Finite Group Databasebeta

← #37584075

Subgroups — #37584075

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×#20259351C211yesmaximal, characteristic, complemented
2#27133320C211yesmaximal, characteristic, complemented
3#27133312C211yesmaximal, characteristic, complemented
4#27133054C211yesmaximal, characteristic, complemented
5#27132906C211yesmaximal, characteristic, complemented
6#27132904C211yesmaximal, characteristic, complemented
7#27132876C211yesmaximal, characteristic, complemented
8#27132871C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132825C211yesmaximal, characteristic, complemented
11#27132817C211yesmaximal, characteristic, complemented
12#27132805C211yesmaximal, characteristic, complemented
13#27132764C211yesmaximal, characteristic, complemented
14#27132734C211yesmaximal, characteristic, complemented
15C2×#135×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18#20259351—24no—
19C2×Dic764C2211yescharacteristic
20C2×#15690277C2211yescharacteristic
21C2×#15690277C2211yescharacteristic
22C2×#15690275C2211yescharacteristic
23C2×#15690275C2211yescharacteristic
24C2×#15690270C2211yescharacteristic
25#20259320C2211yescharacteristic
26#20259309—21no—
27#20259309C2211yescharacteristic
28#20259305—21no—
29#20259303—21no—
30#20259301—21no—
31#20259301C2211yescharacteristic
32#20259301C2211yescharacteristic
33#20259298C2211yescharacteristic
34#20259294—21no—
35#20259293—21no—
36#20259293C2211yescharacteristic
37Q8×Dic191C2211yescharacteristic
38D4×Dic191C2211yescharacteristic
39C4×C22⋊D191—21no—
40C4×C22⋊D191C2211yescharacteristic
41C2×Dic191⋊C4—21no—
42#20259256C2211yescharacteristic
43Dic191⋊C8C2211yescharacteristic
44#20259247C2211yescharacteristic
45#20259247C2211yescharacteristic
46#20259246C2211yescharacteristic
47#20259246C2211yescharacteristic
48#20259243C2211yescharacteristic
49#20259243C2211yescharacteristic
50C22⋊(C191⋊C8)C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic
52#20259232C2211yescharacteristic
53C2×#41×C191C2211yescharacteristic
54C2×#41×C191C2211yescharacteristic
55#135×C191—24no—
56C2×Q16×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic
58C2×SD16×C191C2211yescharacteristic
59C2×D8×C191C2211yescharacteristic
60C22×C8×C191C2211yescharacteristic
61#15690286—22no—
62#15690286—41no—
63C2×Dic382—21no—
64C2×Dic382C2311yescharacteristic
65C2×Dic382C2311yescharacteristic
66Q8×D191—22no—
67Q8×D191—41no—
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191D411yescharacteristic
71C22×Dic191—21no—
72C22×Dic191—21no—
73C2×C4×D191—21no—
74C2×C4×D191C2311yescharacteristic
75Dic764—22no—
76#15690277—22no—
77#15690277—41no—
78#15690275—22no—
79#15690275—41no—
80#15690270—22no—
81C2×C191⋊C8C2311yescharacteristic
82C2×C191⋊C8C2311yescharacteristic
83C4⋊Dic191—21no—
84C4⋊Dic191C2311yescharacteristic
85C4⋊Dic191C2311yescharacteristic
86Dic191⋊C4—21no—
87Dic191⋊C4—21no—
88Dic191⋊C4—21no—
89Dic191⋊C4—21no—
90Dic191⋊C4—21no—
91Dic191⋊C4D411yescharacteristic
92C22⋊Dic191—21no—
93C22⋊Dic191—21no—
94C22⋊Dic191—21no—
95C22⋊Dic191—21no—
96C22⋊Dic191—21no—
97C22⋊Dic191—21no—
98C22⋊Dic191D411yescharacteristic
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4D411yescharacteristic
101C4×Dic191—21no—
102C4×Dic191C2311yescharacteristic
103#41×C191—22no—
104#41×C191—22no—
105#41×C191—41no—
106#41×C191—41no—
107C2×Q8×C191C2311yescharacteristic
108C2×Q8×C191C2311yescharacteristic
109C2×D4×C191—21no—
110C2×D4×C191—21no—
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C22×C4×C191—21no—
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116Q16×C191—22no—
117SD16×C191—22no—
118SD16×C191D414yescomplemented
119D8×C191—22no—
120C2×C8×C191—22no—
121C2×C8×C191C2311yescharacteristic
122C2×C8×C191C2311yescharacteristic
123C22×D191C2×D411yescharacteristic
124Dic382—21no—
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128Dic382—42no—
129C22⋊D191—41no—
130C22⋊D191—42no—
131C22⋊D191—42no—
132C2×Dic191—21no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191—42no—
137C2×Dic191—42no—
138C2×Dic191C2×D411yescharacteristic
139C2×Dic191C2×D411yescharacteristic
140C2×Dic191C2×D411yescharacteristic
141C4×D191—21no—
142C4×D191—21no—
143C4×D191—42no—
144C191⋊C8—21no—
145C191⋊C8—21no—
146C23×C191—21no—
147C23×C191—21no—
148C23×C191C2×D411yescharacteristic
149Q8×C191—21no—
150Q8×C191—21no—
151Q8×C191—21no—
152Q8×C191C2×D412yes—
153D4×C191—21no—
154D4×C191—21no—
155D4×C191—21no—
156D4×C191—42no—
157D4×C191—42no—
158D4×C191C2×D412yes—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—42no—
164C2×C4×C191—42no—
165C2×C4×C191C2×D411yescharacteristic
166C2×C4×C191C2411yescharacteristic
167C8×C191—21no—
168C8×C191C2×D412yes—
169C22×C191—21no—
170C22×C191—22no—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191—41no—
174C22×C191—41no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191C22×D411yescharacteristic
179C2×D191—21no—
180C2×D191—21no—
181C2×D191—21no—
182C4×C191—21no—
183C4×C191—22no—
184C4×C191—41no—
185C4×C191C22×D411yescharacteristic
186C4×C191H(5,𝔽2)11yescharacteristic
187Dic191—21no—
188Dic191—21no—
189Dic191—21no—
190Dic191—41no—
191Dic191—41no—
192Dic191—41no—
193Dic191—41no—
194C2×C191—41no—
195C2×C191—42no—
196C2×C191—81no—
197C2×C191D4211yescharacteristic
198C2×C191#57511yescharacteristic
199C2×C191#57811yescharacteristic
200D191—41no—
201C191#326911yescharacteristic, complemented, Sylow
202#3269—1911nomaximal, Sylow
203C2×#137—1911no—
204C2×#135D19111yescharacteristic, complemented
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#452—1911no—
214#451—1911no—
215#449—1911no—
216#440—1911no—
217#435—1911no—
218C2×#41—1911no—
219C2×#41C2×D19111yescharacteristic
220C2×#41C2×D19111yescharacteristic
221C22×Q8—1911no—
222#137—3824no—
223#135—24no—
224C2×Q16—1911no—
225C2×Q16—1911no—
226C2×Q16—1911no—
227C2×Q16C2×D19111yescharacteristic
228C2×SD16—1911no—
229C2×SD16—1911no—
230C2×SD16C2×D19111yescharacteristic
231C2×SD16C2×D19111yescharacteristic
232C2×D8C2×D19111yescharacteristic
233C2×M16—1911no—
234C22×C8C2×D19111yescharacteristic
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⋊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#97—1911no—
263#41—22no—
264#41—22no—
265#41—41no—
266#41—41no—
267#41—3822no—
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—21no—
278C2×D4—3821no—
279C2×D4—3821no—
280C2×D4—1911no—
281C2×D4C22×D19111yescharacteristic
282C2×D4C22×D19111yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—3821no—
286C22×C4—3821no—
287C22×C4—3821no—
288C22×C4—1911no—
289C22×C4C22×D19111yescharacteristic
290Q16—22no—
291Q16—3822no—
292Q16—3822no—
293Q16—7641no—
294SD16—22no—
295SD16—3822no—
296SD16—7641no—
297SD16C22⋊D19114yescomplemented
298D8—22no—
299M16—3822no—
300C2×C8—22no—
301C2×C8—1911no—
302C2×C8—1911no—
303C2×C8C22×D19111yescharacteristic
304C2×C8C22×D19111yescharacteristic
305C4⋊C4—3821no—
306C4⋊C4—3821no—
307C4⋊C4—3821no—
308C4⋊C4—3821no—
309C4⋊C4—3821no—
310C4⋊C4—3821no—
311C4⋊C4—1911no—
312C4⋊C4—1911no—
313C4⋊C4—1911no—
314C22⋊C4—3821no—
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—21no—
327C23—1911no—
328C23D4×D19111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—3821no—
333Q8—3821no—
334Q8—3821no—
335Q8—3821no—
336Q8—7642no—
337Q8C2×C22⋊D19112yes—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—7641no—
344D4—7642no—
345D4—7642no—
346D4C2×C22⋊D19112yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—3821no—
356C2×C4—3821no—
357C2×C4—3821no—
358C2×C4—3821no—
359C2×C4—1911no—
360C2×C4—1911no—
361C2×C4—1911no—
362C2×C4—7642no—
363C2×C4—7642no—
364C2×C4—7642no—
365C2×C4D4×D19111yescharacteristic
366C2×C4C23×D19111yescharacteristic
367C8—21no—
368C8—3821no—
369C8—3821no—
370C8C2×C22⋊D19112yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3821no—
381C22—3821no—
382C22—3821no—
383C22C2×D4×D19111yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—3821no—
388C4—3821no—
389C4—3821no—
390C4—7641no—
391C4—7641no—
392C4—7641no—
393C4—7641no—
394C4C22×C22⋊D19111yescharacteristic
395C4#2025936711yescharacteristic
396C2—41no—
397C2—42no—
398C2—81no—
399C2—7641no—
400C2D4×C22⋊D19111yescharacteristic
401C2#2713346011yescharacteristic
402C2#2713347711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#135×C191C2
socleC22×C191C22×D4
hypercenterC2×#135D191
hypocenterC191#3269
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#20259351C2semidirect
2. #27133320C2semidirect
3. #27133312C2semidirect
4. #27133054C2semidirect
5. #27132906C2semidirect
6. #27132904C2semidirect
7. #27132876C2semidirect
8. #27132871C2semidirect
9. #27132825C2semidirect
10. #27132825C2semidirect
11. #27132817C2semidirect
12. #27132805C2semidirect
13. #27132764C2semidirect
14. #27132734C2semidirect
15. C2×#135×C191C2semidirect
118. SD16×C191D4semidirect
201. C191#3269semidirect
204. C2×#135D191semidirect
297. SD16C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593512
#271333202
#271333122
#271330542
#271329062
#271329042
#271328762
#271328712
#271328252
#271328172
#271328052
#271327642
#271327342
C2×#135×C1912
#3269191