Finite Group Databasebeta

← #33056091

Subgroups — #33056091

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619042C211yesmaximal, characteristic
2#22619026C211yesmaximal, characteristic, complemented
3#22608803C211yesmaximal, characteristic, complemented
4#22608789C211yesmaximal, characteristic, complemented
5#22608782C211yesmaximal, characteristic, complemented
6#22608768C211yesmaximal, characteristic, complemented
7#22608750C211yesmaximal, characteristic, complemented
8SD16⋊#10596831C211yesmaximal, characteristic, complemented
9Q16×#10596831C211yesmaximal, characteristic, complemented
10SD16×#10596831C211yesmaximal, characteristic, complemented
11#22608716C211yesmaximal, characteristic, complemented
12#22608365C211yesmaximal, characteristic
13#22608334C211yesmaximal, characteristic, complemented
14#22608231C211yesmaximal, characteristic, complemented
15C73×#3107C211yesmaximal, characteristic
16#17988718—21no—
17#17988718—21no—
18#17988718C2211yescharacteristic
19#17988718C2211yescharacteristic
20Q8×#10596831C2211yescharacteristic
21Q8×#10596831C2211yescharacteristic
22Q8×#10596831C2211yescharacteristic
23Q8×#10596831C2211yescharacteristic
24D4×#10596831—21no—
25D4×#10596831—21no—
26D4×#10596831C2211yescharacteristic
27D4×#10596831C2211yescharacteristic
28#17988648—21no—
29#17988648—21no—
30#17988644C2211yescharacteristic
31#17988640—21no—
32#17988640C2211yescharacteristic
33#17988640C2211yescharacteristic
34C2×C4⋊#10596831—21no—
35C2×C4⋊#10596831C2211yescharacteristic
36#17988340C2211yescharacteristic
37#17988340C2211yescharacteristic
38#17988338C2211yescharacteristic
39#17988338C2211yescharacteristic
40Q8⋊#10596831C2211yescharacteristic
41Q8⋊#10596831C2211yescharacteristic
42Q8⋊#10596831C2211yescharacteristic
43Q8⋊#10596831C2211yescharacteristic
44Q8⋊#10596831C2211yescharacteristic
45Q8⋊#10596831C2211yescharacteristic
46D4⋊#10596831C2211yescharacteristic
47D4⋊#10596831C2211yescharacteristic
48#17988309C2211yescharacteristic
49#17988309C2211yescharacteristic
50#17988308C2211yescharacteristic
51C8×#10596831C2211yescharacteristic
52#533×C73C2211yescharacteristic
53C4×#41×C73C2211yescharacteristic
54#485×C73C411yescharacteristic
55#485×C73C411yescharacteristic
56#475×C73C411yescharacteristic
57#474×C73C411yescharacteristic
58#462×C73C411yescharacteristic
59#461×C73C411yescharacteristic
60#452×C73C411yescharacteristic
61#452×C73C411yescharacteristic
62C4×Q16×C73C2211yescharacteristic
63C4×Q16×C73C2211yescharacteristic
64C4×SD16×C73C2211yescharacteristic
65C4×SD16×C73C2211yescharacteristic
66#431×C73C2211yescharacteristic
67C22×#10596831—21no—
68C22×#10596831—21no—
69C22×#10596831—21no—
70C4⋊#10596831—21no—
71C4⋊#10596831—21no—
72C4⋊#10596831—21no—
73C4⋊#10596831—21no—
74C4⋊#10596831—21no—
75C4⋊#10596831—21no—
76C4⋊#10596831—21no—
77C4⋊#10596831—21no—
78C4⋊#10596831D411yescharacteristic
79C4⋊#10596831D411yescharacteristic
80C4⋊#10596831C2311yescharacteristic
81C4⋊#10596831C2311yescharacteristic
82C4⋊#10596831C2311yescharacteristic
83C4⋊#10596831C2311yescharacteristic
84C22⋊#10596831—21no—
85C22⋊#10596831—21no—
86C22⋊#10596831—21no—
87C22⋊#10596831—21no—
88C22⋊#10596831D411yescharacteristic
89C22⋊#10596831D411yescharacteristic
90#14556502—21no—
91#14556502—21no—
92#14556502—21no—
93C4×#10596831—21no—
94C4×#10596831—21no—
95C4×#10596831—21no—
96C4×#10596831—21no—
97C4×#10596831C2311yescharacteristic
98C4×#10596831C2311yescharacteristic
99C4×#10596831C2311yescharacteristic
100C4×#10596831C2311yescharacteristic
101C2×#41×C73C2×C411yescharacteristic
102C2×Q16×C73C2×C411yescharacteristic
103C2×Q16×C73C2×C411yescharacteristic
104C2×SD16×C73C2×C411yescharacteristic
105C2×SD16×C73C2×C411yescharacteristic
106C4⋊Q8×C73C2×C411yescharacteristic
107C4⋊Q8×C73C2×C411yescharacteristic
108#125×C73—21no—
109C22⋊Q8×C73—21no—
110C22⋊Q8×C73C2×C411yescharacteristic
111C22⋊Q8×C73C2×C411yescharacteristic
112C4×Q8×C73—21no—
113C4×Q8×C73C2×C411yescharacteristic
114C4×Q8×C73C2311yescharacteristic
115C4×Q8×C73C2311yescharacteristic
116C4×Q8×C73C2311yescharacteristic
117C4×D4×C73—21no—
118C4×D4×C73—21no—
119C4×D4×C73C2×C411yescharacteristic
120C4×D4×C73C2311yescharacteristic
121#117×C73—21no—
122#117×C73—21no—
123#117×C73C2×C411yescharacteristic
124C2×C42×C73—21no—
125C2×C42×C73C2311yescharacteristic
126#107×C73C2×C411yescharacteristic
127#107×C73C2×C411yescharacteristic
128#106×C73C2×C411yescharacteristic
129#106×C73C2×C411yescharacteristic
130C4⋊C8×C73C2×C411yescharacteristic
131C4⋊C8×C73C2×C411yescharacteristic
132#103×C73C2×C411yescharacteristic
133#103×C73C2×C411yescharacteristic
134#103×C73C2×C411yescharacteristic
135#103×C73C2×C411yescharacteristic
136#103×C73C2×C411yescharacteristic
137#103×C73C2×C411yescharacteristic
138#102×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73C2×C411yescharacteristic
141C22⋊C8×C73C2×C411yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—81no—
151C2×#10596831—81no—
152C2×#10596831—81no—
153C2×#10596831C2×D411yescharacteristic
154C2×#10596831C2×D411yescharacteristic
155C2×#10596831C2×D411yescharacteristic
156C2×#10596831C2×D411yescharacteristic
157#41×C73—22no—
158#41×C73—22no—
159C2×Q8×C73—21no—
160C2×Q8×C73C22×C411yescharacteristic
161C2×Q8×C73C22×C411yescharacteristic
162C2×Q8×C73C22×C411yescharacteristic
163C2×D4×C73—21no—
164C2×D4×C73C22×C411yescharacteristic
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73C22×C411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73#4111yescharacteristic
171Q16×C73—22no—
172Q16×C73—22no—
173SD16×C73—22no—
174SD16×C73—22no—
175C2×C8×C73C22×C411yescharacteristic
176C2×C8×C73C22×C411yescharacteristic
177C2×C8×C73C22×C411yescharacteristic
178C2×C8×C73C22×C411yescharacteristic
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73C22×C411yescharacteristic
189C4⋊C4×C73C22×C411yescharacteristic
190C4⋊C4×C73C22×C411yescharacteristic
191C4⋊C4×C73C22×C411yescharacteristic
192C22⋊C4×C73—21no—
193C22⋊C4×C73—21no—
194C22⋊C4×C73—21no—
195C22⋊C4×C73—21no—
196C22⋊C4×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73C22×C411yescharacteristic
203C42×C73C2×D411yescharacteristic
204C42×C73#4111yescharacteristic
205C42×C73C2411yescharacteristic
206C23×C73—21no—
207C23×C73C4×D411yescharacteristic
208Q8×C73—21no—
209Q8×C73—21no—
210Q8×C73—21no—
211Q8×C73—21no—
212Q8×C73—21no—
213Q8×C73—81no—
214Q8×C73#13112yes—
215D4×C73—24no—
216D4×C73—21no—
217D4×C73#13112yes—
218C2×C4×C73—24no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—81no—
234C2×C4×C73C4×D411yescharacteristic
235C2×C4×C73C4×D411yescharacteristic
236C2×C4×C73C4×D411yescharacteristic
237C2×C4×C73C23×C411yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C8×C73—21no—
241C8×C73—21no—
242C8×C73—21no—
243C8×C73—21no—
244#10596831—21no—
245#10596831—21no—
246#10596831—21no—
247#10596831—21no—
248#10596831—41no—
249#10596831—41no—
250#10596831—41no—
251#10596831—41no—
252#10596831—41no—
253#10596831—41no—
254C22×C73—21no—
255C22×C73—22no—
256C22×C73—21no—
257C22×C73—21no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73C2×C4×D411yescharacteristic
261C4×C73—21no—
262C4×C73—22no—
263C4×C73—21no—
264C4×C73—21no—
265C4×C73—21no—
266C4×C73—41no—
267C4×C73—41no—
268C4×C73—41no—
269C4×C73—41no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73C2×#13111yescharacteristic
273C4×C73C2×#13111yescharacteristic
274C4×C73#57711yescharacteristic
275C4×C73#57811yescharacteristic
276#17921—731nomaximal, Sylow
277C2×C73—41no—
278C2×C73—42no—
279C2×C73#292911yescharacteristic
280C2×C73#294711yescharacteristic
281C2×C73#295711yescharacteristic
282#3107D7311yescharacteristic
283#2988—731no—
284#2985—731no—
285#1587—731no—
286#1585—731no—
287#1584—731no—
288#1582—731no—
289#1579—731no—
290SD16⋊C8—731no—
291C8×Q16—731no—
292C8×SD16—731no—
293#1573—731no—
294#1488—731no—
295#1481—731no—
296#1454—731no—
297C73#1792111yescharacteristic, complemented, Sylow
298#533C2×D7311yescharacteristic
299C4×#41C2×D7311yescharacteristic
300#485Dic7311yescharacteristic
301#485Dic7311yescharacteristic
302#475Dic7311yescharacteristic
303#474Dic7311yescharacteristic
304#462Dic7311yescharacteristic
305#461Dic7311yescharacteristic
306#452Dic7311yescharacteristic
307#452Dic7311yescharacteristic
308#446—731no—
309#446—731no—
310#446—1461no—
311#446—1461no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8—731no—
316C4×Q16C2×D7311yescharacteristic
317C4×Q16C2×D7311yescharacteristic
318C4×SD16C2×D7311yescharacteristic
319C4×SD16C2×D7311yescharacteristic
320C8×D4—731no—
321C8×D4—731no—
322C8×D4—1461no—
323C8×D4—1461no—
324#433—1461no—
325#433—1461no—
326#432—731no—
327#431—731no—
328#431—731no—
329#431—1461no—
330#431C2×D7311yescharacteristic
331C2×C4⋊C8—731no—
332C2×C4⋊C8—1461no—
333#335—731no—
334#335—731no—
335#334—731no—
336#334—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343D4⋊C8—731no—
344D4⋊C8—731no—
345#322—731no—
346#322—731no—
347#322—731no—
348C82—731no—
349C2×#41C2×Dic7311yescharacteristic
350C2×Q16C2×Dic7311yescharacteristic
351C2×Q16C2×Dic7311yescharacteristic
352C2×SD16C2×Dic7311yescharacteristic
353C2×SD16C2×Dic7311yescharacteristic
354C22×C8—1461no—
355C22×C8—1461no—
356C22×C8—1461no—
357C4⋊Q8C2×Dic7311yescharacteristic
358C4⋊Q8C2×Dic7311yescharacteristic
359#125—21no—
360C22⋊Q8—21no—
361C22⋊Q8C2×Dic7311yescharacteristic
362C22⋊Q8C2×Dic7311yescharacteristic
363C4×Q8—21no—
364C4×Q8C2×Dic7311yescharacteristic
365C4×Q8C22×D7311yescharacteristic
366C4×Q8C22×D7311yescharacteristic
367C4×Q8C22×D7311yescharacteristic
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C2×Dic7311yescharacteristic
371C4×D4C22×D7311yescharacteristic
372#117—21no—
373#117—21no—
374#117C2×Dic7311yescharacteristic
375C2×C42—21no—
376C2×C42C22×D7311yescharacteristic
377#107C2×Dic7311yescharacteristic
378#107C2×Dic7311yescharacteristic
379#106C2×Dic7311yescharacteristic
380#106C2×Dic7311yescharacteristic
381C4⋊C8—731no—
382C4⋊C8—731no—
383C4⋊C8—731no—
384C4⋊C8—731no—
385C4⋊C8—731no—
386C4⋊C8—731no—
387C4⋊C8—1461no—
388C4⋊C8—1461no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8C2×Dic7311yescharacteristic
396C4⋊C8C2×Dic7311yescharacteristic
397#103C2×Dic7311yescharacteristic
398#103C2×Dic7311yescharacteristic
399#103C2×Dic7311yescharacteristic
400#103C2×Dic7311yescharacteristic
401#103C2×Dic7311yescharacteristic
402#103C2×Dic7311yescharacteristic
403#102C2×Dic7311yescharacteristic
404#102C2×Dic7311yescharacteristic
405C22⋊C8—731no—
406C22⋊C8—731no—
407C22⋊C8—1461no—
408C22⋊C8—1461no—
409C22⋊C8—1461no—
410C22⋊C8—1461no—
411C22⋊C8C2×Dic7311yescharacteristic
412C22⋊C8C2×Dic7311yescharacteristic
413#97—1461no—
414#97—1461no—
415#97—1461no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—1461no—
421C4×C8—1461no—
422C4×C8—1461no—
423C4×C8—1461no—
424C4×C8C22×D7311yescharacteristic
425C4×C8C22×D7311yescharacteristic
426#41—22no—
427#41—22no—
428C2×Q8—21no—
429C2×Q8C22×Dic7311yescharacteristic
430C2×Q8C22×Dic7311yescharacteristic
431C2×Q8C22×Dic7311yescharacteristic
432C2×D4—21no—
433C2×D4C22×Dic7311yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C22×Dic7311yescharacteristic
438C22×C4D4×D7311yescharacteristic
439C22×C4#1192402711yescharacteristic
440Q16—22no—
441Q16—22no—
442SD16—22no—
443SD16—22no—
444C2×C8—5841no—
445C2×C8—5841no—
446C2×C8—5841no—
447C2×C8—731no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8C22×Dic7311yescharacteristic
458C2×C8C22×Dic7311yescharacteristic
459C2×C8C22×Dic7311yescharacteristic
460C2×C8C22×Dic7311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C22×Dic7311yescharacteristic
471C4⋊C4C22×Dic7311yescharacteristic
472C4⋊C4C22×Dic7311yescharacteristic
473C4⋊C4C22×Dic7311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42C22×Dic7311yescharacteristic
485C42D4×D7311yescharacteristic
486C42#1192402711yescharacteristic
487C42C23×D7311yescharacteristic
488C23—21no—
489C23D4×Dic7311yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—81no—
496Q8#1455659812yes—
497D4—24no—
498D4—21no—
499D4#1455659812yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4D4×Dic7311yescharacteristic
517C2×C4D4×Dic7311yescharacteristic
518C2×C4D4×Dic7311yescharacteristic
519C2×C4C23×Dic7311yescharacteristic
520C2×C4C2×D4×D7311yescharacteristic
521C2×C4C2×#1192402711yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—2921no—
527C8—2921no—
528C8—2921no—
529C8—2921no—
530C8—2921no—
531C8—2921no—
532C8—1461no—
533C8—1461no—
534C8—1461no—
535C8—1461no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22C2×D4×Dic7311yescharacteristic
543C4—21no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4C2×#1455659811yescharacteristic
555C4C2×#1455659811yescharacteristic
556C4#1798939611yescharacteristic
557C4#1798940311yescharacteristic
558C2—41no—
559C2—42no—
560C2#2261862611yescharacteristic
561C2#2261877511yescharacteristic
562C2#2261884811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#3107C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C2×C4
cosocleC42×C73C24
hypercenter#3107D73
hypocenterC73#17921
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #22619026C2semidirect
3. #22608803C2semidirect
4. #22608789C2semidirect
5. #22608782C2semidirect
6. #22608768C2semidirect
7. #22608750C2semidirect
8. SD16⋊#10596831C2semidirect
9. Q16×#10596831C2semidirect
10. SD16×#10596831C2semidirect
11. #22608716C2semidirect
13. #22608334C2semidirect
14. #22608231C2semidirect
297. C73#17921semidirect

Maximal subgroups

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

subgroupindex
#226190422
#226190262
#226088032
#226087892
#226087822
#226087682
#226087502
SD16⋊#105968312
Q16×#105968312
SD16×#105968312
#226087162
#226083652
#226083342
#226082312
C73×#31072
#1792173