Finite Group Databasebeta

← #37584074

Subgroups — #37584074

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×#20259343C211yesmaximal, characteristic, complemented
3#27133321C211yesmaximal, characteristic, complemented
4#27133316C211yesmaximal, characteristic, complemented
5#27132906C211yesmaximal, characteristic, complemented
6#27132901C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132870C211yesmaximal, characteristic, complemented
9#27132824C211yesmaximal, characteristic, complemented
10#27132822C211yesmaximal, characteristic, complemented
11#27132817C211yesmaximal, characteristic, complemented
12#27132807C211yesmaximal, characteristic, complemented
13#27132765C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#497×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690288C2211yescharacteristic
18C2×#15690286C2211yescharacteristic
19C2×Q8×D191C2211yescharacteristic
20#20259354—24no—
21#20259343—24no—
22C2×#15690277C2211yescharacteristic
23C2×#15690277C2211yescharacteristic
24C2×#15690276C2211yescharacteristic
25C2×#15690274C2211yescharacteristic
26C2×#15690272C2211yescharacteristic
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259308—21no—
30#20259307—21no—
31#20259307—21no—
32#20259299—21no—
33#20259298C2211yescharacteristic
34#20259298C2211yescharacteristic
35#20259298C2211yescharacteristic
36#20259297—21no—
37#20259295—21no—
38#20259293C2211yescharacteristic
39C4×Dic382C2211yescharacteristic
40C4×D764C2211yescharacteristic
41C4×D764C2211yescharacteristic
42(C22⋊D191)⋊C4—21no—
43C4⋊C4×D191—21no—
44#20259256C2211yescharacteristic
45C4⋊(C191⋊C8)C2211yescharacteristic
46#20259247C2211yescharacteristic
47#20259246C2211yescharacteristic
48#20259246C2211yescharacteristic
49#20259246C2211yescharacteristic
50#20259244C2211yescharacteristic
51#20259243C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53(C2×D191)⋊C8C2211yescharacteristic
54C2×Q16×C191C2211yescharacteristic
55C2×Q16×C191C2211yescharacteristic
56C2×SD16×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic
58C4⋊Q8×C191C2211yescharacteristic
59#124×C191C2211yescharacteristic
60#97×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—22no—
63#15690288—41no—
64#15690288—41no—
65#15690286—22no—
66#15690286—41no—
67C2×Dic382—21no—
68C2×Dic382C2311yescharacteristic
69Q8×D191—22no—
70Q8×D191—41no—
71C2×D764—21no—
72C2×D764—21no—
73C2×D764D411yescharacteristic
74C2×D764C2311yescharacteristic
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C22×Dic191—21no—
78C2×C4×D191—21no—
79C2×C4×D191—21no—
80C2×C4×D191—21no—
81C2×C4×D191—21no—
82C2×C4×D191C2311yescharacteristic
83C2×C4×D191C2311yescharacteristic
84#15690277—22no—
85#15690277—22no—
86#15690276D414yescomplemented
87#15690274—22no—
88#15690272—22no—
89#15690270—22no—
90C2×C191⋊C8C2311yescharacteristic
91C2×C191⋊C8C2311yescharacteristic
92C8×D191—22no—
93C4⋊Dic191D411yescharacteristic
94C4⋊Dic191C2311yescharacteristic
95C4⋊Dic191C2311yescharacteristic
96Dic191⋊C4—21no—
97Dic191⋊C4—21no—
98Dic191⋊C4—21no—
99Dic191⋊C4—21no—
100C22⋊Dic191—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4D411yescharacteristic
106(C2×D191)⋊C4D411yescharacteristic
107C4×Dic191—21no—
108C2×Q8×C191C2311yescharacteristic
109C2×Q8×C191C2311yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191C2311yescharacteristic
112Q16×C191—22no—
113Q16×C191—41no—
114SD16×C191—22no—
115SD16×C191—41no—
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C4⋊C4×C191—21no—
119C4⋊C4×C191—21no—
120C22⋊C4×C191—21no—
121C22⋊C4×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191C2×D411yescharacteristic
125C22×D191C2×D411yescharacteristic
126Dic382—21no—
127Dic382—42no—
128Dic382C2×D412yes—
129D764—21no—
130D764—41no—
131D764—42no—
132D764—42no—
133D764C2×D412yes—
134C22⋊D191—42no—
135C22⋊D191—42no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—42no—
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C4×D191—21no—
143C4×D191—21no—
144C4×D191—21no—
145C4×D191—21no—
146C4×D191—42no—
147C4×D191—42no—
148C4×D191—42no—
149C4×D191—42no—
150C191⋊C8—21no—
151C191⋊C8—21no—
152C23×C191—21no—
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191—21no—
156Q8×C191—21no—
157Q8×C191—21no—
158Q8×C191—21no—
159D4×C191—21no—
160D4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2×D411yescharacteristic
166C2×C4×C191C2411yescharacteristic
167C8×C191—21no—
168C8×C191C2×D412yes—
169C22×C191—41no—
170C22×C191—41no—
171C22×C191—41no—
172C22×C191C22×D411yescharacteristic
173C2×D191—21no—
174C2×D191—21no—
175C2×D191—22no—
176C2×D191—21no—
177C2×D191—21no—
178C2×D191—21no—
179C2×D191—21no—
180C2×D191—41no—
181C2×D191—41no—
182C4×C191—21no—
183C4×C191—21no—
184C4×C191—41no—
185C4×C191—41no—
186C4×C191—41no—
187C4×C191C22×D411yescharacteristic
188C4×C191H(5,𝔽2)11yescharacteristic
189Dic191—21no—
190Dic191—21no—
191Dic191—22no—
192Dic191—41no—
193Dic191—41no—
194C2×C191—81no—
195C2×C191D4211yescharacteristic
196C2×C191#57511yescharacteristic
197C2×C191#57811yescharacteristic
198D191—41no—
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#497D19111yescharacteristic, complemented
208#468—1911no—
209#467—1911no—
210#463—1911no—
211#462—1911no—
212#452—1911no—
213#452—1911no—
214#451—1911no—
215#449—1911no—
216#440—1911no—
217#435—1911no—
218C2×#41—1911no—
219C2×#41—1911no—
220C2×#41—1911no—
221C22×Q8—1911no—
222#137—3824no—
223#135—3824no—
224C2×Q16—1911no—
225C2×Q16—1911no—
226C2×Q16C2×D19111yescharacteristic
227C2×Q16C2×D19111yescharacteristic
228C2×SD16—1911no—
229C2×SD16—1911no—
230C2×SD16C2×D19111yescharacteristic
231C2×SD16C2×D19111yescharacteristic
232C2×D8—1911no—
233C2×M16—1911no—
234C22×C8—1911no—
235C4⋊Q8C2×D19111yescharacteristic
236#124—3821no—
237#124C2×D19111yescharacteristic
238#123—3821no—
239#123—3821no—
240C22⋊Q8—3821no—
241C22⋊Q8—1911no—
242C22⋊Q8—1911no—
243C22⋊Q8—1911no—
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#103—1911no—
258#102—1911no—
259#102—1911no—
260C22⋊C8—1911no—
261C22⋊C8—1911no—
262#97C2×D19111yescharacteristic
263#41—3822no—
264#41—3822no—
265#41—3822no—
266#41—7641no—
267#41—7641no—
268#41—7641no—
269C2×Q8—3821no—
270C2×Q8—3822no—
271C2×Q8—1911no—
272C2×Q8—7641no—
273C2×Q8C22×D19111yescharacteristic
274C2×Q8C22×D19111yescharacteristic
275C2×Q8C22×D19111yescharacteristic
276C2×D4—3821no—
277C2×D4—3821no—
278C2×D4—3821no—
279C2×D4—3821no—
280C2×D4—1911no—
281C2×D4—1911no—
282C2×D4C22×D19111yescharacteristic
283C22×C4—3821no—
284C22×C4—3821no—
285C22×C4—3821no—
286C22×C4—3821no—
287C22×C4—3821no—
288C22×C4—1911no—
289C22×C4—1911no—
290Q16—22no—
291Q16—41no—
292Q16—3822no—
293Q16—3822no—
294SD16—22no—
295SD16—41no—
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—21no—
307C4⋊C4—3821no—
308C4⋊C4—3821no—
309C4⋊C4—3821no—
310C4⋊C4—3821no—
311C4⋊C4—1911no—
312C4⋊C4—1911no—
313C4⋊C4—1911no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—3821no—
317C22⋊C4—3821no—
318C22⋊C4—3821no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—1911no—
322C22⋊C4—1911no—
323C42—3821no—
324C42C22×D19111yescharacteristic
325C23—21no—
326C23—3821no—
327C23—1911no—
328C23—1911no—
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—3821no—
336Q8—1912no—
337Q8—7642no—
338D4—21no—
339D4—21no—
340D4—3821no—
341D4—1912no—
342D4—7641no—
343D4—7642no—
344D4—7642no—
345D4—7642no—
346D4—7642no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—3821no—
356C2×C4—3821no—
357C2×C4—1911no—
358C2×C4—1911no—
359C2×C4—7642no—
360C2×C4—7642no—
361C2×C4—7642no—
362C2×C4—7642no—
363C2×C4—7642no—
364C2×C4D4×D19111yescharacteristic
365C2×C4D4×D19111yescharacteristic
366C2×C4C23×D19111yescharacteristic
367C8—21no—
368C8—3821no—
369C8—3821no—
370C8D4×D19112yes—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—3821no—
375C22—3821no—
376C22—3821no—
377C22—3821no—
378C22—3821no—
379C22—3821no—
380C22—3822no—
381C22—7641no—
382C22—7641no—
383C22C2×D4×D19111yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—3821no—
390C4—3821no—
391C4—3822no—
392C4—7641no—
393C4—7641no—
394C4C2×D4×D19111yescharacteristic
395C4#2025936711yescharacteristic
396C2—81no—
397C2—7641no—
398C2—7641no—
399C2—7642no—
400C2#2713329811yescharacteristic
401C2#2713346211yescharacteristic
402C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#497×C191C2
socleC22×C191C22×D4
hypercenter#497D191
hypocenterC191#3269
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#20259354C2semidirect
2. C2×#20259343C2semidirect
3. #27133321C2semidirect
4. #27133316C2semidirect
5. #27132906C2semidirect
6. #27132901C2semidirect
7. #27132875C2semidirect
8. #27132870C2semidirect
9. #27132824C2semidirect
10. #27132822C2semidirect
11. #27132817C2semidirect
12. #27132807C2semidirect
13. #27132765C2semidirect
14. #27132736C2semidirect
15. #497×C191C2semidirect
86. #15690276D4semidirect
201. C191#3269semidirect
207. #497D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593542
C2×#202593432
#271333212
#271333162
#271329062
#271329012
#271328752
#271328702
#271328242
#271328222
#271328172
#271328072
#271327652
#271327362
#497×C1912
#3269191