Finite Group Databasebeta

← #33073843

Subgroups — #33073843

Order 18688.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620872C211yesmaximal, characteristic, complemented
2#22620853C211yesmaximal, characteristic, complemented
3#22620486C211yesmaximal, characteristic, complemented
4#22620477C211yesmaximal, characteristic, complemented
5C2×#17988933C211yesmaximal, characteristic, complemented
6#22609557C212yesmaximal, complemented
7#22609550C212yesmaximal, complemented
8#22609530C212yesmaximal, complemented
9#22609520C212yesmaximal, complemented
10M16×#10596831C211yesmaximal, characteristic, complemented
11C73×#3151C211yesmaximal, characteristic
12#17989144C2211yescharacteristic
13C2×#14556591C2211yescharacteristic
14#17988996—22no—
15#17988992C2212yescomplemented
16#17988985C2212yescomplemented
17#17988983C2211yescharacteristic, complemented
18#17988959C2212yes—
19#17988959C2211yescharacteristic, complemented
20#17988959C2212yescomplemented
21#17988939C2211yescharacteristic, complemented
22#17988939C2212yescomplemented
23#17988933—22no—
24#17988933C2212yes—
25#17988933C2212yes—
26#17988933C2212yes—
27#17988933C2211yescharacteristic, complemented
28#17988814—22no—
29#17988792—22no—
30#17988784—22no—
31#17988738—22no—
32#17988725—22no—
33C4×#11924016—22no—
34#17988640C2211yescharacteristic
35#17988629—22no—
36C2×#14556512—22no—
37C2×C4×#10596831C2211yescharacteristic
38#17988308C2212yescomplemented
39C8×#10596831C2212yescomplemented
40C2×AGL(1,ℤ/8)×C73—22no—
41#532×C73C2211yescharacteristic
42C2×#127×C73C2211yescharacteristic
43#496×C73—22no—
44#493×C73C2212yes—
45#492×C73C2212yes—
46C4×M16×C73C2211yescharacteristic
47C22×Dic146—21no—
48C2×#11924016—22no—
49C2×#11924016—24no—
50C2×#11924016—24no—
51C22×#10596831D412yes—
52#14556591—21no—
53#14556591—21no—
54#14556591—21no—
55#14556591C2312yes—
56#14556591C2311yescharacteristic
57#14556591C2311yescharacteristic
58#14556583—21no—
59#14556570—21no—
60#14556570—21no—
61C4×Dic146—21no—
62C4×Dic146—21no—
63#14556550—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66#14556525—24no—
67#14556523—24no—
68#14556522—22no—
69C4⋊#10596831D412yes—
70#14556514—22no—
71#14556514—24no—
72#14556512—22no—
73#14556512—24no—
74#14556512—24no—
75#14556512—24no—
76C22⋊#10596831D412yes—
77C4×#10596831D412yes—
78C4×#10596831C2312yes—
79C4×#10596831C2311yescharacteristic
80C4×#10596831C2311yescharacteristic
81C2×#41×C73—21no—
82C22×D4×C73—21no—
83C22×D4×C73—21no—
84C22×D4×C73—21no—
85AGL(1,ℤ/8)×C73—48no—
86C2×SD16×C73—24no—
87C2×D8×C73—24no—
88C2×M16×C73D412yes—
89C4⋊Q8×C73C2311yescharacteristic
90#127×C73—21no—
91#127×C73—21no—
92#127×C73—21no—
93#127×C73C2312yes—
94#127×C73C2311yescharacteristic
95#124×C73—21no—
96#121×C73—21no—
97#121×C73—21no—
98C4×D4×C73—21no—
99C4×D4×C73—21no—
100C2×C42×C73C2311yescharacteristic
101#97×C73D412yes—
102C4×C8×C73C2312yes—
103C2×Dic146—21no—
104C2×Dic146—22no—
105C2×Dic146—21no—
106C2×Dic146—41no—
107C2×Dic146—82no—
108C22×Dic73—41no—
109#11924016—44no—
110#11924016—84no—
111#11924016—84no—
112C2×#10596831—42no—
113C2×#10596831C2×D412yes—
114C2×#10596831C2×D412yes—
115C2×#10596831C2×D414yes—
116C4⋊Dic73—22no—
117C4⋊Dic73—21no—
118C4⋊Dic73—21no—
119C4⋊Dic73—21no—
120C4⋊Dic73—22no—
121C4⋊Dic73—21no—
122C4⋊Dic73—41no—
123C4⋊Dic73—41no—
124#11924007—41no—
125#11924007—41no—
126C22⋊Dic73—41no—
127C4×Dic73—41no—
128C24×C73—41no—
129#41×C73—44no—
130C2×Q8×C73—21no—
131C2×D4×C73—21no—
132C2×D4×C73—21no—
133C2×D4×C73—21no—
134C2×D4×C73—22no—
135C2×D4×C73—22no—
136C2×D4×C73—22no—
137C2×D4×C73—44no—
138C2×D4×C73—41no—
139C2×D4×C73—41no—
140C2×D4×C73—41no—
141C2×D4×C73—41no—
142C2×D4×C73—44no—
143C2×D4×C73—82no—
144C22×C4×C73—41no—
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73C2×D411yescharacteristic
148SD16×C73—48no—
149D8×C73—48no—
150M16×C73SD1618yescomplemented
151C2×C8×C73C2×D414yes—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—21no—
154C22⋊C4×C73—41no—
155C22⋊C4×C73—41no—
156C42×C73C2×D411yescharacteristic
157C42×C73C2×D411yescharacteristic
158C42×C73C2×D411yescharacteristic
159C42×C73C2411yescharacteristic
160C23×C73—41no—
161C23×C73—42no—
162C23×C73—82no—
163C23×C73—82no—
164C23×C73—82no—
165C23×C73#12711yescharacteristic
166Q8×C73—42no—
167D4×C73—42no—
168D4×C73—44no—
169D4×C73—44no—
170D4×C73—42no—
171D4×C73—44no—
172D4×C73—42no—
173D4×C73—82no—
174D4×C73—82no—
175D4×C73—82no—
176D4×C73—82no—
177C2×C4×C73—22no—
178C2×C4×C73—22no—
179C2×C4×C73—41no—
180C2×C4×C73—82no—
181C2×C4×C73#12711yescharacteristic
182C2×C4×C73#12711yescharacteristic
183C2×C4×C73#12711yescharacteristic
184C2×C4×C73C2×SD1614yes—
185C2×C4×C73C22×D411yescharacteristic
186C2×C4×C73C22×D411yescharacteristic
187C2×C4×C73C22×D411yescharacteristic
188C2×Dic73—41no—
189C2×Dic73—41no—
190C2×Dic73—42no—
191C2×Dic73—82no—
192Dic146—42no—
193Dic146—42no—
194Dic146—44no—
195Dic146—82no—
196C8×C73C2×SD1618yes—
197#10596831—22no—
198#10596831—22no—
199#10596831—24no—
200C22×C73—21no—
201C22×C73—21no—
202C22×C73—81no—
203C22×C73—82no—
204C22×C73—82no—
205C22×C73—81no—
206C22×C73—82no—
207C22×C73—81no—
208C22×C73—164no—
209C22×C73#49212yes—
210C22×C73C2×#12711yescharacteristic
211C4×C73—21no—
212C4×C73—21no—
213C4×C73—81no—
214C4×C73#49212yes—
215C4×C73C22×SD1612yes—
216C4×C73#57512yes—
217C4×C73#57612yes—
218Dic73—81no—
219Dic73—82no—
220Dic73—81no—
221#19623—731nomaximal, Sylow
222C2×C73—22no—
223C2×C73—161no—
224C2×C73—162no—
225C2×C73C2×#49211yescharacteristic
226C2×C73#315311yescharacteristic
227C2×C73#315511yescharacteristic
228#3208—731no—
229#3206—731no—
230#3167—731no—
231#3165—731no—
232#3151D7311yescharacteristic
233C2×#486—731no—
234#1715—732no—
235#1714—732no—
236#1708—732no—
237#1706—732no—
238C8×M16—731no—
239C73#1962311yescharacteristic, complemented, Sylow
240C2×AGL(1,ℤ/8)—22no—
241#533—731no—
242#532C2×D7311yescharacteristic
243C2×C4⋊Q8—731no—
244C2×#127C2×D7311yescharacteristic
245#501—1462no—
246#500—732no—
247#498—731no—
248#498—732no—
249#496—22no—
250#493C2×D7312yes—
251#492—731no—
252#492—732no—
253#492—732no—
254#492C2×D7312yes—
255#487—731no—
256#487—732no—
257#486—732no—
258#486—732no—
259#486—732no—
260#486—731no—
261#486—1462no—
262#465—1462no—
263#461—1462no—
264#460—1462no—
265#450—1462no—
266#448—1462no—
267C4×SD16—1462no—
268#431—731no—
269#428—1462no—
270C2×#102—1462no—
271C4×M16C2×D7311yescharacteristic
272C2×C4×C8—731no—
273#322—732no—
274C82—732no—
275C2×#41—21no—
276C22×Q8—1461no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280AGL(1,ℤ/8)—48no—
281C2×SD16—24no—
282C2×SD16—1462no—
283C2×SD16—1464no—
284C2×SD16—1464no—
285C2×D8—24no—
286C2×M16C22⋊D7312yes—
287C22×C8—732no—
288C4⋊Q8—732no—
289C4⋊Q8—731no—
290C4⋊Q8—731no—
291C4⋊Q8—1461no—
292C4⋊Q8—1461no—
293C4⋊Q8—1461no—
294C4⋊Q8C22×D7311yescharacteristic
295#127—21no—
296#127—21no—
297#127—21no—
298#127C22×D7312yes—
299#127C22×D7311yescharacteristic
300#125—1461no—
301#124—21no—
302C22⋊Q8—1461no—
303C22⋊Q8—1461no—
304#121—21no—
305#121—21no—
306C4×Q8—1461no—
307C4×Q8—1461no—
308C4×D4—21no—
309C4×D4—21no—
310#117—1461no—
311C2×C4⋊C4—1461no—
312C2×C4⋊C4—1461no—
313C2×C42C22×D7311yescharacteristic
314#107—1464no—
315#106—1462no—
316#106—1464no—
317C4⋊C8—732no—
318#103—1462no—
319#103—1464no—
320#102—1464no—
321#102—1464no—
322#102—1462no—
323#102—1464no—
324C22⋊C8—732no—
325#97C22⋊D7312yes—
326C4×C8—732no—
327C4×C8—731no—
328C4×C8—732no—
329C4×C8—731no—
330C4×C8C22×D7312yes—
331C24—41no—
332#41—44no—
333C2×Q8—21no—
334C2×Q8—2921no—
335C2×Q8—5842no—
336C2×Q8—1461no—
337C2×Q8—1462no—
338C2×Q8—1461no—
339C2×D4—22no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—44no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—44no—
351C2×D4—82no—
352C22×C4—41no—
353C22×C4—2921no—
354C22×C4D4×D7311yescharacteristic
355C22×C4D4×D7311yescharacteristic
356C22×C4C2×C22⋊D7311yescharacteristic
357SD16—48no—
358SD16—2924no—
359SD16—5844no—
360SD16—5844no—
361D8—48no—
362M16#1192401618yescomplemented
363C2×C8—2922no—
364C2×C8—732no—
365C2×C8—732no—
366C2×C8—734no—
367C2×C8C2×C22⋊D7314yes—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—2921no—
371C4⋊C4—2921no—
372C4⋊C4—2921no—
373C4⋊C4—2921no—
374C4⋊C4—1462no—
375C4⋊C4—1461no—
376C4⋊C4—1461no—
377C4⋊C4—1461no—
378C4⋊C4—1462no—
379C4⋊C4—1461no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—2921no—
383C42—2921no—
384C42D4×D7311yescharacteristic
385C42D4×D7311yescharacteristic
386C42C2×C22⋊D7311yescharacteristic
387C42C23×D7311yescharacteristic
388C23—41no—
389C23—42no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23#1455658711yescharacteristic
394Q8—42no—
395Q8—2922no—
396Q8—2922no—
397Q8—2924no—
398Q8—5842no—
399D4—42no—
400D4—44no—
401D4—42no—
402D4—44no—
403D4—42no—
404D4—44no—
405D4—82no—
406D4—82no—
407D4—82no—
408D4—82no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—82no—
413C2×C4—2921no—
414C2×C4—2921no—
415C2×C4—2922no—
416C2×C4—5842no—
417C2×C4#1455658711yescharacteristic
418C2×C4#1455658711yescharacteristic
419C2×C4#1455658711yescharacteristic
420C2×C4C2×#1192401614yes—
421C2×C4C2×D4×D7311yescharacteristic
422C2×C4C2×D4×D7311yescharacteristic
423C2×C4C22×C22⋊D7311yescharacteristic
424C8—1462no—
425C8—1462no—
426C8—1464no—
427C8C2×#1192401618yes—
428C22—21no—
429C22—21no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22—82no—
434C22—82no—
435C22—81no—
436C22—164no—
437C22#1798895912yes—
438C22C2×#1455658711yescharacteristic
439C4—21no—
440C4—21no—
441C4—81no—
442C4—5842no—
443C4—5841no—
444C4—5841no—
445C4#1798895912yes—
446C4C22×#1192401612yes—
447C4#1798938312yes—
448C4#1798939112yes—
449C2—22no—
450C2—162no—
451C2—161no—
452C2C2×#1798895911yescharacteristic
453C2#2262039311yescharacteristic
454C2#2262041611yescharacteristic
subgroupG/H
centerC22C2×#14556587
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3151C2
socleC22×C73C2×#127
hypercenter#3151D73
hypocenterC73#19623
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #22620872C2semidirect
2. #22620853C2semidirect
3. #22620486C2semidirect
4. #22620477C2semidirect
5. C2×#17988933C2semidirect
6. #22609557C2semidirect
7. #22609550C2semidirect
8. #22609530C2semidirect
9. #22609520C2semidirect
10. M16×#10596831C2semidirect
15. #17988992C22semidirect
16. #17988985C22semidirect
17. #17988983C22semidirect
19. #17988959C22semidirect
20. #17988959C22semidirect
21. #17988939C22semidirect
22. #17988939C22semidirect
27. #17988933C22semidirect
38. #17988308C22semidirect
39. C8×#10596831C22semidirect
150. M16×C73SD16semidirect
239. C73#19623semidirect
362. M16#11924016semidirect

Maximal subgroups

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

subgroupindex
#226208722
#226208532
#226204862
#226204772
C2×#179889332
#226095572
#226095502
#226095302
#226095202
M16×#105968312
C73×#31512
#1962373