Finite Group Databasebeta

← #26111006

Subgroups — #26111006

Order 12032.

476 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252520C211yesmaximal, characteristic
2#20244366C211yesmaximal, characteristic, complemented
3C4×#13093951C211yesmaximal, characteristic
4#20241809C211yesmaximal, characteristic, complemented
5#20241279C211yesmaximal, characteristic, complemented
6#20241220C211yesmaximal, characteristic, complemented
7C47×#3107C211yesmaximal, characteristic
8Q8×C47⋊C8—21no—
9#15689201—22no—
10#15689201—21no—
11#15689176—21no—
12#15689176C2211yescharacteristic
13#15689147—21no—
14#15689147C2211yescharacteristic
15C2×#13093951—21no—
16C4×#10625329C2211yescharacteristic
17C42×Dic47C2211yescharacteristic
18#15688882—22no—
19#15688882—21no—
20Q8⋊(C47⋊C8)—21no—
21Q8⋊(C47⋊C8)—21no—
22C47×#533C2211yescharacteristic
23C4×#41×C47C2211yescharacteristic
24C47×#485—21no—
25C47×#475C411yescharacteristic, complemented
26C47×#474C411yescharacteristic, complemented
27C47×#462C411yescharacteristic, complemented
28C47×#461C411yescharacteristic, complemented
29C47×#452—21no—
30C4×Q16×C47—21no—
31C4×SD16×C47—21no—
32C47×#431C2211yescharacteristic
33C2×#10625329—22no—
34C2×#10625329—21no—
35C2×#10625329D412yes—
36C2×C4×Dic47—21no—
37C2×C4×Dic47—21no—
38C2×C4×Dic47—21no—
39C4⋊(C47⋊C8)—22no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)D411yescharacteristic
43C4⋊(C47⋊C8)D411yescharacteristic
44#13093951—24no—
45C22⋊(C47⋊C8)—22no—
46C22⋊(C47⋊C8)—21no—
47#13093935—21no—
48C4×C47⋊C8—22no—
49C4×C47⋊C8—21no—
50C4×C47⋊C8—21no—
51C2×#41×C47D411yescharacteristic
52C2×Q16×C47—21no—
53C2×SD16×C47—21no—
54C4⋊Q8×C47C2×C411yescharacteristic
55C4⋊Q8×C47C2×C411yescharacteristic
56#125×C47—21no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C4×Q8×C47—21no—
60C4×Q8×C47—21no—
61C4×Q8×C47C2×C411yescharacteristic
62C4×Q8×C47D411yescharacteristic
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47C2×C411yescharacteristic
66C4×D4×C47D411yescharacteristic
67#117×C47—21no—
68#117×C47—21no—
69#117×C47D411yescharacteristic
70C2×C42×C47—21no—
71C2×C42×C47C2311yescharacteristic
72#107×C47—21no—
73#106×C47—21no—
74C4⋊C8×C47C2×C411yescharacteristic
75C4⋊C8×C47C2×C411yescharacteristic
76#103×C47—21no—
77#103×C47—21no—
78#103×C47—21no—
79#102×C47—21no—
80C22⋊C8×C47—21no—
81C4×C8×C47—21no—
82#41×C47—22no—
83#41×C47—41no—
84C2×Q8×C47—21no—
85C2×Q8×C47—21no—
86C2×Q8×C47C22⋊C411yescharacteristic
87C2×D4×C47—21no—
88C2×D4×C47C22⋊C411yescharacteristic
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47#4111yescharacteristic
95C22×Dic47—41no—
96Q16×C47—42no—
97SD16×C47—42no—
98C2×C8×C47—21no—
99C2×C8×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47C22⋊C411yescharacteristic
109C4⋊C4×C47C22⋊C411yescharacteristic
110C22⋊C4×C47—21no—
111C22⋊C4×C47—21no—
112C22⋊C4×C47—21no—
113C22⋊C4×C47—41no—
114C42×C47—21no—
115C42×C47—21no—
116C42×C47—21no—
117C42×C47—41no—
118C42×C47C22×C411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47#4111yescharacteristic
122C4×Dic47—22no—
123C4×Dic47—22no—
124C4×Dic47—22no—
125C4×Dic47—41no—
126C4×Dic47—41no—
127C4×Dic47—41no—
128#10625329—22no—
129#10625329—41no—
130#10625329—42no—
131#10625329—82no—
132C2×C47⋊C8—21no—
133C2×C47⋊C8—22no—
134C2×C47⋊C8—21no—
135C2×C47⋊C8—41no—
136C23×C47—21no—
137C23×C47#12111yescharacteristic
138Q8×C47—21no—
139Q8×C47—41no—
140Q8×C47—41no—
141Q8×C47—41no—
142Q8×C47—41no—
143Q8×C47#10412yes—
144D4×C47—21no—
145D4×C47—21no—
146D4×C47—42no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—42no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47C2×C22⋊C411yescharacteristic
164C2×C4×C47C4×D411yescharacteristic
165C2×C4×C47C4×D411yescharacteristic
166C2×C4×C47#12011yescharacteristic
167C2×C4×C47#12111yescharacteristic
168C2×C4×C47#12311yescharacteristic
169C2×Dic47—42no—
170C2×Dic47—42no—
171C2×Dic47—42no—
172C8×C47—41no—
173C8×C47—41no—
174C47⋊C8—42no—
175C47⋊C8—42no—
176C47⋊C8—41no—
177C47⋊C8—41no—
178#8972—471nomaximal, Sylow
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—41no—
183C22×C47—41no—
184C22×C47—41no—
185C22×C47#38611yescharacteristic
186C4×C47—21no—
187C4×C47—22no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—41no—
191C4×C47—41no—
192C4×C47—81no—
193C4×C47—81no—
194C4×C47C2×#10411yescharacteristic
195C4×C47#42111yescharacteristic
196C4×C47#44311yescharacteristic
197C4×C47#44411yescharacteristic
198C4×C47#45511yescharacteristic
199C4×C47#45611yescharacteristic
200Dic47—44no—
201#3107D4711yescharacteristic
202#2991—471no—
203#1994—471no—
204C4×#104—471no—
205#1485—471no—
206#1299—471no—
207#1276—471no—
208C2×C47—41no—
209C2×C47—81no—
210C2×C47#187111yescharacteristic
211C2×C47#187611yescharacteristic
212C2×C47#188211yescharacteristic
213#533C2×D4711yescharacteristic
214C4×#41C2×D4711yescharacteristic
215#485—21no—
216#475Dic4711yescharacteristic, complemented
217#474Dic4711yescharacteristic, complemented
218#462Dic4711yescharacteristic, complemented
219#461Dic4711yescharacteristic, complemented
220#452—21no—
221C8×Q8—941no—
222C4×Q16—21no—
223C4×SD16—21no—
224#436—942no—
225#436—941no—
226#431—471no—
227#431—941no—
228#431C2×D4711yescharacteristic
229#423—471no—
230#423—941no—
231C2×#104—941no—
232C4×M16—471no—
233C43—471no—
234#339—942no—
235#339—941no—
236Q8⋊C8—941no—
237Q8⋊C8—941no—
238C47#897211yescharacteristic, complemented, Sylow
239C2×#41C22⋊D4711yescharacteristic
240C2×Q16—21no—
241C2×SD16—21no—
242C2×M16—472no—
243C2×M16—942no—
244C2×M16—941no—
245C4⋊Q8C2×Dic4711yescharacteristic
246C4⋊Q8C2×Dic4711yescharacteristic
247#125—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8C2×Dic4711yescharacteristic
253C4×Q8C22⋊D4711yescharacteristic
254C4×D4—21no—
255C4×D4—21no—
256C4×D4C2×Dic4711yescharacteristic
257C4×D4C22⋊D4711yescharacteristic
258#117—21no—
259#117—21no—
260#117C22⋊D4711yescharacteristic
261C2×C42—21no—
262C2×C42—941no—
263C2×C42—941no—
264C2×C42—941no—
265C2×C42C22×D4711yescharacteristic
266#107—21no—
267#106—21no—
268C4⋊C8—471no—
269C4⋊C8—471no—
270C4⋊C8—942no—
271C4⋊C8—941no—
272C4⋊C8—941no—
273C4⋊C8C2×Dic4711yescharacteristic
274C4⋊C8C2×Dic4711yescharacteristic
275#104—944no—
276#103—21no—
277#103—21no—
278#103—21no—
279#102—21no—
280C22⋊C8—21no—
281C22⋊C8—942no—
282C22⋊C8—941no—
283#97—941no—
284C4×C8—21no—
285C4×C8—942no—
286C4×C8—941no—
287C4×C8—941no—
288#41—22no—
289#41—41no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8C22⋊Dic4711yescharacteristic
293C2×D4—21no—
294C2×D4C22⋊Dic4711yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—1881no—
299C22×C4D4×D4711yescharacteristic
300C22×C4#1062535211yescharacteristic
301C22×C4C2×C22⋊D4711yescharacteristic
302Q16—42no—
303SD16—42no—
304M16—942no—
305M16—3762no—
306M16—1881no—
307M16—1882no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—941no—
311C2×C8—941no—
312C2×C8—942no—
313C2×C8—1881no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4C22⋊Dic4711yescharacteristic
323C4⋊C4C22⋊Dic4711yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—942no—
333C42—942no—
334C42—942no—
335C42—1881no—
336C42—1881no—
337C42—1881no—
338C42D4×D4711yescharacteristic
339C42#1062535211yescharacteristic
340C42C22×Dic4711yescharacteristic
341C42C2×C22⋊D4711yescharacteristic
342C23—21no—
343C23#1309399611yescharacteristic
344Q8—21no—
345Q8—41no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349Q8#1309395112yes—
350D4—21no—
351D4—21no—
352D4—42no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—1882no—
370C2×C4—1882no—
371C2×C4—1882no—
372C2×C4C2×C22⋊Dic4711yescharacteristic
373C2×C4D4×Dic4711yescharacteristic
374C2×C4D4×Dic4711yescharacteristic
375C2×C4#1309399211yescharacteristic
376C2×C4#1309399611yescharacteristic
377C2×C4#1309400511yescharacteristic
378C8—41no—
379C8—41no—
380C8—1882no—
381C8—1881no—
382C8—1882no—
383C8—1881no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22#1568900311yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—81no—
399C4—1884no—
400C4C2×#1309395111yescharacteristic
401C4#1568914011yescharacteristic
402C4#1568924011yescharacteristic
403C4#1568924511yescharacteristic
404C4#1568929911yescharacteristic
405C4#1568930411yescharacteristic
406C2—41no—
407C2—81no—
408C2#2024369111yescharacteristic
409C2#2024371711yescharacteristic
410C2#2024375911yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3107C2
socleC22×C47#386
prime radicalC2×#41×C47D4
cosocleC2×C42×C47C23
hypercenter#3107D47
hypocenterC47#8972
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #20244366C2semidirect
4. #20241809C2semidirect
5. #20241279C2semidirect
6. #20241220C2semidirect
25. C47×#475C4semidirect
26. C47×#474C4semidirect
27. C47×#462C4semidirect
28. C47×#461C4semidirect
216. #475Dic47semidirect
217. #474Dic47semidirect
218. #462Dic47semidirect
219. #461Dic47semidirect
238. C47#8972semidirect

Maximal subgroups

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

subgroupindex
#202525202
#202443662
C4×#130939512
#202418092
#202412792
#202412202
C47×#31072
#897247