Finite Group Databasebeta

← #37584076

Subgroups — #37584076

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×#20259353C211yesmaximal, characteristic, complemented
2C2×#20259342C211yesmaximal, characteristic, complemented
3#27133318C211yesmaximal, characteristic, complemented
4#27133058C211yesmaximal, characteristic, complemented
5#27132907C211yesmaximal, characteristic, complemented
6#27132902C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132873C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132823C211yesmaximal, characteristic, complemented
11#27132819C211yesmaximal, characteristic, complemented
12#27132806C211yesmaximal, characteristic, complemented
13#27132761C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15#548×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×#15690287C2211yescharacteristic
18#20259353—24no—
19#20259342—24no—
20C2×#15690277C2211yescharacteristic
21C2×#15690277C2211yescharacteristic
22C2×#15690277C2211yescharacteristic
23C2×#15690277C2211yescharacteristic
24C2×#15690275C2211yescharacteristic
25C2×#15690275C2211yescharacteristic
26C2×#15690274C2211yescharacteristic
27C2×#15690274C2211yescharacteristic
28C2×#15690272C2211yescharacteristic
29C2×#15690271C2211yescharacteristic
30C22×C191⋊C8C2211yescharacteristic
31#20259322C2211yescharacteristic
32#20259304—21no—
33#20259304—21no—
34#20259301C2211yescharacteristic
35#20259301C2211yescharacteristic
36#20259293—21no—
37C4×D764C2211yescharacteristic
38C4⋊C4×D191—21no—
39#20259255C2211yescharacteristic
40C4⋊(C191⋊C8)C2211yescharacteristic
41#20259246C2211yescharacteristic
42#20259246C2211yescharacteristic
43#20259245C2211yescharacteristic
44#20259245C2211yescharacteristic
45#20259243C2211yescharacteristic
46#20259242C2211yescharacteristic
47C22⋊(C191⋊C8)C2211yescharacteristic
48C22⋊(C191⋊C8)C2211yescharacteristic
49#20259233C2211yescharacteristic
50C2×#41×C191C2211yescharacteristic
51C22×Q8×C191C2211yescharacteristic
52#124×C191—21no—
53#124×C191C2211yescharacteristic
54C22⋊Q8×C191—21no—
55C22⋊Q8×C191C2211yescharacteristic
56#121×C191—21no—
57#121×C191C2211yescharacteristic
58C4×Q8×C191C2211yescharacteristic
59C4×D4×C191—21no—
60C4×D4×C191C2211yescharacteristic
61#15690287—22no—
62#15690287—22no—
63#15690287—41no—
64#15690287—41no—
65C2×Dic382C2311yescharacteristic
66C2×Dic382C2311yescharacteristic
67C2×D764C2311yescharacteristic
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C4×D191—21no—
71C2×C4×D191—21no—
72C2×C4×D191—21no—
73#15690277—22no—
74#15690277—22no—
75#15690277—22no—
76#15690277—41no—
77#15690275—22no—
78#15690275—41no—
79#15690274—22no—
80#15690274D414yescomplemented
81#15690272—22no—
82#15690271—22no—
83C2×C191⋊C8—22no—
84C2×C191⋊C8C2311yescharacteristic
85C2×C191⋊C8C2311yescharacteristic
86C2×C191⋊C8C2311yescharacteristic
87C2×C191⋊C8C2311yescharacteristic
88C4⋊Dic191—21no—
89C4⋊Dic191—21no—
90C4⋊Dic191C2311yescharacteristic
91Dic191⋊C4—21no—
92Dic191⋊C4—21no—
93C22⋊Dic191—21no—
94C22⋊Dic191—21no—
95(C2×D191)⋊C4—21no—
96#41×C191—22no—
97#41×C191—41no—
98C2×Q8×C191—21no—
99C2×Q8×C191—22no—
100C2×Q8×C191—41no—
101C2×Q8×C191C2311yescharacteristic
102C2×Q8×C191C2311yescharacteristic
103C2×D4×C191—21no—
104C2×D4×C191—21no—
105C2×D4×C191D411yescharacteristic
106C2×D4×C191C2311yescharacteristic
107C22×C4×C191—21no—
108C22×C4×C191—21no—
109C22×C4×C191C2311yescharacteristic
110C22×C4×C191C2311yescharacteristic
111C4⋊C4×C191—21no—
112C4⋊C4×C191—21no—
113C4⋊C4×C191D411yescharacteristic
114C4⋊C4×C191C2311yescharacteristic
115C22⋊C4×C191—21no—
116C22⋊C4×C191—21no—
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C22⋊C4×C191D411yescharacteristic
120C22⋊C4×C191D411yescharacteristic
121C42×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124Dic382—21no—
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128D764—21no—
129D764C2×D412yes—
130C22⋊D191—42no—
131C22⋊D191—42no—
132C2×Dic191—21no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C4×D191—42no—
136C4×D191—42no—
137C4×D191—42no—
138C191⋊C8—21no—
139C191⋊C8—21no—
140C191⋊C8—21no—
141C191⋊C8C2×D412yes—
142C23×C191—21no—
143C23×C191C2×D411yescharacteristic
144C23×C191C2×D411yescharacteristic
145Q8×C191—21no—
146Q8×C191—21no—
147Q8×C191—21no—
148Q8×C191—42no—
149Q8×C191C2×D412yes—
150D4×C191—21no—
151D4×C191—21no—
152D4×C191—41no—
153D4×C191—42no—
154D4×C191—42no—
155C2×C4×C191—21no—
156C2×C4×C191—21no—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—42no—
163C2×C4×C191—42no—
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2×D411yescharacteristic
166C2×C4×C191C2×D411yescharacteristic
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2411yescharacteristic
169C22×C191—21no—
170C22×C191—21no—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191—21no—
174C22×C191—21no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191C22×D411yescharacteristic
179C2×D191—22no—
180C2×D191—41no—
181C2×D191—41no—
182C4×C191—21no—
183C4×C191—21no—
184C4×C191—22no—
185C4×C191—21no—
186C4×C191—21no—
187C4×C191—41no—
188C4×C191—41no—
189C4×C191C22×D411yescharacteristic
190C4×C191H(5,𝔽2)11yescharacteristic
191Dic191—41no—
192Dic191—41no—
193Dic191—41no—
194C2×C191—41no—
195C2×C191—41no—
196C2×C191—81no—
197C2×C191D4211yescharacteristic
198C2×C191#57511yescharacteristic
199C2×C191#57811yescharacteristic
200D191—42no—
201C191#326911yescharacteristic, complemented, Sylow
202#3269—1911nomaximal, Sylow
203C2×#137—1911no—
204C2×#135—1911no—
205#548D19111yescharacteristic, complemented
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×#41—1911no—
220C2×#41C2×D19111yescharacteristic
221C22×Q8C2×D19111yescharacteristic
222#137—3824no—
223#135—3824no—
224C2×Q16—1911no—
225C2×Q16—1911no—
226C2×Q16—1911no—
227C2×Q16—1911no—
228C2×SD16—1911no—
229C2×SD16—1911no—
230C2×SD16—1911no—
231C2×SD16—1911no—
232C2×D8—1911no—
233C2×M16—1911no—
234C22×C8—1911no—
235C4⋊Q8—1911no—
236#124—21no—
237#124C2×D19111yescharacteristic
238#123—3821no—
239#123—3821no—
240C22⋊Q8—21no—
241C22⋊Q8—1911no—
242C22⋊Q8—1911no—
243C22⋊Q8C2×D19111yescharacteristic
244#121—21no—
245#121—3821no—
246#121C2×D19111yescharacteristic
247C4×Q8C2×D19111yescharacteristic
248C4×D4—21no—
249C4×D4—1911no—
250C4×D4C2×D19111yescharacteristic
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—41no—
265#41—3822no—
266#41—3822no—
267#41—7641no—
268#41—7641no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1911no—
273C2×Q8—1911no—
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×C4C22×D19111yescharacteristic
289C22×C4C22×D19111yescharacteristic
290Q16—3822no—
291Q16—3822no—
292Q16—3822no—
293Q16—7641no—
294SD16—3822no—
295SD16—3822no—
296SD16—1914no—
297SD16—7641no—
298D8—3822no—
299M16—3822no—
300C2×C8—3822no—
301C2×C8—1911no—
302C2×C8—1911no—
303C2×C8—1911no—
304C2×C8—1911no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—3821no—
308C4⋊C4—3821no—
309C4⋊C4—3821no—
310C4⋊C4—3821no—
311C4⋊C4—1911no—
312C4⋊C4C22⋊D19111yescharacteristic
313C4⋊C4C22×D19111yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—3821no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4C22⋊D19111yescharacteristic
322C22⋊C4C22⋊D19111yescharacteristic
323C42—21no—
324C42C22×D19111yescharacteristic
325C23—21no—
326C23—3821no—
327C23C2×C22⋊D19111yescharacteristic
328C23C2×C22⋊D19111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—3821no—
334Q8—3821no—
335Q8—3821no—
336Q8—3821no—
337Q8D4×D19112yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—3821no—
344D4—1912no—
345D4—7642no—
346D4—7642no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—3821no—
357C2×C4—3821no—
358C2×C4—3821no—
359C2×C4—7642no—
360C2×C4—7642no—
361C2×C4—7642no—
362C2×C4C2×C22⋊D19111yescharacteristic
363C2×C4C2×C22⋊D19111yescharacteristic
364C2×C4C2×C22⋊D19111yescharacteristic
365C2×C4C2×C22⋊D19111yescharacteristic
366C2×C4C23×D19111yescharacteristic
367C8—3821no—
368C8—3821no—
369C8—3821no—
370C8—1912no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3822no—
381C22—7641no—
382C22—7641no—
383C22C22×C22⋊D19111yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—7641no—
392C4—7641no—
393C4—7641no—
394C4C2×D4×D19111yescharacteristic
395C4#2025936711yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—7642no—
400C2D4×C22⋊D19111yescharacteristic
401C2#2713346111yescharacteristic
402C2#2713347811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#548×C191C2
socleC22×C191C22×D4
hypercenter#548D191
hypocenterC191#3269
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#20259353C2semidirect
2. C2×#20259342C2semidirect
3. #27133318C2semidirect
4. #27133058C2semidirect
5. #27132907C2semidirect
6. #27132902C2semidirect
7. #27132875C2semidirect
8. #27132873C2semidirect
9. #27132825C2semidirect
10. #27132823C2semidirect
11. #27132819C2semidirect
12. #27132806C2semidirect
13. #27132761C2semidirect
14. #27132735C2semidirect
15. #548×C191C2semidirect
80. #15690274D4semidirect
201. C191#3269semidirect
205. #548D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593532
C2×#202593422
#271333182
#271330582
#271329072
#271329022
#271328752
#271328732
#271328252
#271328232
#271328192
#271328062
#271327612
#271327352
#548×C1912
#3269191