Finite Group Databasebeta

← #26198998

Subgroups — #26198998

Order 12032.

596 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253974C211yesmaximal, characteristic, complemented
2#20253176C212yesmaximal
3#20242970C212yesmaximal, complemented
4#20242963C212yesmaximal, complemented
5#20242916C211yesmaximal, characteristic, complemented
6#20242911C211yesmaximal, characteristic, complemented
7#20242882C211yesmaximal, characteristic, complemented
8#20242877C211yesmaximal, characteristic, complemented
9C2×#15688874C211yesmaximal, characteristic, complemented
10#20242043C212yesmaximal, complemented
11C47×#2965C211yesmaximal, characteristic
12#15689680C2212yes—
13#15689528C2211yescharacteristic
14#15689503C2211yescharacteristic
15#15689474C2212yes—
16#15689430—22no—
17#15689423—22no—
18#15689418C2212yes—
19#15689413C2212yes—
20#15689408C2212yes—
21#15689403C2212yes—
22#15689364—21no—
23#15689364—21no—
24C4×Dic188—21no—
25C4×Dic188—21no—
26#15689159—22no—
27C2×C4⋊(C47⋊C8)C2212yes—
28#15689119—22no—
29#15688874C2212yes—
30#15688874C2211yescharacteristic
31#15688874C2211yescharacteristic
32#15688871C2212yes—
33#15688871C2212yes—
34#15688870C2212yes—
35#15688870C2212yes—
36C4×#41×C47C2211yescharacteristic
37C47×#446C2211yescharacteristic
38C47×#446C2211yescharacteristic
39C47×#436C2211yescharacteristic
40C47×#436C2211yescharacteristic
41C47×#435—21no—
42C47×#435—21no—
43C47×#431—21no—
44C47×#431C2211yescharacteristic
45C4×M16×C47—21no—
46C2×C4×C8×C47C2211yescharacteristic
47C2×Dic188—21no—
48C2×Dic188—21no—
49C22×C47⋊C8—22no—
50#13094024C2312yes—
51#13094024C2312yes—
52#13094016—22no—
53#13094003—22no—
54#13094003—22no—
55C4×Dic94—22no—
56C4×Dic94—22no—
57#13093983—22no—
58#13093958—21no—
59#13093958—21no—
60#13093957—22no—
61#13093957—22no—
62#13093955—22no—
63#13093955—22no—
64C4⋊(C47⋊C8)—22no—
65C4⋊(C47⋊C8)C2312yes—
66C4⋊(C47⋊C8)C2312yes—
67#13093949—22no—
68#13093949—22no—
69#13093948—22no—
70#13093948—22no—
71#13093945—22no—
72#13093945—22no—
73C2×#41×C47D411yescharacteristic
74C2×M16×C47—21no—
75C2×M16×C47—21no—
76C22×C8×C47D411yescharacteristic
77C22×C8×C47D411yescharacteristic
78C4×Q8×C47D411yescharacteristic
79C4×Q8×C47C2311yescharacteristic
80C4×D4×C47—21no—
81C4×D4×C47—21no—
82C4×D4×C47D411yescharacteristic
83C4×D4×C47C2311yescharacteristic
84#117×C47—21no—
85#117×C47D411yescharacteristic
86C2×C42×C47—21no—
87C2×C42×C47C2311yescharacteristic
88C4⋊C8×C47—21no—
89C4⋊C8×C47—21no—
90C4⋊C8×C47C2311yescharacteristic
91C4⋊C8×C47C2311yescharacteristic
92C22⋊C8×C47—21no—
93C22⋊C8×C47—21no—
94C22⋊C8×C47—21no—
95#97×C47—21no—
96#97×C47—21no—
97C4×C8×C47—21no—
98C4×C8×C47D411yescharacteristic
99C4×C8×C47D411yescharacteristic
100C4×C8×C47C2311yescharacteristic
101C4×C8×C47C2311yescharacteristic
102#41×C47—24no—
103C2×Q8×C47C2×D411yescharacteristic
104C2×D4×C47—21no—
105C2×D4×C47C2×D411yescharacteristic
106C22×C4×C47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C2×Dic94—22no—
113C2×Dic94—22no—
114M16×C47—42no—
115M16×C47—42no—
116C2×C8×C47—22no—
117C2×C8×C47—21no—
118C2×C8×C47—21no—
119C2×C8×C47—41no—
120C2×C8×C47C2×D411yescharacteristic
121C2×C8×C47C2×D411yescharacteristic
122C2×C8×C47C2×D411yescharacteristic
123C2×C8×C47C2×D411yescharacteristic
124C4⋊C4×C47—21no—
125C4⋊C4×C47—21no—
126C4⋊C4×C47C2×D411yescharacteristic
127C4⋊C4×C47C2×D411yescharacteristic
128C22⋊C4×C47—21no—
129C22⋊C4×C47—21no—
130C22⋊C4×C47—21no—
131C42×C47—21no—
132C42×C47—21no—
133C42×C47—21no—
134C42×C47C2×D411yescharacteristic
135C42×C47C2×D411yescharacteristic
136C42×C47#4111yescharacteristic
137C42×C47C2411yescharacteristic
138C22⋊Dic47—42no—
139C4⋊Dic47—22no—
140C4⋊Dic47—22no—
141C4⋊Dic47—22no—
142C4⋊Dic47—22no—
143Dic47⋊C4—42no—
144Dic47⋊C4—42no—
145C4×Dic47—42no—
146C2×C47⋊C8—22no—
147C2×C47⋊C8—22no—
148C2×C47⋊C8—44no—
149Dic188—42no—
150Dic188—42no—
151C23×C47—21no—
152C23×C47#12111yescharacteristic
153Q8×C47—22no—
154D4×C47—22no—
155D4×C47—42no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—41no—
162C2×C4×C47—42no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—42no—
166C2×C4×C47#12111yescharacteristic
167C2×C4×C47#12111yescharacteristic
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47#13714yes—
170C2×C4×C47C22×D411yescharacteristic
171C2×C4×C47C22×D411yescharacteristic
172C2×C4×C47C2×#4111yescharacteristic
173C2×Dic47—42no—
174C2×Dic47—42no—
175Dic94—42no—
176Dic94—42no—
177Dic94—82no—
178C8×C47—21no—
179C8×C47—21no—
180C8×C47—21no—
181C8×C47—41no—
182C8×C47—41no—
183C8×C47#13512yes—
184C47⋊C8—42no—
185C47⋊C8—42no—
186#19158—471nomaximal, Sylow
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—41no—
190C22×C47—41no—
191C22×C47—41no—
192C22×C47#47012yes—
193C22×C47C2×#12111yescharacteristic
194C4×C47—21no—
195C4×C47—21no—
196C4×C47—21no—
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—41no—
200C4×C47#47012yes—
201C4×C47C2×#13511yescharacteristic
202C4×C47C2×#13712yes—
203C4×C47#57511yescharacteristic
204Dic47—82no—
205Dic47—82no—
206#3170—471no—
207#3076—472no—
208#2965D4711yescharacteristic
209#1704—472no—
210#1703—472no—
211#1696—471no—
212#1695—471no—
213#1690—471no—
214#1689—471no—
215C2×#334—471no—
216#1542—472no—
217C2×C47—22no—
218C2×C47—81no—
219C2×C47#303511yescharacteristic
220C2×C47C2×#47011yescharacteristic
221C2×C47#306711yescharacteristic
222#533—472no—
223C4×#41C2×D4711yescharacteristic
224#500—471no—
225#494—471no—
226#487—472no—
227#479—942no—
228#478—942no—
229#477—472no—
230#476—472no—
231#475—472no—
232#474—472no—
233#467—941no—
234#467—941no—
235#446C2×D4711yescharacteristic
236#446C2×D4711yescharacteristic
237C4×Q16—941no—
238C4×Q16—941no—
239#436C2×D4711yescharacteristic
240#436C2×D4711yescharacteristic
241#435—21no—
242#435—21no—
243#431—21no—
244#431C2×D4711yescharacteristic
245#427—942no—
246C2×C4⋊C8—472no—
247#416—942no—
248C4×M16—21no—
249C2×C4×C8C2×D4711yescharacteristic
250#334—471no—
251#334—471no—
252#334—472no—
253#333—472no—
254#333—472no—
255#332—472no—
256#332—472no—
257C47#1915811yescharacteristic, complemented, Sylow
258C2×#41D18811yescharacteristic
259C2×Q16—941no—
260C2×Q16—941no—
261C2×M16—21no—
262C2×M16—21no—
263C22×C8—942no—
264C22×C8C22⋊D4711yescharacteristic
265C22×C8C22⋊D4711yescharacteristic
266C4⋊Q8—472no—
267C4⋊Q8—472no—
268#125—942no—
269C22⋊Q8—942no—
270C22⋊Q8—942no—
271C4×Q8—942no—
272C4×Q8—942no—
273C4×Q8D18811yescharacteristic
274C4×Q8C22×D4711yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4D18811yescharacteristic
278C4×D4C22×D4711yescharacteristic
279#117—21no—
280#117—942no—
281#117D18811yescharacteristic
282C2×C42—21no—
283C2×C42C22×D4711yescharacteristic
284#107—941no—
285#107—941no—
286#107—942no—
287#107—942no—
288#106—942no—
289#106—942no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—472no—
293C4⋊C8—472no—
294C4⋊C8—942no—
295C4⋊C8C22×D4711yescharacteristic
296C4⋊C8C22×D4711yescharacteristic
297#103—942no—
298#103—942no—
299#103—942no—
300#103—942no—
301#102—942no—
302#102—942no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306#97—21no—
307#97—21no—
308C4×C8—21no—
309C4×C8C22⋊D4711yescharacteristic
310C4×C8C22⋊D4711yescharacteristic
311C4×C8C22×D4711yescharacteristic
312C4×C8C22×D4711yescharacteristic
313#41—24no—
314C2×Q8—942no—
315C2×Q8—942no—
316C2×Q8C2×D18811yescharacteristic
317C2×D4—21no—
318C2×D4C2×D18811yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4C2×D18811yescharacteristic
323C22×C4#1062535011yescharacteristic
324C22×C4C2×C22⋊D4711yescharacteristic
325Q16—1882no—
326Q16—1882no—
327M16—42no—
328M16—42no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—942no—
334C2×C8—942no—
335C2×C8—1884no—
336C2×C8C2×C22⋊D4711yescharacteristic
337C2×C8C2×C22⋊D4711yescharacteristic
338C2×C8C2×C22⋊D4711yescharacteristic
339C2×C8C2×C22⋊D4711yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—942no—
343C4⋊C4—942no—
344C4⋊C4—942no—
345C4⋊C4—942no—
346C4⋊C4—1882no—
347C4⋊C4—1882no—
348C4⋊C4C2×D18811yescharacteristic
349C4⋊C4C2×D18811yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—1882no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—1882no—
358C42C2×D18811yescharacteristic
359C42#1062535011yescharacteristic
360C42C2×C22⋊D4711yescharacteristic
361C42C23×D4711yescharacteristic
362C23—21no—
363C23#1309399811yescharacteristic
364Q8—22no—
365Q8—3762no—
366Q8—1882no—
367Q8—1882no—
368D4—22no—
369D4—42no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—1882no—
381C2×C4—1882no—
382C2×C4#1309399811yescharacteristic
383C2×C4#1309399811yescharacteristic
384C2×C4#1309399811yescharacteristic
385C2×C4#1309405714yes—
386C2×C4C22×C22⋊D4711yescharacteristic
387C2×C4C22×D18811yescharacteristic
388C2×C4C2×#1062535011yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—1882no—
395C8—1882no—
396C8#1309404412yes—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#1568938112yes—
403C22C2×#1309399811yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—3762no—
411C4—3762no—
412C4#1568938112yes—
413C4C2×#1309404411yescharacteristic
414C4C2×#1309405712yes—
415C4#1568992111yescharacteristic
416C2—22no—
417C2—81no—
418C2#2025284311yescharacteristic
419C2C2×#1568938111yescharacteristic
420C2#2025310811yescharacteristic
subgroupG/H
centerC2×C4#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2965C2
socleC22×C47C2×#121
prime radicalC2×#41×C47D4
hypercenter#2965D47
hypocenterC47#19158
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #20253974C2semidirect
3. #20242970C2semidirect
4. #20242963C2semidirect
5. #20242916C2semidirect
6. #20242911C2semidirect
7. #20242882C2semidirect
8. #20242877C2semidirect
9. C2×#15688874C2semidirect
10. #20242043C2semidirect
257. C47#19158semidirect

Maximal subgroups

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

subgroupindex
#202539742
#202531762
#202429702
#202429632
#202429162
#202429112
#202428822
#202428772
C2×#156888742
#202420432
C47×#29652
#1915847