22×Dic146C2×C4116yes—
86C22×Dic146C2312yes—
87C22×Dic146C2312yes—
88C22×Dic146C2312yes—
89C22×Dic146C2312yes—
90C23×Dic73C2311yescharacteristic
91C23×Dic73C2312yes—
92C23×Dic73C2311yescharacteristic
93#14556590C2×C418yes—
94#14556590C2×C418yes—
95#14556590C2×C4116yes—
96#14556590C2×C418yes—
97#14556590C2×C418yes—
98#14556590C2×C4116yes—
99C4×Dic146—48no—
100#14556556—48no—
101#14556547—24no—
102#14556547—28no—
103#14556547Q818yescomplemented
104C2×C4⋊Dic73C2×C418yes—
105C2×C4⋊Dic73C2312yes—
106C2×C4⋊Dic73C2314yes—
107C2×C4⋊Dic73C2311yescharacteristic
108C2×C4⋊Dic73C2311yescharacteristic
109C2×C4⋊Dic73C2312yes—
110C2×#11924007C2×C4116yes—
111C2×#11924007C2314yes—
112C2×#11924007C2312yes—
113C2×#11924007C2314yes—
114C2×#11924007C2314yes—
115C2×#11924007C2314yes—
116C2×#11924007C2312yes—
117C2×#11924007C2312yes—
118C2×#11924007C2312yes—
119C2×#11924007C2312yes—
120C2×#11924007C2312yes—
121C2×C22⋊Dic73C2312yes—
122C2×C22⋊Dic73C2311yescharacteristic
123C2×C22⋊Dic73C2311yescharacteristic
124C2×C4×Dic73C2×C418yes—
125C2×C4×Dic73C2312yes—
126C2×C4×Dic73C2312yes—
127C2×C4×Dic73C2312yes—
128C2×C4×Dic73C2312yes—
129C2×C4×Dic73C2314yes—
130C2×C4×Dic73C2314yes—
131C2×C4×Dic73C2314yes—
132C2×C4×Dic73C2314yes—
133C2×C4×Dic73C2311yescharacteristic
134C2×C4×Dic73C2311yescharacteristic
135C2×C4×Dic73C2311yescharacteristic
136C2×C4×Dic73C2311yescharacteristic
137C2×C4×Dic73C2312yes—
138#14556498C2312yes—
139#14556498C2312yes—
140#14556498C2312yes—
141#14556498C2314yes—
142#14556498C2314yes—
143#14556498C2314yes—
144#14556498C2312yes—
145#14556498C2314yes—
146#14556498C2314yes—
147#14556498C2314yes—
148#14556498C2314yes—
149C23×C4×C73C2311yescharacteristic
150C23×C4×C73C2312yes—
151C2×C4⋊C4×C73C2×C418yes—
152C2×C4⋊C4×C73C2312yes—
153C2×C4⋊C4×C73C2314yes—
154C2×C4⋊C4×C73C2311yescharacteristic
155C2×C4⋊C4×C73C2312yes—
156C2×C4⋊C4×C73C2311yescharacteristic
157C2×C22⋊C4×C73C2312yes—
158C2×C22⋊C4×C73C2311yescharacteristic
159C2×C22⋊C4×C73C2311yescharacteristic
160C2×C42×C73C2312yes—
161C2×C42×C73C2314yes—
162#95×C73C2312yes—
163#95×C73C2314yes—
164#95×C73C2314yes—
165#95×C73C2312yes—
166C2×Dic146—28no—
167C2×Dic146—28no—
168C2×Dic146—432no—
169C2×Dic146—44no—
170C2×Dic146—44no—
171C2×Dic146C22×C418yes—
172C2×Dic146C22×C418yes—
173C2×Dic146C22×C418yes—
174C2×Dic146C22×C418yes—
175C22×Dic73—24no—
176C22×Dic73—44no—
177C22×Dic73C22×C414yes—
178C22×Dic73C22×C418yes—
179C22×Dic73C22×C414yes—
180C22×Dic73C2×Q814yes—
181C22×Dic73#4114yes—
182C22×Dic73C2412yes—
183C22×Dic73C2414yes—
184C22×Dic73C2411yescharacteristic
185C22×Dic73C2412yes—
186C22×Dic73C2412yes—
187C22×Dic73C2412yes—
188C22×Dic73C2411yescharacteristic
189C22×Dic73C2411yescharacteristic
190C22×Dic73C2411yescharacteristic
191C4⋊Dic73—24no—
192C4⋊Dic73—44no—
193C4⋊Dic73C22×C418yes—
194C4⋊Dic73C22×C414yes—
195C4⋊Dic73C22×C414yes—
196#11924007—24no—
197#11924007—28no—
198#11924007—28no—
199#11924007—44no—
200#11924007—44no—
201#11924007C22×C418yes—
202#11924007C22×C418yes—
203#11924007C22×C418yes—
204#11924007C22×C418yes—
205#11924007C2×Q818yescomplemented
206C22⋊Dic73—24no—
207C22⋊Dic73—22no—
208C22⋊Dic73C2×Q814yes—
209C4×Dic73—24no—
210C4×Dic73—28no—
211C4×Dic73—24no—
212C4×Dic73—24no—
213C4×Dic73—24no—
214C4×Dic73—44no—
215C4×Dic73—44no—
216C4×Dic73—44no—
217C4×Dic73C22×C414yes—
218C4×Dic73C22×C414yes—
219C4×Dic73C22×C414yes—
220C4×Dic73C22×C414yes—
221C4×Dic73C2×Q818yescomplemented
222C24×C73C2411yescharacteristic
223C22×C4×C73—24no—
224C22×C4×C73—44no—
225C22×C4×C73C22×C414yes—
226C22×C4×C73C22×C418yes—
227C22×C4×C73C2414yes—
228C22×C4×C73C2412yes—
229C22×C4×C73C2411yescharacteristic
230C22×C4×C73C2412yes—
231C22×C4×C73C2412yes—
232C22×C4×C73C2412yes—
233C22×C4×C73C2411yescharacteristic
234C4⋊C4×C73—24no—
235C4⋊C4×C73—44no—
236C4⋊C4×C73C22×C418yes—
237C4⋊C4×C73C22×C414yes—
238C4⋊C4×C73C22×C414yes—
239C22⋊C4×C73—24no—
240C22⋊C4×C73—22no—
241C22⋊C4×C73C2×Q814yes—
242C42×C73—24no—
243C42×C73—44no—
244C23×C73—22no—
245C23×C73—22no—
246C23×C73C23×C412yes—
247C23×C73C22×Q812yes—
248C23×C73C2×#4112yes—
249C23×C73C2511yescharacteristic
250C2×C4×C73—22no—
251C2×C4×C73—24no—
252C2×C4×C73—24no—
253C2×C4×C73—22no—
254C2×C4×C73—22no—
255C2×C4×C73—28no—
256C2×C4×C73—24no—
257C2×C4×C73—22no—
258C2×C4×C73—24no—
259C2×C4×C73—22no—
260C2×C4×C73—22no—
261C2×C4×C73—24no—
262C2×C4×C73—48no—
263C2×C4×C73C23×C412yes—
264C2×C4×C73C23×C414yes—
265C2×C4×C73C23×C414yes—
266C2×C4×C73C23×C412yes—
267C2×C4×C73C22×Q814yes—
268C2×C4×C732−1+414yes—
269C2×Dic73—22no—
270C2×Dic73—24no—
271C2×Dic73—24no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—28no—
275C2×Dic73—22no—
276C2×Dic73—24no—
277C2×Dic73—22no—
278C2×Dic73—24no—
279C2×Dic73—22no—
280C2×Dic73—24no—
281C2×Dic73—48no—
282C2×Dic73C4×Q8116yescomplemented
283C2×Dic73C23×C412yes—
284C2×Dic73C23×C414yes—
285C2×Dic73C23×C414yes—
286C2×Dic73C23×C412yes—
287C2×Dic73C23×C412yes—
288C2×Dic73C23×C412yes—
289C2×Dic73C22×Q814yes—
290C2×Dic73C22×Q812yes—
291C2×Dic73C22×Q812yes—
292C2×Dic73C2×#4112yes—
293C2×Dic73C2×#4112yes—
294C2×Dic73H(5,𝔽2)14yes—
295Dic146—216no—
296Dic146—216no—
297Dic146—48no—
298Dic146—48no—
299C22×C73—22no—
300C22×C73—22no—
301C22×C73—24no—
302C22×C73—24no—
303C22×C73C2×C4×Q814yes—
304C22×C73C24×C411yescharacteristic
305C22×C73C23×Q811yescharacteristic
306C22×C73C22×#4111yescharacteristic
307C22×C73C2×H(5,𝔽2)12yes—
308C22×C73C2×2−1+412yes—
309C4×C73—24no—
310C4×C73—24no—
311C4×C73—22no—
312C4×C73—24no—
313C4×C73—44no—
314C4×C73—42no—
315C4×C73#51914yes—
316Dic73—24no—
317Dic73—24no—
318Dic73—22no—
319Dic73—24no—
320Dic73—42no—
321Dic73—44no—
322Dic73C2×C4×Q814yes—
323Dic73C2×C4×Q814yes—
324Dic73#51814yes—
325#34216—731nomaximal, Sylow
326C2×C73—24no—
327C2×C73C22×C4×Q811yescharacteristic
328C2×C73C2×#51811yescharacteristic
329C2×C73C2×#51911yescharacteristic
330C2×C73#350312yes—
331C2×C73#351112yes—
332C22×C4⋊Q8—731no—
333#2369—732no—
334#2369—732no—
335#2365—734no—
336#2345—731no—
337#2345—731no—
338#2345—731no—
339#2345D7311yescharacteristic
340Q8×C22⋊C4—732no—
341Q8×C22⋊C4—732no—
342#2331—734no—
343#2316—732no—
344C4×C4⋊Q8—732no—
345C4×C4⋊Q8—732no—
346#2285—731no—
347#2274—732no—
348C4×#117—731no—
349C73#3421611yescharacteristic, complemented, Sylow
350C23×Q8—732no—
351C2×C4⋊Q8—7316no—
352C2×C4⋊Q8—731no—
353C2×C4⋊Q8—731no—
354C2×C4⋊Q8—732no—
355C2×C4⋊Q8—732no—
356C2×C4⋊Q8—731no—
357C2×C4⋊Q8—731no—
358C2×C4×Q8—734no—
359C2×C4×Q8—734no—
360C2×#117—731no—
361C2×#117—731no—
362C2×#117—732no—
363C22×C4⋊C4—732no—
364C22×C4⋊C4—731no—
365C22×C4⋊C4C2×D7311yescharacteristic
366C22×C42—731no—
367#391—734no—
368#391—734no—
369#391—734no—
370#391—734no—
371#391—734no—
372#391—734no—
373#389—734no—
374#389—732no—
375#389—732no—
376#389—734no—
377#389—732no—
378#389—734no—
379#389—734no—
380#389C2×D7312yes—
381#387—734no—
382#387—734no—
383#387—734no—
384#387C2×D7314yes—
385#384—732no—
386#382—732no—
387#380—732no—
388#380—731no—
389#380—732no—
390#380—732no—
391#380—732no—
392#380C2×D7312yes—
393#380C2×D7311yescharacteristic
394C4×C4⋊C4—732no—
395C4×C4⋊C4—732no—
396C4×C4⋊C4—732no—
397C4×C4⋊C4—734no—
398C4×C4⋊C4—732no—
399C4×C4⋊C4—734no—
400C4×C4⋊C4—732no—
401C4×C4⋊C4—734no—
402C4×C4⋊C4—732no—
403C4×C4⋊C4—732no—
404C4×C4⋊C4—734no—
405C4×C4⋊C4C2×D7312yes—
406C4×C22⋊C4—732no—
407C4×C22⋊C4—731no—
408C4×C22⋊C4—731no—
409C4×C22⋊C4—731no—
410C4×C22⋊C4—732no—
411C4×C22⋊C4—732no—
412C4×C22⋊C4C2×D7312yes—
413C4×C22⋊C4C2×D7311yescharacteristic
414#376—732no—
415C43—732no—
416C22×Q8—2928no—
417C22×Q8—7316no—
418C22×Q8—732no—
419C22×Q8—732no—
420C22×Q8—732no—
421C22×Q8—732no—
422C23×C4—732no—
423C23×C4—731no—
424C23×C4—731no—
425C23×C4C22×D7311yescharacteristic
426C23×C4C22×D7312yes—
427C4⋊Q8—7316no—
428C4⋊Q8—738no—
429C4⋊Q8—738no—
430C4⋊Q8—738no—
431C4⋊Q8—738no—
432C4⋊Q8—7316no—
433C4×Q8—2928no—
434C4×Q8—2928no—
435#117—738no—
436#117—1464no—
437#117—1468no—
438C2×C4⋊C4—734no—
439C2×C4⋊C4—734no—
440C2×C4⋊C4—734no—
441C2×C4⋊C4—7316no—
442C2×C4⋊C4—738no—
443C2×C4⋊C4—732no—
444C2×C4⋊C4—732no—
445C2×C4⋊C4—732no—
446C2×C4⋊C4—732no—
447C2×C4⋊C4—731no—
448C2×C4⋊C4—731no—
449C2×C4⋊C4—732no—
450C2×C4⋊C4—732no—
451C2×C4⋊C4—734no—
452C2×C4⋊C4—732no—
453C2×C4⋊C4—734no—
454C2×C4⋊C4—732no—
455C2×C4⋊C4C4×D7318yes—
456C2×C4⋊C4C22×D7312yes—
457C2×C4⋊C4C22×D7314yes—
458C2×C4⋊C4C22×D7311yescharacteristic
459C2×C4⋊C4C22×D7311yescharacteristic
460C2×C4⋊C4C22×D7312yes—
461C2×C22⋊C4—731no—
462C2×C22⋊C4—731no—
463C2×C22⋊C4—732no—
464C2×C22⋊C4C22×D7311yescharacteristic
465C2×C22⋊C4C22×D7312yes—
466C2×C22⋊C4C22×D7311yescharacteristic
467C2×C42—734no—
468C2×C42—734no—
469C2×C42—734no—
470C2×C42—738no—
471C2×C42—732no—
472C2×C42—731no—
473C2×C42—731no—
474C2×C42—731no—
475C2×C42—731no—
476C2×C42—732no—
477C2×C42—734no—
478C2×C42—732no—
479C2×C42—732no—
480C2×C42—732no—
481C2×C42C22×D7312yes—
482C2×C42C22×D7314yes—
483#95—732no—
484#95—732no—
485#95—732no—
486#95—734no—
487#95—734no—
488#95—734no—
489#95—732no—
490#95—734no—
491#95—734no—
492#95—734no—
493#95—734no—
494#95C22×D7312yes—
495#95C22×D7314yes—
496#95C22×D7312yes—
497#95C22×D7314yes—
498C24C23×D7311yescharacteristic
499C2×Q8—2924no—
500C2×Q8—2924no—
501C2×Q8—29232no—
502C2×Q8—738no—
503C2×Q8—738no—
504C2×Q8—738no—
505C2×Q8—738no—
506C2×Q8—1468no—
507C2×Q8—1468no—
508C22×C4—24no—
509C22×C4—44no—
510C22×C4—2924no—
511C22×C4—738no—
512C22×C4—734no—
513C22×C4—734no—
514C22×C4—734no—
515C22×C4—731no—
516C22×C4—731no—
517C22×C4—731no—
518C22×C4—734no—
519C22×C4—731no—
520C22×C4—732no—
521C22×C4—732no—
522C22×C4—732no—
523C22×C4—734no—
524C22×C4—732no—
525C22×C4—1464no—
526C22×C4C2×C4×D7314yes—
527C22×C4C2×C4×D7318yes—
528C22×C4C23×D7314yes—
529C22×C4C23×D7312yes—
530C22×C4C23×D7311yescharacteristic
531C22×C4C23×D7312yes—
532C22×C4C23×D7312yes—
533C22×C4C23×D7312yes—
534C22×C4C23×D7311yescharacteristic
535C4⋊C4—24no—
536C4⋊C4—44no—
537C4⋊C4—2924no—
538C4⋊C4—2924no—
539C4⋊C4—2924no—
540C4⋊C4—738no—
541C4⋊C4—738no—
542C4⋊C4—738no—
543C4⋊C4—738no—
544C4⋊C4—738no—
545C4⋊C4—734no—
546C4⋊C4—734no—
547C4⋊C4—738no—
548C4⋊C4—1464no—
549C4⋊C4—1468no—
550C4⋊C4—1468no—
551C4⋊C4—1464no—
552C4⋊C4C2×C4×D7318yes—
553C4⋊C4C2×C4×D7314yes—
554C4⋊C4C2×C4×D7314yes—
555C22⋊C4—22no—
556C22⋊C4—24no—
557C22⋊C4—734no—
558C22⋊C4—1462no—
559C22⋊C4—1464no—
560C22⋊C4Q8×D7314yes—
561C42—24no—
562C42—44no—
563C42—2924no—
564C42—2924no—
565C42—2924no—
566C42—734no—
567C42—734no—
568C42—734no—
569C42—734no—
570C42—738no—
571C42—1468no—
572C42—1464no—
573C42—1464no—
574C42—1464no—
575C42—1464no—
576C23—22no—
577C23—22no—
578C23C22×C4×D7312yes—
579C23C2×Q8×D7312yes—
580C23#41×D7312yes—
581C23C24×D7311yescharacteristic
582Q8—2928no—
583Q8—2928no—
584Q8—14616no—
585Q8—14616no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—48no—
599C2×C4—2928no—
600C2×C4—732no—
601C2×C4—732no—
602C2×C4—732no—
603C2×C4—732no—
604C2×C4—732no—
605C2×C4—732no—
606C2×C4—732no—
607C2×C4—734no—
608C2×C4—7316no—
609C2×C4—732no—
610C2×C4—734no—
611C2×C4—734no—
612C2×C4—734no—
613C2×C4—1464no—
614C2×C4—1462no—
615C2×C4—1464no—
616C2×C4—1462no—
617C2×C4—1462no—
618C2×C4—1464no—
619C2×C4—1468no—
620C2×C4—1462no—
621C2×C4—1464no—
622C2×C4—1464no—
623C2×C4—1462no—
624C2×C4—1462no—
625C2×C4C22×C4×D7312yes—
626C2×C4C22×C4×D7314yes—
627C2×C4C22×C4×D7314yes—
628C2×C4C22×C4×D7312yes—
629C2×C4C2×Q8×D7314yes—
630C2×C4#1455663814yes—
631C22—24no—
632C22—24no—
633C22—22no—
634C22—22no—
635C22C4×Q8×D7314yes—
636C22C23×C4×D7311yescharacteristic
637C22C22×Q8×D7311yescharacteristic
638C22C2×#41×D7311yescharacteristic
639C22H(5,𝔽2)×D7312yes—
640C22C2×#1455663812yes—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—22no—
645C4—44no—
646C4—42no—
647C4—2924no—
648C4—2922no—
649C4—734no—
650C4—734no—
651C4—734no—
652C4—1464no—
653C4—1464no—
654C4—1464no—
655C4—1462no—
656C4#1798906414yes—
657C2—24no—
658C2C2×C4×Q8×D7311yescharacteristic
659C2#518×D7311yescharacteristic
660C2C2×#1798906411yescharacteristic
661C2#2262374512yes—
662C2#2262383312yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2345C2
socle, cosocleC23×C73C25
prime radicalC24×C73C24
hypercenter#2345D73
hypocenterC73#34216
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2345

Complements

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

NH ≅ G/Ntype
2. #22613018C2semidirect
3. #22613017C2semidirect
4. #22612986C2semidirect
10. #22612676C2semidirect
12. #22612502C2semidirect
13. C4×#14556590C2semidirect
19. C2×#14556590C4semidirect
103. #14556547Q8semidirect
205. #11924007C2×Q8semidirect
221. C4×Dic73C2×Q8semidirect
282. C2×Dic73C4×Q8semidirect
349. C73#34216semidirect

Maximal subgroups

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

subgroupindex
C22×#145565902
#226130182
#226130172
#226129862
#226128102
#226128082
#226128062
C22⋊C4×Dic1462
#226127632
#226126762
#226125482
#226125022
C4×#145565902
#226123332
#226122862
C4×#145565472
C73×#23452
#3421673