Finite Group Databasebeta

← #37584068

Subgroups — #37584068

Order 24448.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259354C211yesmaximal, characteristic, complemented
2C2×#20259342C211yesmaximal, characteristic, complemented
3#27133320C211yesmaximal, characteristic, complemented
4#27133315C211yesmaximal, characteristic, complemented
5#27133054C211yesmaximal, characteristic, complemented
6#27132908C211yesmaximal, characteristic, complemented
7#27132902C211yesmaximal, characteristic, complemented
8#27132877C211yesmaximal, characteristic, complemented
9#27132871C211yesmaximal, characteristic, complemented
10#27132827C211yesmaximal, characteristic, complemented
11#27132820C211yesmaximal, characteristic, complemented
12#27132807C211yesmaximal, characteristic, complemented
13#27132766C211yesmaximal, characteristic, complemented
14#27132731C211yesmaximal, characteristic, complemented
15#452×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690287C2211yescharacteristic
18C2×Q8×D191C2211yescharacteristic
19#20259354—24no—
20#20259342—24no—
21C2×Dic764C2211yescharacteristic
22C2×#15690277C2211yescharacteristic
23C2×#15690277C2211yescharacteristic
24C2×#15690276C2211yescharacteristic
25C2×#15690275C2211yescharacteristic
26C2×#15690274C2211yescharacteristic
27C2×#15690272C2211yescharacteristic
28C2×#15690270C2211yescharacteristic
29C22×C191⋊C8C2211yescharacteristic
30#20259320C2211yescharacteristic
31#20259309—21no—
32#20259309C2211yescharacteristic
33#20259307—21no—
34#20259304—21no—
35#20259302C2211yescharacteristic
36#20259301—21no—
37#20259301C2211yescharacteristic
38#20259295—21no—
39#20259293—21no—
40#20259293C2211yescharacteristic
41Q8×Dic191C2211yescharacteristic
42#20259284C2211yescharacteristic
43C4×C22⋊D191—21no—
44C4×C22⋊D191C2211yescharacteristic
45C4⋊C4×D191—21no—
46#20259255C2211yescharacteristic
47Dic191⋊C8C2211yescharacteristic
48#20259247C2211yescharacteristic
49#20259246C2211yescharacteristic
50#20259245C2211yescharacteristic
51#20259244C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53#20259232C2211yescharacteristic
54C2×#41×C191C2211yescharacteristic
55C2×Q16×C191C2211yescharacteristic
56C2×SD16×C191C2211yescharacteristic
57C22⋊Q8×C191C2211yescharacteristic
58#103×C191C2211yescharacteristic
59#102×C191C2211yescharacteristic
60C22⋊C8×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—41no—
63#15690287—22no—
64#15690287—41no—
65C2×Dic382—21no—
66C2×Dic382C2311yescharacteristic
67C2×Dic382C2311yescharacteristic
68Q8×D191—22no—
69Q8×D191—41no—
70C2×D764—21no—
71C2×D764C2311yescharacteristic
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191D411yescharacteristic
75C2×C4×D191—21no—
76C2×C4×D191—21no—
77C2×C4×D191—21no—
78C2×C4×D191—21no—
79C2×C4×D191C2311yescharacteristic
80Dic764—41no—
81#15690277—22no—
82#15690277—22no—
83#15690276—22no—
84#15690275—22no—
85#15690274D414yescomplemented
86#15690272—22no—
87#15690270—22no—
88C2×C191⋊C8—22no—
89C2×C191⋊C8C2311yescharacteristic
90C2×C191⋊C8C2311yescharacteristic
91C4⋊Dic191—21no—
92C4⋊Dic191—21no—
93C4⋊Dic191C2311yescharacteristic
94Dic191⋊C4—21no—
95Dic191⋊C4—21no—
96Dic191⋊C4—21no—
97Dic191⋊C4D411yescharacteristic
98C22⋊Dic191—21no—
99C22⋊Dic191—21no—
100C22⋊Dic191—21no—
101C22⋊Dic191—21no—
102C22⋊Dic191D411yescharacteristic
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4D411yescharacteristic
106C4×Dic191—21no—
107C4×Dic191C2311yescharacteristic
108#41×C191—22no—
109#41×C191—41no—
110C2×Q8×C191C2311yescharacteristic
111C2×Q8×C191C2311yescharacteristic
112C2×D4×C191—21no—
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116Q16×C191—22no—
117SD16×C191—41no—
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
120C4⋊C4×C191—21no—
121C4⋊C4×C191C2311yescharacteristic
122C22⋊C4×C191—21no—
123C22×D191—21no—
124C22×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128Dic382—21no—
129Dic382—42no—
130D764—21no—
131D764—42no—
132D764C2×D412yes—
133C22⋊D191—41no—
134C22⋊D191—42no—
135C22⋊D191—42no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191C2×D411yescharacteristic
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C4×D191—21no—
143C4×D191—21no—
144C4×D191—42no—
145C4×D191—42no—
146C4×D191—42no—
147C4×D191—42no—
148C191⋊C8—21no—
149C191⋊C8C2×D412yes—
150C23×C191—21no—
151C23×C191C2×D411yescharacteristic
152Q8×C191—21no—
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191C2×D412yes—
156D4×C191—21no—
157D4×C191—21no—
158D4×C191—42no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—42no—
165C2×C4×C191C2×D411yescharacteristic
166C2×C4×C191C2411yescharacteristic
167C8×C191—21no—
168C8×C191—21no—
169C22×C191—21no—
170C22×C191—21no—
171C22×C191—21no—
172C22×C191—41no—
173C22×C191—41no—
174C22×C191—41no—
175C22×C191C22×D411yescharacteristic
176C2×D191—21no—
177C2×D191—22no—
178C2×D191—21no—
179C2×D191—21no—
180C2×D191—41no—
181C2×D191—41no—
182C4×C191—21no—
183C4×C191—22no—
184C4×C191—41no—
185C4×C191—41no—
186C4×C191C22×D411yescharacteristic
187C4×C191H(5,𝔽2)11yescharacteristic
188Dic191—21no—
189Dic191—21no—
190Dic191—21no—
191Dic191—41no—
192Dic191—41no—
193Dic191—41no—
194C2×C191—41no—
195C2×C191—81no—
196C2×C191D4211yescharacteristic
197C2×C191#57511yescharacteristic
198C2×C191#57811yescharacteristic
199D191—41no—
200D191—42no—
201C191#326911yescharacteristic, complemented, Sylow
202#3269—1911nomaximal, Sylow
203C2×#137—1911no—
204C2×#135—1911no—
205#548—1911no—
206#547—1911no—
207#497—1911no—
208#468—1911no—
209#467—1911no—
210#463—1911no—
211#462—1911no—
212#452—1911no—
213#452D19111yescharacteristic, complemented
214#451—1911no—
215#449—1911no—
216#440—1911no—
217#435—1911no—
218C2×#41—1911no—
219C2×#41—1911no—
220C2×#41C2×D19111yescharacteristic
221C22×Q8—1911no—
222#137—3824no—
223#135—3824no—
224C2×Q16—1911no—
225C2×Q16—1911no—
226C2×Q16—1911no—
227C2×Q16C2×D19111yescharacteristic
228C2×SD16—1911no—
229C2×SD16—1911no—
230C2×SD16—1911no—
231C2×SD16C2×D19111yescharacteristic
232C2×D8—1911no—
233C2×M16—1911no—
234C22×C8—1911no—
235C4⋊Q8—1911no—
236#124—3821no—
237#124—1911no—
238#123—3821no—
239#123—3821no—
240C22⋊Q8—3821no—
241C22⋊Q8—1911no—
242C22⋊Q8—1911no—
243C22⋊Q8C2×D19111yescharacteristic
244#121—3821no—
245#121—3821no—
246#121—1911no—
247C4×Q8—1911no—
248C4×D4—3821no—
249C4×D4—1911no—
250C4×D4—1911no—
251C2×C4⋊C4—3821no—
252#107—1911no—
253C4⋊C8—1911no—
254#103—1911no—
255#103—1911no—
256#103—1911no—
257#103C2×D19111yescharacteristic
258#102—1911no—
259#102C2×D19111yescharacteristic
260C22⋊C8—1911no—
261C22⋊C8C2×D19111yescharacteristic
262#97—1911no—
263#41—22no—
264#41—41no—
265#41—3822no—
266#41—3822no—
267#41—7641no—
268#41—7641no—
269C2×Q8—3821no—
270C2×Q8—3822no—
271C2×Q8—1911no—
272C2×Q8—1911no—
273C2×Q8—7641no—
274C2×Q8C22×D19111yescharacteristic
275C2×Q8C22×D19111yescharacteristic
276C2×D4—21no—
277C2×D4—3821no—
278C2×D4—3821no—
279C2×D4—3821no—
280C2×D4—1911no—
281C2×D4—1911no—
282C2×D4C22×D19111yescharacteristic
283C22×C4—21no—
284C22×C4—3821no—
285C22×C4—3821no—
286C22×C4—3821no—
287C22×C4—3821no—
288C22×C4—1911no—
289C22×C4C22×D19111yescharacteristic
290Q16—22no—
291Q16—3822no—
292Q16—3822no—
293Q16—7641no—
294SD16—41no—
295SD16—3822no—
296SD16—3822no—
297SD16—1914no—
298D8—3822no—
299M16—3822no—
300C2×C8—3822no—
301C2×C8—1911no—
302C2×C8—1911no—
303C2×C8C22×D19111yescharacteristic
304C2×C8C22×D19111yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—3821no—
307C4⋊C4—3821no—
308C4⋊C4—3821no—
309C4⋊C4—3821no—
310C4⋊C4—3821no—
311C4⋊C4—1911no—
312C4⋊C4—1911no—
313C4⋊C4C22×D19111yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—3821no—
316C22⋊C4—3821no—
317C22⋊C4—3821no—
318C22⋊C4—3821no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—1911no—
322C22⋊C4—1911no—
323C42—3821no—
324C42—1911no—
325C23—21no—
326C23—3821no—
327C23—1911no—
328C23D4×D19111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—3821no—
333Q8—3821no—
334Q8—3821no—
335Q8—3821no—
336Q8—7642no—
337Q8D4×D19112yes—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—3821no—
342D4—1912no—
343D4—7642no—
344D4—7641no—
345D4—7642no—
346D4—7642no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—3821no—
356C2×C4—3821no—
357C2×C4—3821no—
358C2×C4—1911no—
359C2×C4—1911no—
360C2×C4—1911no—
361C2×C4—7642no—
362C2×C4—7642no—
363C2×C4—7642no—
364C2×C4—7642no—
365C2×C4D4×D19111yescharacteristic
366C2×C4C23×D19111yescharacteristic
367C8—21no—
368C8—21no—
369C8—3821no—
370C8—1912no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3821no—
378C22—3821no—
379C22—3821no—
380C22—3822no—
381C22—7641no—
382C22—7641no—
383C22C2×D4×D19111yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—3821no—
389C4—3821no—
390C4—3821no—
391C4—7641no—
392C4—7641no—
393C4—7641no—
394C4C2×D4×D19111yescharacteristic
395C4#2025936711yescharacteristic
396C2—41no—
397C2—81no—
398C2—7641no—
399C2—7642no—
400C2#2713329511yescharacteristic
401C2#2713346011yescharacteristic
402C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#452×C191C2
socleC22×C191C22×D4
hypercenter#452D191
hypocenterC191#3269
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#20259354C2semidirect
2. C2×#20259342C2semidirect
3. #27133320C2semidirect
4. #27133315C2semidirect
5. #27133054C2semidirect
6. #27132908C2semidirect
7. #27132902C2semidirect
8. #27132877C2semidirect
9. #27132871C2semidirect
10. #27132827C2semidirect
11. #27132820C2semidirect
12. #27132807C2semidirect
13. #27132766C2semidirect
14. #27132731C2semidirect
15. #452×C191C2semidirect
85. #15690274D4semidirect
201. C191#3269semidirect
213. #452D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593542
C2×#202593422
#271333202
#271333152
#271330542
#271329082
#271329022
#271328772
#271328712
#271328272
#271328202
#271328072
#271327662
#271327312
#452×C1912
#3269191