Finite Group Databasebeta

← #26316231

Subgroups — #26316231

Order 12032.

612 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253900C211yesmaximal, characteristic, complemented
2C2×#15689240C211yesmaximal, characteristic
3#20244562C212yesmaximal, complemented
4#20244558C212yesmaximal, complemented
5#20244287C211yesmaximal, characteristic
6#20243512C211yesmaximal, characteristic, complemented
7#20243511C211yesmaximal, characteristic, complemented
8#20243440C212yesmaximal, complemented
9#20243432C212yesmaximal, complemented
10#20243268C211yesmaximal, characteristic, complemented
11C47×#1976C211yesmaximal, characteristic
12C2×#13094031C2212yes—
13#15689676C2212yes—
14#15689527—21no—
15#15689524—21no—
16#15689523—21no—
17#15689520—21no—
18#15689240—21no—
19#15689240—21no—
20#15689240—21no—
21#15689240—21no—
22C2×#13093960C2211yescharacteristic
23C2×#13093959C2211yescharacteristic
24#15689161C2211yescharacteristic
25#15689159C2211yescharacteristic
26#15689159C2212yes—
27#15689140C2212yes—
28C2×#13093951C2212yes—
29#15689097C2212yes—
30#15689078C2211yescharacteristic
31C2×C8×Dic47C2211yescharacteristic
32#15688894C2212yes—
33#15688893C2212yes—
34#15688884C2211yescharacteristic
35#15688882C2211yescharacteristic
36#15688879C2212yes—
37#15688879C2212yes—
38C2×#135×C47C2211yescharacteristic
39C47×#427C2211yescharacteristic
40C47×#421C2212yes—
41C47×#410C2212yes—
42C47×#405C2211yescharacteristic
43#13094031—22no—
44#13094031—22no—
45#13094031—22no—
46#13094031—22no—
47C2×#10625329—22no—
48C2×#10625329C2311yescharacteristic
49C2×#10625329C2312yes—
50C22×C47⋊C8—22no—
51C22×C47⋊C8C2311yescharacteristic
52#13094023—22no—
53#13094015—22no—
54#13094001—42no—
55C4×Dic94—42no—
56#13093983C2311yescharacteristic
57#13093983C2312yes—
58C2×C4×Dic47C2311yescharacteristic
59#13093960—21no—
60#13093960—21no—
61#13093959—21no—
62#13093959—21no—
63#13093958—21no—
64#13093957—22no—
65#13093957—21no—
66#13093956—21no—
67#13093955—22no—
68#13093955—21no—
69#13093951—22no—
70#13093951—22no—
71#13093951—42no—
72#13093939—42no—
73#13093934—21no—
74C8×Dic47—21no—
75C8×Dic47D411yescharacteristic
76C8×Dic47D411yescharacteristic
77C8×Dic47D411yescharacteristic
78C8×Dic47D411yescharacteristic
79C2×#41×C47C2312yes—
80#135×C47—21no—
81#135×C47—21no—
82#135×C47—21no—
83#135×C47—21no—
84C2×Q16×C47Q811yescharacteristic, complemented
85C2×SD16×C47Q812yescomplemented
86C2×D8×C47Q811yescharacteristic, complemented
87C2×M16×C47C2311yescharacteristic
88C22×C8×C47C2311yescharacteristic
89#117×C47C2311yescharacteristic
90#117×C47C2312yes—
91#107×C47—21no—
92#106×C47—21no—
93#104×C47—42no—
94#99×C47—42no—
95#97×C47—21no—
96C4×C8×C47—21no—
97#41×C47—22no—
98#41×C47—22no—
99#41×C47—22no—
100#41×C47—22no—
101C2×Q8×C47C2×Q812yes—
102C2×D4×C47—22no—
103C2×D4×C47C2×Q812yes—
104C22×C4×C47—22no—
105C22×C4×C47C2411yescharacteristic
106C22×Dic47—21no—
107C2×Dic94—42no—
108Q16×C47—21no—
109Q16×C47—21no—
110SD16×C47—22no—
111SD16×C47—22no—
112D8×C47—21no—
113D8×C47—21no—
114M16×C47—21no—
115M16×C47—21no—
116C2×C8×C47—21no—
117C2×C8×C47C2×D411yescharacteristic
118C2×C8×C47C2×D411yescharacteristic
119C2×C8×C47C2×Q811yescharacteristic
120C2×C8×C47C2×Q811yescharacteristic
121C2×C8×C47#4111yescharacteristic
122C2×C8×C47#4111yescharacteristic
123C4⋊C4×C47—22no—
124C4⋊C4×C47—21no—
125C22⋊C4×C47—22no—
126C22⋊C4×C47—21no—
127C42×C47—21no—
128C42×C47—22no—
129C22⋊Dic47—21no—
130C22⋊Dic47—22no—
131C4⋊Dic47—22no—
132C4⋊Dic47—21no—
133C4⋊Dic47—42no—
134Dic47⋊C4—42no—
135Dic47⋊C4—42no—
136C4×Dic47—21no—
137C4×Dic47—21no—
138C4×Dic47—22no—
139C4×Dic47C2×D411yescharacteristic
140C4×Dic47C2×D411yescharacteristic
141#10625329—22no—
142#10625329—21no—
143#10625329—22no—
144#10625329—21no—
145#10625329—42no—
146#10625329—42no—
147C2×C47⋊C8—21no—
148C2×C47⋊C8—22no—
149C2×C47⋊C8—42no—
150C2×C47⋊C8—42no—
151C2×C47⋊C8C2×D411yescharacteristic
152C2×C47⋊C8C2×D411yescharacteristic
153C2×C47⋊C8#4111yescharacteristic
154C2×C47⋊C8#4111yescharacteristic
155C2×C47⋊C8#4111yescharacteristic
156C2×C47⋊C8#4111yescharacteristic
157C23×C47—22no—
158C23×C47H(5,𝔽2)11yescharacteristic
159Q8×C47—22no—
160Q8×C47—22no—
161D4×C47—22no—
162D4×C47—22no—
163D4×C47—42no—
164D4×C47—42no—
165C2×C4×C47—22no—
166C2×C4×C47—42no—
167C2×C4×C47—42no—
168C2×C4×C47—42no—
169C2×C4×C47—41no—
170C2×C4×C47C22×D411yescharacteristic
171C2×C4×C47C22×Q811yescharacteristic
172C2×C4×C47C2×#4111yescharacteristic
173C2×C4×C47C2×#4111yescharacteristic
174C2×C4×C47C2×#4111yescharacteristic
175C2×C4×C47H(5,𝔽2)11yescharacteristic
176C2×Dic47—21no—
177C2×Dic47—21no—
178C2×Dic47—42no—
179C2×Dic47—42no—
180C2×Dic47—41no—
181Dic94—84no—
182C8×C47—41no—
183C8×C47C22⋊Q811yescharacteristic
184C8×C47C22⋊Q811yescharacteristic
185C8×C47C22⋊Q811yescharacteristic
186C8×C47C22⋊Q811yescharacteristic
187C47⋊C8—42no—
188C47⋊C8—41no—
189C47⋊C8#12411yescharacteristic
190C47⋊C8#12411yescharacteristic
191C47⋊C8#12411yescharacteristic
192C47⋊C8#12411yescharacteristic
193#29154—471nomaximal, Sylow
194C22×C47—41no—
195C22×C47—42no—
196C22×C47—82no—
197C22×C47#53511yescharacteristic
198C22×C47#55411yescharacteristic
199C22×C47#55511yescharacteristic
200C4×C47—42no—
201C4×C47—82no—
202C4×C47—81no—
203C4×C47C2×C22⋊Q811yescharacteristic
204C4×C47C2×#12411yescharacteristic
205C4×C47#55411yescharacteristic
206C4×C47#55511yescharacteristic
207Dic47—22no—
208Dic47—41no—
209Dic47—82no—
210Dic47—81no—
211#3160—471no—
212C2×#443—471no—
213#2031—472no—
214#2031—472no—
215#1976—471no—
216#1976D4711yescharacteristic
217#1833—471no—
218#1833—471no—
219#1814—472no—
220#1812—472no—
221#1772—471no—
222C2×C47—21no—
223C2×C47—21no—
224C2×C47—21no—
225C2×C47—82no—
226C2×C47#256311yescharacteristic
227C2×#135C2×D4711yescharacteristic
228C2×#131—472no—
229#533—472no—
230#500—941no—
231#499—941no—
232#499—941no—
233#498—941no—
234#443—941no—
235#443—941no—
236#443—941no—
237#443—941no—
238C2×#108—471no—
239C2×#108—471no—
240#427—471no—
241#427—471no—
242#427—472no—
243#427C2×D4711yescharacteristic
244#421—472no—
245#421C2×D4712yes—
246C2×#104—472no—
247#410—472no—
248#410C2×D4712yes—
249#405—471no—
250#405C2×D4711yescharacteristic
251C2×C4×C8—471no—
252#344—472no—
253#344—472no—
254#339—471no—
255#339—471no—
256H(3,ℤ/4)—472no—
257H(3,ℤ/4)—472no—
258C47#2915411yescharacteristic, complemented, Sylow
259C2×#41C22×D4712yes—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—21no—
264C2×Q16Dic9411yescharacteristic, complemented
265C2×SD16Dic9412yescomplemented
266C2×D8Dic9411yescharacteristic, complemented
267#131—942no—
268#131—942no—
269#131—942no—
270#131—942no—
271C2×M16—472no—
272C2×M16—471no—
273C2×M16—942no—
274C2×M16C22×D4711yescharacteristic
275C22×C8—471no—
276C22×C8—942no—
277C22×C8C22×D4711yescharacteristic
278C4⋊Q8—942no—
279#125—942no—
280C22⋊Q8—1882no—
281C4×Q8—1882no—
282#117—472no—
283#117—471no—
284#117C22×D4712yes—
285#117C22×D4711yescharacteristic
286C2×C42—471no—
287#108—941no—
288#108—941no—
289#108—941no—
290#108—941no—
291#107—21no—
292#107—941no—
293#107—942no—
294#107—941no—
295#106—21no—
296#106—941no—
297#106—942no—
298#106—941no—
299#104—42no—
300#104—942no—
301#104—942no—
302#104—1882no—
303#99—42no—
304#99—1882no—
305#97—21no—
306#97—941no—
307C4×C8—21no—
308C4×C8—471no—
309C4×C8—471no—
310C4×C8—471no—
311C4×C8—471no—
312C4×C8—941no—
313#41—22no—
314#41—22no—
315#41—22no—
316#41—22no—
317C2×Q8—1882no—
318C2×Q8C2×Dic9412yes—
319C2×D4—22no—
320C2×D4C2×Dic9412yes—
321C22×C4—22no—
322C22×C4—941no—
323C22×C4C23×D4711yescharacteristic
324Q16—21no—
325Q16—21no—
326SD16—22no—
327SD16—22no—
328D8—21no—
329D8—21no—
330M16—21no—
331M16—21no—
332M16—942no—
333M16—941no—
334M16—942no—
335M16—941no—
336M16—1882no—
337M16—1882no—
338C2×C8—21no—
339C2×C8—471no—
340C2×C8—471no—
341C2×C8—471no—
342C2×C8—471no—
343C2×C8—471no—
344C2×C8—471no—
345C2×C8—942no—
346C2×C8—941no—
347C2×C8—1882no—
348C2×C8—1882no—
349C2×C8C2×Dic9411yescharacteristic
350C2×C8C2×Dic9411yescharacteristic
351C2×C8D4×D4711yescharacteristic
352C2×C8D4×D4711yescharacteristic
353C2×C8#1062535211yescharacteristic
354C2×C8#1062535211yescharacteristic
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—942no—
358C4⋊C4—941no—
359C4⋊C4—1882no—
360C4⋊C4—1882no—
361C4⋊C4—1882no—
362C22⋊C4—21no—
363C22⋊C4—22no—
364C22⋊C4—941no—
365C22⋊C4—942no—
366C42—21no—
367C42—22no—
368C42—471no—
369C42—471no—
370C42—941no—
371C42—941no—
372C42—942no—
373C23—22no—
374C23#1309406911yescharacteristic
375Q8—22no—
376Q8—22no—
377Q8—3764no—
378D4—22no—
379D4—22no—
380D4—42no—
381D4—42no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—941no—
388C2×C4—941no—
389C2×C4—1882no—
390C2×C4—1881no—
391C2×C4—1882no—
392C2×C4C2×D4×D4711yescharacteristic
393C2×C4C22×Dic9411yescharacteristic
394C2×C4#41×D4711yescharacteristic
395C2×C4#41×D4711yescharacteristic
396C2×C4C2×#1062535211yescharacteristic
397C2×C4#1309406911yescharacteristic
398C8—41no—
399C8—471no—
400C8—471no—
401C8—471no—
402C8—471no—
403C8—1881no—
404C8—1882no—
405C8#1309400111yescharacteristic
406C8#1309400111yescharacteristic
407C8#1309400111yescharacteristic
408C8#1309400111yescharacteristic
409C22—41no—
410C22—42no—
411C22—82no—
412C22#1568968611yescharacteristic
413C22#1568983211yescharacteristic
414C22#1568983711yescharacteristic
415C4—42no—
416C4—82no—
417C4—81no—
418C4—942no—
419C4—3762no—
420C4—3761no—
421C4—1881no—
422C4C2×#1309400111yescharacteristic
423C4#124×D4711yescharacteristic
424C4#1568983211yescharacteristic
425C4#1568983711yescharacteristic
426C2—21no—
427C2—21no—
428C2—21no—
429C2—82no—
430C2#2024855911yescharacteristic
subgroupG/H
centerC4#15689837
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1976C2
socleC2×C47#2563
prime radicalC2×#135×C47C22
hypercenter#1976D47
hypocenterC47#29154
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
1. #20253900C2semidirect
3. #20244562C2semidirect
4. #20244558C2semidirect
6. #20243512C2semidirect
7. #20243511C2semidirect
8. #20243440C2semidirect
9. #20243432C2semidirect
10. #20243268C2semidirect
84. C2×Q16×C47Q8semidirect
85. C2×SD16×C47Q8semidirect
86. C2×D8×C47Q8semidirect
258. C47#29154semidirect
264. C2×Q16Dic94semidirect
265. C2×SD16Dic94semidirect
266. C2×D8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202539002
C2×#156892402
#202445622
#202445582
#202442872
#202435122
#202435112
#202434402
#202434322
#202432682
C47×#19762
#2915447