Finite Group Databasebeta

← #26369694

Subgroups — #26369694

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256522C211yesmaximal, characteristic
2#20252305C211yesmaximal, characteristic, complemented
3#20252304C211yesmaximal, characteristic, complemented
4#20245271C214yesmaximal, complemented
5#20245266C214yesmaximal, complemented
6#20245182C211yesmaximal, characteristic, complemented
7C2×#15688931C211yesmaximal, characteristic, complemented
8#20244984C211yesmaximal, characteristic, complemented
9C2×C47×#359C211yesmaximal, characteristic, complemented
10C2×#13094046C2211yescharacteristic
11#15689890—21no—
12#15689889—21no—
13#15689564C412yescomplemented
14#15689563C412yescomplemented
15#15689559C412yescomplemented
16#15689558C412yescomplemented
17#15689553C2214yes—
18C2×#13093964C2211yescharacteristic
19#15689245—22no—
20#15689244—22no—
21#15689240C2214yescomplemented
22#15689239C2214yescomplemented
23C2×#13093960C2211yescharacteristic
24#15689140—21no—
25#15689139—21no—
26#15689078C2211yescharacteristic
27#15688952C2214yescomplemented
28#15688931C2212yescomplemented
29#15688931C2212yescomplemented
30#15688898C2212yescomplemented
31C16×Dic47C2212yescomplemented
32C2×#135×C47C2211yescharacteristic
33C22×C16×C47C2211yescharacteristic
34C2×#108×C47C2211yescharacteristic
35C47×#359C2212yescomplemented
36C47×#359C2212yescomplemented
37C2×#10625350—21no—
38#13094046—21no—
39#13094046—21no—
40#13094046C2×C412yes—
41#13094046C2312yes—
42C2×Dic188C2×C411yescharacteristic
43C2×#10625325—21no—
44C2×D376C2×C411yescharacteristic
45#13094032—24no—
46#13094031—24no—
47#13094031—42no—
48#13094030—42no—
49C2×#10625329—21no—
50C2×#10625329—42no—
51C2×#10625329D411yescharacteristic
52M16×D47—42no—
53#13093983D411yescharacteristic
54#13093970C2×C414yes—
55#13093968C2×C414yes—
56#13093967C2×C414yes—
57#13093965C2×C414yes—
58#13093964C2×C412yes—
59#13093964C2×C412yes—
60C2×C47⋊C16C2312yes—
61#13093960—21no—
62#13093960C2312yes—
63#13093951—24no—
64#13093950—24no—
65#13093934D412yes—
66C8×Dic47C2312yes—
67C2×#41×C47—21no—
68#135×C47—21no—
69#135×C47—21no—
70#135×C47C2×C412yes—
71#135×C47C2312yes—
72C2×Q16×C47C2×C411yescharacteristic
73C2×SD16×C47—21no—
74C2×D8×C47C2×C411yescharacteristic
75C2×M16×C47—21no—
76C22×C8×C47C2311yescharacteristic
77C2×C16×C47—21no—
78C2×C16×C47—21no—
79C2×C16×C47C2311yescharacteristic
80C2×C16×C47C2311yescharacteristic
81#108×C47—21no—
82#108×C47C2312yes—
83#41×C47—22no—
84#41×C47—22no—
85#41×C47—41no—
86#41×C47—41no—
87C2×Q8×C47—21no—
88C2×D4×C47—21no—
89C2×D4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47C2×D411yescharacteristic
92C2×C22⋊D47—41no—
93#10625350—22no—
94#10625350—22no—
95#10625350—41no—
96#10625350—41no—
97C2×D188—21no—
98C2×C4×D47—41no—
99C2×Dic94—21no—
100Q16×C47—21no—
101Q16×C47C22×C412yes—
102SD16×C47—22no—
103SD16×C47—22no—
104D8×C47—21no—
105D8×C47C22×C412yes—
106M16×C47—22no—
107M16×C47—41no—
108C2×C8×C47—41no—
109C2×C8×C47C22×C411yescharacteristic
110C2×C8×C47C22×C411yescharacteristic
111C2×C8×C47C2×D411yescharacteristic
112C2×C8×C47#4111yescharacteristic
113C2×C8×C47#4111yescharacteristic
114C2×C8×C47C2411yescharacteristic
115C22⋊Dic47—21no—
116C4⋊Dic47—21no—
117C4×Dic47C2×D412yes—
118#10625329—22no—
119#10625329—41no—
120#10625329—44no—
121#10625329D812yes—
122#10625329D812yes—
123C2×C47⋊C8—41no—
124C2×C47⋊C8—44no—
125C2×C47⋊C8C2×D412yes—
126Dic188—21no—
127Dic188C22×C412yes—
128D376—21no—
129D376C22×C412yes—
130#10625325—22no—
131#10625325—22no—
132#10625324—44no—
133C8×D47—44no—
134C16×C47—21no—
135C16×C47—21no—
136C47⋊C16C22×C414yes—
137C23×C47—41no—
138C23×C47#13511yescharacteristic
139C22×D47—41no—
140Q8×C47—22no—
141Q8×C47—41no—
142D4×C47—22no—
143D4×C47—42no—
144D4×C47—42no—
145D4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47C4×D411yescharacteristic
150C2×C4×C47C4×D411yescharacteristic
151C2×C4×C47C2×D811yescharacteristic
152C2×C4×C47C2×D811yescharacteristic
153C2×C4×C47#13511yescharacteristic
154C2×C4×C47C22×D411yescharacteristic
155C22⋊D47—42no—
156C22⋊D47—42no—
157C2×Dic47—22no—
158C2×Dic47—41no—
159D188—22no—
160D188—41no—
161C4×D47—42no—
162C4×D47—42no—
163Dic94—22no—
164Dic94—41no—
165C8×C47—42no—
166C8×C47C4×D411yescharacteristic
167C8×C47C4×D411yescharacteristic
168C8×C47C23×C411yescharacteristic
169C8×C47C2×#4111yescharacteristic
170C47⋊C8—42no—
171C47⋊C8C2×D814yes—
172#33464—471nomaximal, Sylow
173C22×C47—41no—
174C22×C47—42no—
175C22×C47—161no—
176C22×C47C4×D811yescharacteristic
177C22×C47C4×D811yescharacteristic
178C22×C47C2×#13511yescharacteristic
179C2×D47—42no—
180C2×D47—161no—
181C4×C47—42no—
182C4×C47C4×D811yescharacteristic
183C4×C47C4×D811yescharacteristic
184C4×C47C2×C4×D411yescharacteristic
185C4×C47C22×D811yescharacteristic
186Dic47—42no—
187Dic47—42no—
188#3414—471no—
189#2957—471no—
190#2957—471no—
191#2179—474no—
192#2178—474no—
193#2161—471no—
194C2×#359—471no—
195C2×#359D4711yescharacteristic, complemented
196#2108—471no—
197C2×C47—21no—
198C2×C47—21no—
199C2×C47—21no—
200C2×C47—82no—
201C2×C47C2×C4×D811yescharacteristic
202D47—82no—
203C2×#135—471no—
204C2×#135C2×D4711yescharacteristic
205#568—941no—
206#568—941no—
207#510—472no—
208#510—472no—
209#509—472no—
210#509—472no—
211#508—474no—
212C2×M32—471no—
213C22×C16C2×D4711yescharacteristic
214#444—942no—
215#444—942no—
216#443—474no—
217#443—474no—
218C2×#108—471no—
219C2×#108C2×D4711yescharacteristic
220#421—941no—
221#421—941no—
222#405—471no—
223#368—474no—
224#359—472no—
225#359—472no—
226#359C2×D4712yescomplemented
227#359C2×D4712yescomplemented
228#346—472no—
229C4×C16—472no—
230C47#3346411yescharacteristic, complemented, Sylow
231C2×#41—21no—
232C2×#41—941no—
233#135—21no—
234#135—21no—
235#135—472no—
236#135—472no—
237#135—941no—
238#135—941no—
239#135C4×D4712yes—
240#135C22×D4712yes—
241C2×Q16—471no—
242C2×Q16C4×D4711yescharacteristic
243C2×SD16—21no—
244C2×SD16—941no—
245C2×D8—471no—
246C2×D8C4×D4711yescharacteristic
247#131—944no—
248#131—944no—
249#131—1882no—
250#131—1882no—
251C2×M16—21no—
252C2×M16—471no—
253C2×M16—941no—
254C2×M16—1882no—
255C2×M16—1882no—
256C22×C8C22×D4711yescharacteristic
257#117—471no—
258Q32—474no—
259SD32—474no—
260SD32—474no—
261D16—474no—
262M32—472no—
263M32—472no—
264C2×C16—21no—
265C2×C16—21no—
266C2×C16—472no—
267C2×C16C22×D4711yescharacteristic
268C2×C16C22×D4711yescharacteristic
269#108—21no—
270#108—472no—
271#108—941no—
272#108C22×D4712yes—
273#104—944no—
274#104—944no—
275#97—472no—
276C4×C8—472no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—942no—
282#41—942no—
283#41—1881no—
284#41—1881no—
285C2×Q8—21no—
286C2×Q8—941no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—941no—
290C2×D4—1881no—
291C22×C4—41no—
292C22×C4—1881no—
293C22×C4D4×D4711yescharacteristic
294Q16—21no—
295Q16—472no—
296Q16—941no—
297Q16C2×C4×D4712yes—
298SD16—22no—
299SD16—22no—
300SD16—942no—
301SD16—942no—
302D8—21no—
303D8—472no—
304D8—941no—
305D8C2×C4×D4712yes—
306M16—22no—
307M16—41no—
308M16—472no—
309M16—472no—
310M16—942no—
311M16—1884no—
312M16—1884no—
313M16—1881no—
314C2×C8—41no—
315C2×C8—472no—
316C2×C8—1884no—
317C2×C8—1884no—
318C2×C8—1881no—
319C2×C8C2×C4×D4711yescharacteristic
320C2×C8C2×C4×D4711yescharacteristic
321C2×C8D4×D4711yescharacteristic
322C2×C8#1062535211yescharacteristic
323C2×C8#1062535211yescharacteristic
324C2×C8C23×D4711yescharacteristic
325C4⋊C4—941no—
326C22⋊C4—941no—
327C42—472no—
328C16—21no—
329C16—21no—
330C16—474no—
331C23—41no—
332C23—1881no—
333C23#1309404511yescharacteristic
334Q8—22no—
335Q8—41no—
336Q8—942no—
337Q8—1881no—
338D4—22no—
339D4—42no—
340D4—42no—
341D4—41no—
342D4—942no—
343D4—1882no—
344D4—1882no—
345D4—1881no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—942no—
350C2×C4—1881no—
351C2×C4—1882no—
352C2×C4—1882no—
353C2×C4(C22⋊D47)⋊C411yescharacteristic
354C2×C4(C22⋊D47)⋊C411yescharacteristic
355C2×C4D8×D4711yescharacteristic
356C2×C4D8×D4711yescharacteristic
357C2×C4#1309404511yescharacteristic
358C2×C4C2×D4×D4711yescharacteristic
359C8—42no—
360C8—474no—
361C8—1882no—
362C8(C22⋊D47)⋊C411yescharacteristic
363C8(C22⋊D47)⋊C411yescharacteristic
364C8C22×C4×D4711yescharacteristic
365C8C2×#1062535211yescharacteristic
366C22—41no—
367C22—42no—
368C22—161no—
369C22—1882no—
370C22—7521no—
371C22#1568920311yescharacteristic
372C22#1568920311yescharacteristic
373C22C2×#1309404511yescharacteristic
374C4—42no—
375C4—1882no—
376C4—1882no—
377C4#1568920311yescharacteristic
378C4#1568920311yescharacteristic
379C4C2×(C22⋊D47)⋊C411yescharacteristic
380C4C2×D8×D4711yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—82no—
385C2—3762no—
386C2C2×#1568920311yescharacteristic
subgroupG/H
centerC4C2×D8×D47
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC2×C47×#359C2
socleC2×C47C2×C4×D8
prime radical#20256522C2
cosocleC2×C8×C47C24
hypercenterC2×#359D47
hypocenterC47#33464
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#359

Complements

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

NH ≅ G/Ntype
2. #20252305C2semidirect
3. #20252304C2semidirect
4. #20245271C2semidirect
5. #20245266C2semidirect
6. #20245182C2semidirect
7. C2×#15688931C2semidirect
8. #20244984C2semidirect
9. C2×C47×#359C2semidirect
13. #15689564C4semidirect
14. #15689563C4semidirect
15. #15689559C4semidirect
16. #15689558C4semidirect
21. #15689240C22semidirect
22. #15689239C22semidirect
27. #15688952C22semidirect
28. #15688931C22semidirect
29. #15688931C22semidirect
30. #15688898C22semidirect
31. C16×Dic47C22semidirect
35. C47×#359C22semidirect
36. C47×#359C22semidirect
195. C2×#359D47semidirect
226. #359C2×D47semidirect
227. #359C2×D47semidirect
230. C47#33464semidirect

Maximal subgroups

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

subgroupindex
#202565222
#202523052
#202523042
#202452712
#202452662
#202451822
C2×#156889312
#202449842
C2×C47×#3592
#3346447