Finite Group Databasebeta

← #26200255

Subgroups — #26200255

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253197C211yesmaximal, characteristic
2#20253197C211yesmaximal, characteristic
3C8×#10625350C211yesmaximal, characteristic
4#20242843C214yesmaximal, complemented
5#20242836C214yesmaximal, complemented
6#20242147C211yesmaximal, characteristic, complemented
7#20242073C211yesmaximal, characteristic, complemented
8#20242069C211yesmaximal, characteristic, complemented
9C2×C47×#487C211yesmaximal, characteristic, complemented
10C4×#10625350C2211yescharacteristic
11#15689415—21no—
12#15689415C2212yes—
13#15689410—21no—
14#15689410C2212yes—
15#15689405—21no—
16#15689405C2212yes—
17#15689400—21no—
18#15689400C2212yes—
19C8×Dic94C2212yes—
20C8×D188C2212yes—
21C8×C22⋊D47—22no—
22#15689176C2211yescharacteristic
23#15689174—21no—
24#15689163—21no—
25#15689163—21no—
26#15689147C2211yescharacteristic
27#15689147C2211yescharacteristic
28#15689122—21no—
29#15689122—21no—
30C4×C8×D47—21no—
31#15688876C2214yes—
32#15688872C2214yes—
33#15688869C2214yes—
34C2×C4⋊Q8×C47C2211yescharacteristic
35C2×C4⋊Q8×C47C2211yescharacteristic
36C47×#487—21no—
37C47×#487—21no—
38C47×#487C2214yes—
39C2×#103×C47—22no—
40C2×C4×C8×C47C2211yescharacteristic
41C2×#10625350D411yescharacteristic
42C2×#10625329—21no—
43C2×#10625329—21no—
44C2×C8×D47—22no—
45C4×Dic94D411yescharacteristic
46C4×Dic94C2311yescharacteristic
47C4×D188D411yescharacteristic
48C4×D188C2311yescharacteristic
49C4×C22⋊D47—21no—
50C4×C22⋊D47—21no—
51#13093983D411yescharacteristic
52#13093981—21no—
53C42×D47—21no—
54#13093957—22no—
55#13093957—22no—
56#13093955—22no—
57#13093955—22no—
58C4⋊(C47⋊C8)—21no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)C2312yes—
61C4⋊(C47⋊C8)C2312yes—
62C4⋊(C47⋊C8)C2312yes—
63Dic47⋊C8—22no—
64#13093947—22no—
65#13093947—22no—
66#13093944—22no—
67#13093944—22no—
68C22⋊(C47⋊C8)—21no—
69(C2×D47)⋊C8—22no—
70C4×C47⋊C8—21no—
71C8×Dic47—22no—
72C22×Q8×C47—21no—
73C22×Q8×C47—21no—
74C22×C8×C47D412yes—
75C4⋊Q8×C47—21no—
76C4⋊Q8×C47—21no—
77C4⋊Q8×C47—21no—
78C4⋊Q8×C47—21no—
79C4⋊Q8×C47—21no—
80C4⋊Q8×C47—21no—
81C4⋊Q8×C47C2312yes—
82C4⋊Q8×C47C2312yes—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47—21no—
87C2×C42×C47C2311yescharacteristic
88#103×C47—24no—
89#103×C47—24no—
90C4×C8×C47D411yescharacteristic
91C4×C8×C47D411yescharacteristic
92C4×C8×C47C2312yes—
93C2×Q8×C47—22no—
94C2×Q8×C47—22no—
95C2×Q8×C47—44no—
96C2×Q8×C47—41no—
97C2×Q8×C47—41no—
98C2×Q8×C47—82no—
99C22×C4×C47—41no—
100C22×C4×C47—41no—
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47#4111yescharacteristic
104C2×C22⋊D47—21no—
105#10625350—22no—
106#10625350Q1614yes—
107C2×D188C2×D411yescharacteristic
108C2×C4×D47—21no—
109C2×C4×D47—21no—
110C2×C4×D47—21no—
111C2×Dic94C2×D411yescharacteristic
112C2×C8×C47—42no—
113C2×C8×C47C2×D412yes—
114C2×C8×C47C2×D412yes—
115C4⋊C4×C47—22no—
116C4⋊C4×C47—22no—
117C4⋊C4×C47—22no—
118C4⋊C4×C47—22no—
119C4⋊C4×C47—41no—
120C4⋊C4×C47—41no—
121C4⋊C4×C47—41no—
122C4⋊C4×C47—41no—
123C42×C47C2×D411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47#4111yescharacteristic
126C42×C47C2411yescharacteristic
127C22⋊Dic47—21no—
128(C2×D47)⋊C4—21no—
129(C2×D47)⋊C4—21no—
130C4⋊Dic47C2×D411yescharacteristic
131C4⋊Dic47C2×D411yescharacteristic
132Dic47⋊C4—21no—
133Dic47⋊C4—21no—
134C4×Dic47—21no—
135C4×Dic47—21no—
136C4×Dic47—21no—
137#10625329—42no—
138#10625329—42no—
139C2×C47⋊C8—22no—
140C2×C47⋊C8—22no—
141C2×C47⋊C8—22no—
142C8×D47—44no—
143C23×C47#12111yescharacteristic
144C22×D47—21no—
145Q8×C47—44no—
146Q8×C47—82no—
147Q8×C47—82no—
148Q8×C47—82no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—82no—
154C2×C4×C47—82no—
155C2×C4×C47#12111yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47#12111yescharacteristic
158C2×C4×C47C2×Q1612yes—
159C2×C4×C47#13512yes—
160C2×C4×C47C22×D411yescharacteristic
161C2×C4×C47C22×D411yescharacteristic
162C2×C4×C47C2×#4111yescharacteristic
163C22⋊D47—24no—
164C2×Dic47—21no—
165C2×Dic47—21no—
166C2×Dic47—21no—
167D188—21no—
168D188C2×Q1612yes—
169C4×D47—24no—
170C4×D47—42no—
171C4×D47—42no—
172Dic94—21no—
173Dic94C2×Q1612yes—
174C8×C47—22no—
175C8×C47—22no—
176C47⋊C8—42no—
177C47⋊C8—42no—
178C47⋊C8—42no—
179#19277—471nomaximal, Sylow
180C22×C47—21no—
181C22×C47—21no—
182C22×C47#46712yes—
183C22×C47C2×#12111yescharacteristic
184C2×D47—22no—
185C2×D47—41no—
186C2×D47—41no—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—82no—
190C4×C47—82no—
191C4×C47#46712yes—
192C4×C47#47112yes—
193C4×C47C22×Q1611yescharacteristic
194C4×C47C2×#13511yescharacteristic
195C4×C47#57512yes—
196Dic47—22no—
197Dic47—41no—
198Dic47—41no—
199C2×#487D4711yescharacteristic, complemented
200#3079—471no—
201#3079—471no—
202C8×#41—471no—
203#1684—474no—
204#1683—474no—
205#1567—471no—
206#1549—471no—
207#1548—471no—
208C2×C47—22no—
209C2×C47#304011yescharacteristic
210C2×C47C2×#46711yescharacteristic
211C2×C47C2×#47111yescharacteristic
212D47—42no—
213C2×C4⋊Q8C2×D4711yescharacteristic
214C2×C4⋊Q8C2×D4711yescharacteristic
215C4×#41—471no—
216#487—21no—
217#487—21no—
218#487C2×D4714yes—
219#477—472no—
220#477—941no—
221#476—472no—
222#476—941no—
223#475—472no—
224#475—941no—
225#474—472no—
226#474—941no—
227C8×Q8—472no—
228C8×D4—472no—
229C8×D4—942no—
230#431—471no—
231#431—941no—
232#428—941no—
233#428—941no—
234#423—471no—
235#423—471no—
236#417—941no—
237#417—941no—
238C2×#103—22no—
239C2×C4×C8—941no—
240C2×C4×C8C2×D4711yescharacteristic
241#335—474no—
242#333—474no—
243#332—474no—
244C47#1927711yescharacteristic, complemented, Sylow
245C2×#41—471no—
246C22×Q8—21no—
247C22×Q8—21no—
248C2×M16—941no—
249C2×M16—941no—
250C22×C8—942no—
251C22×C8C22⋊D4712yes—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C22×D4712yes—
259C4⋊Q8C22×D4712yes—
260C4×Q8—471no—
261C4×Q8—471no—
262C4×D4—471no—
263C4×D4—471no—
264C4×D4—941no—
265C4×D4—941no—
266#117—471no—
267#117—941no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C42—941no—
273C2×C42C22×D4711yescharacteristic
274#107—942no—
275#107—942no—
276#106—942no—
277#106—942no—
278C4⋊C8—472no—
279C4⋊C8—472no—
280C4⋊C8—472no—
281C4⋊C8—942no—
282C4⋊C8—941no—
283C4⋊C8—941no—
284#103—24no—
285#103—24no—
286#103—942no—
287#103—942no—
288#102—942no—
289#102—942no—
290C22⋊C8—941no—
291C22⋊C8—942no—
292C4×C8—942no—
293C4×C8—941no—
294C4×C8C22⋊D4711yescharacteristic
295C4×C8C22⋊D4711yescharacteristic
296C4×C8C22×D4712yes—
297#41—474no—
298#41—942no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—82no—
305C2×Q8—471no—
306C2×D4—471no—
307C2×D4—941no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—941no—
311C22×C4—941no—
312C22×C4—941no—
313C22×C4D4×D4711yescharacteristic
314C22×C4#1062535211yescharacteristic
315C22×C4C2×C22⋊D4711yescharacteristic
316M16—1882no—
317M16—1882no—
318C2×C8—42no—
319C2×C8—942no—
320C2×C8—942no—
321C2×C8—942no—
322C2×C8—1884no—
323C2×C8C2×C22⋊D4712yes—
324C2×C8C2×C22⋊D4712yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—471no—
334C4⋊C4—471no—
335C4⋊C4—941no—
336C4⋊C4—941no—
337C22⋊C4—941no—
338C22⋊C4—941no—
339C22⋊C4—941no—
340C42—941no—
341C42—941no—
342C42—941no—
343C42D4×D4711yescharacteristic
344C42#1062535211yescharacteristic
345C42C2×C22⋊D4711yescharacteristic
346C42C23×D4711yescharacteristic
347C23—941no—
348C23#1309399511yescharacteristic
349Q8—44no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353Q8—472no—
354Q8—941no—
355D4—472no—
356D4—944no—
357D4—941no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—941no—
365C2×C4—944no—
366C2×C4—941no—
367C2×C4—941no—
368C2×C4—1882no—
369C2×C4—1882no—
370C2×C4#1309399511yescharacteristic
371C2×C4#1309399511yescharacteristic
372C2×C4#1309399511yescharacteristic
373C2×C4Q16×D4712yes—
374C2×C4#1309404712yes—
375C2×C4C2×D4×D4711yescharacteristic
376C2×C4C22×C22⋊D4711yescharacteristic
377C2×C4C2×#1062535211yescharacteristic
378C8—22no—
379C8—22no—
380C8—1882no—
381C8—1882no—
382C8—1882no—
383C22—21no—
384C22—21no—
385C22—942no—
386C22—1881no—
387C22—1881no—
388C22#1568936112yes—
389C22C2×#1309399511yescharacteristic
390C4—21no—
391C4—21no—
392C4—82no—
393C4—82no—
394C4—942no—
395C4—1881no—
396C4—1881no—
397C4#1568936112yes—
398C4#1568938312yes—
399C4C2×Q16×D4711yescharacteristic
400C4C2×#1309404711yescharacteristic
401C4#1568992112yes—
402C2—22no—
403C2—1882no—
404C2#2025287811yescharacteristic
405C2C2×#1568936111yescharacteristic
406C2C2×#1568938311yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#487C2
socleC22×C47C2×#121
prime radicalC2×#10625350D4
hypercenterC2×#487D47
hypocenterC47#19277
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #20242843C2semidirect
5. #20242836C2semidirect
6. #20242147C2semidirect
7. #20242073C2semidirect
8. #20242069C2semidirect
9. C2×C47×#487C2semidirect
199. C2×#487D47semidirect
244. C47#19277semidirect

Maximal subgroups

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

subgroupindex
#202531972
C8×#106253502
#202428432
#202428362
#202421472
#202420732
#202420692
C2×C47×#4872
#1927747