Finite Group Databasebeta

← #37581812

Subgroups — #37581812

Order 24448.

700 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259369C211yesmaximal, characteristic
2C2×#20259351C211yesmaximal, characteristic, complemented
3C2×Q16×D191C211yesmaximal, characteristic, complemented
4#27133236C211yesmaximal, characteristic, complemented
5#27132826C211yesmaximal, characteristic, complemented
6#27132825C211yesmaximal, characteristic, complemented
7#27132822C211yesmaximal, characteristic, complemented
8#27132821C211yesmaximal, characteristic, complemented
9#27132820C211yesmaximal, characteristic, complemented
10#27132818C211yesmaximal, characteristic, complemented
11#27132817C211yesmaximal, characteristic, complemented
12#27132670C211yesmaximal, characteristic, complemented
13#27132656C211yesmaximal, characteristic, complemented
14#27132628C211yesmaximal, characteristic, complemented
15#452×C191C211yesmaximal, characteristic, complemented
16#20259369—24no—
17#20259369—24no—
18C2×#15690287—21no—
19C2×#15690287C2211yescharacteristic
20C2×#15690286—21no—
21C2×#15690286—21no—
22C2×#15690286C2211yescharacteristic
23C2×#15690286C2211yescharacteristic
24C22×Dic382—21no—
25C22×Dic382C2211yescharacteristic
26C2×Q8×D191C2211yescharacteristic
27C2×Q8×D191C2211yescharacteristic
28C2×Q8×D191C2211yescharacteristic
29#20259351—24no—
30C2×Dic764C2211yescharacteristic
31C2×Dic764C2211yescharacteristic
32C2×#15690277C2211yescharacteristic
33C2×#15690277C2211yescharacteristic
34C2×#15690277C2211yescharacteristic
35Q16×D191—24no—
36C2×#15690275C2211yescharacteristic
37C2×#15690270C2211yescharacteristic
38C2×C8×D191C2211yescharacteristic
39#20259321—21no—
40#20259308—21no—
41#20259307—21no—
42#20259304—21no—
43#20259302C2211yescharacteristic
44#20259301C2211yescharacteristic
45#20259298C2211yescharacteristic
46#20259280C2211yescharacteristic
47#20259247C2211yescharacteristic
48#20259247C2211yescharacteristic
49#20259246C2211yescharacteristic
50#20259245C2211yescharacteristic
51#20259245C2211yescharacteristic
52#20259243C2211yescharacteristic
53C22⋊(C191⋊C8)C2211yescharacteristic
54(C2×D191)⋊C8C2211yescharacteristic
55(C2×D191)⋊C8C2211yescharacteristic
56C2×#41×C191C2211yescharacteristic
57C2×Q16×C191C2211yescharacteristic
58C2×SD16×C191C2211yescharacteristic
59C22⋊Q8×C191C2211yescharacteristic
60#103×C191C2211yescharacteristic
61#102×C191C2211yescharacteristic
62C22⋊C8×C191C2211yescharacteristic
63#15690287—28no—
64#15690287—22no—
65#15690287—22no—
66#15690286—28no—
67#15690286—28no—
68#15690286—22no—
69#15690286—22no—
70#15690286—22no—
71#15690286D414yes—
72C2×Dic382—28no—
73C2×Dic382—22no—
74C2×Dic382—22no—
75C2×Dic382—21no—
76C2×Dic382—21no—
77C2×Dic382—21no—
78C2×Dic382—21no—
79C2×Dic382D411yescharacteristic
80C2×Dic382C2311yescharacteristic
81C2×Dic382C2311yescharacteristic
82C2×Dic382C2311yescharacteristic
83Q8×D191—22no—
84Q8×D191—22no—
85Q8×D191—22no—
86Q8×D191—22no—
87Q8×D191—22no—
88Q8×D191D414yes—
89C2×D764D411yescharacteristic
90C2×C22⋊D191—21no—
91C2×C22⋊D191—21no—
92C2×C22⋊D191D411yescharacteristic
93C2×C22⋊D191D411yescharacteristic
94C22×Dic191—21no—
95C22×Dic191—21no—
96C22×Dic191—21no—
97C2×C4×D191—21no—
98C2×C4×D191—21no—
99C2×C4×D191—21no—
100C2×C4×D191C2311yescharacteristic
101C2×C4×D191C2311yescharacteristic
102Dic764—22no—
103Dic764—22no—
104#15690277—22no—
105#15690277—22no—
106#15690277—22no—
107#15690275—22no—
108#15690270—22no—
109C2×C191⋊C8C2311yescharacteristic
110C2×C191⋊C8C2311yescharacteristic
111C8×D191—22no—
112C4⋊Dic191—21no—
113C4⋊Dic191C2311yescharacteristic
114Dic191⋊C4—21no—
115Dic191⋊C4—21no—
116C22⋊Dic191—21no—
117(C2×D191)⋊C4—21no—
118(C2×D191)⋊C4—21no—
119(C2×D191)⋊C4—21no—
120C4×Dic191—21no—
121#41×C191—22no—
122#41×C191—22no—
123C2×Q8×C191C2311yescharacteristic
124C2×Q8×C191C2311yescharacteristic
125C2×D4×C191—21no—
126C2×D4×C191C2311yescharacteristic
127C22×C4×C191—21no—
128C22×C4×C191C2311yescharacteristic
129Q16×C191—22no—
130SD16×C191—22no—
131C2×C8×C191C2311yescharacteristic
132C2×C8×C191C2311yescharacteristic
133C4⋊C4×C191—21no—
134C4⋊C4×C191C2311yescharacteristic
135C22⋊C4×C191—21no—
136C22×D191C2×D411yescharacteristic
137C22×D191C2×D411yescharacteristic
138Dic382—24no—
139Dic382—24no—
140Dic382—21no—
141Dic382—21no—
142Dic382—24no—
143Dic382—22no—
144Dic382—21no—
145Dic382—21no—
146Dic382—42no—
147Dic382C2×D412yes—
148Dic382C2×D412yes—
149D764—22no—
150C22⋊D191—24no—
151C22⋊D191—24no—
152C22⋊D191—22no—
153C22⋊D191—22no—
154C2×Dic191—24no—
155C2×Dic191—24no—
156C2×Dic191—21no—
157C2×Dic191—21no—
158C2×Dic191—24no—
159C2×Dic191—21no—
160C2×Dic191—21no—
161C2×Dic191C2×D411yescharacteristic
162C2×Dic191C2×D411yescharacteristic
163C4×D191—24no—
164C4×D191—24no—
165C4×D191—21no—
166C4×D191—21no—
167C4×D191—42no—
168C4×D191C2×D412yes—
169C4×D191C2×D412yes—
170C191⋊C8—21no—
171C191⋊C8—21no—
172C23×C191—21no—
173C23×C191C2×D411yescharacteristic
174Q8×C191—21no—
175Q8×C191—21no—
176Q8×C191—21no—
177Q8×C191C2×D412yes—
178D4×C191—24no—
179D4×C191—21no—
180D4×C191C2×D412yes—
181C2×C4×C191—24no—
182C2×C4×C191—21no—
183C2×C4×C191—21no—
184C2×C4×C191—21no—
185C2×C4×C191—21no—
186C2×C4×C191—21no—
187C2×C4×C191C2×D411yescharacteristic
188C2×C4×C191C2411yescharacteristic
189C8×C191—21no—
190C8×C191—21no—
191C22×C191—21no—
192C22×C191—22no—
193C22×C191—21no—
194C22×C191—21no—
195C22×C191—41no—
196C22×C191—41no—
197C22×C191C22×D411yescharacteristic
198C2×D191—21no—
199C2×D191—21no—
200C2×D191—21no—
201C2×D191—21no—
202C2×D191—21no—
203C2×D191#12012yes—
204C4×C191—21no—
205C4×C191—22no—
206C4×C191—41no—
207C4×C191—41no—
208C4×C191C22×D411yescharacteristic
209C4×C191C22×D411yescharacteristic
210Dic191—22no—
211Dic191—22no—
212Dic191—21no—
213Dic191—41no—
214Dic191—41no—
215Dic191#12012yes—
216C2×C191—41no—
217C2×C191—42no—
218C2×C191C2×#12011yescharacteristic
219C2×C191#57811yescharacteristic
220C2×C191#57811yescharacteristic
221D191—22no—
222D191—41no—
223C191#301111yescharacteristic, complemented, Sylow
224#3011—1911nomaximal, Sylow
225C2×2−1+4—1911no—
226C2×#137—1911no—
227C22×Q16—1911no—
228#536—1911no—
229#452—1911no—
230#452—1911no—
231#452—1911no—
232#452D19111yescharacteristic, complemented
233#451—1911no—
234#451—1911no—
235#451—1911no—
236#451—1911no—
237#419—1911no—
238#416—1911no—
239#408—1911no—
2402−1+4—3824no—
2412−1+4—3824no—
242C2×#41—3821no—
243C2×#41—3821no—
244C2×#41—3821no—
245C2×#41—1911no—
246C2×#41—1911no—
247C2×#41—1911no—
248C2×#41C2×D19111yescharacteristic
249C22×Q8—3821no—
250C22×Q8—1911no—
251C22×Q8—1911no—
252C22×Q8—1911no—
253C22×Q8—1911no—
254#137—3824no—
255C2×Q16—3824no—
256C2×Q16—1911no—
257C2×Q16—1911no—
258C2×Q16—1911no—
259C2×Q16—1911no—
260C2×Q16—1911no—
261C2×Q16C2×D19111yescharacteristic
262C2×SD16—1911no—
263C2×SD16C2×D19111yescharacteristic
264C2×M16—1911no—
265C22×C8—1911no—
266C4⋊Q8—3821no—
267#124—3821no—
268#123—3821no—
269#123—3821no—
270C22⋊Q8—1911no—
271C22⋊Q8—1911no—
272C22⋊Q8—1911no—
273C22⋊Q8C2×D19111yescharacteristic
274#117—1911no—
275#103—1911no—
276#103—1911no—
277#103—1911no—
278#103—1911no—
279#103—1911no—
280#103C2×D19111yescharacteristic
281#102—1911no—
282#102C2×D19111yescharacteristic
283C22⋊C8—1911no—
284C22⋊C8—1911no—
285C22⋊C8—1911no—
286C22⋊C8C2×D19111yescharacteristic
287#41—22no—
288#41—22no—
289#41—3828no—
290#41—3822no—
291#41—3822no—
292#41—3828no—
293#41—3828no—
294#41—3822no—
295#41—3822no—
296#41—3822no—
297#41—1914no—
298C2×Q8—3828no—
299C2×Q8—3822no—
300C2×Q8—3822no—
301C2×Q8—3821no—
302C2×Q8—3821no—
303C2×Q8—3821no—
304C2×Q8—3822no—
305C2×Q8—3822no—
306C2×Q8—3822no—
307C2×Q8—3821no—
308C2×Q8—3822no—
309C2×Q8—3822no—
310C2×Q8—1914no—
311C2×Q8—1911no—
312C2×Q8—1911no—
313C2×Q8—1911no—
314C2×Q8—1911no—
315C2×Q8C22×D19111yescharacteristic
316C2×Q8C22×D19111yescharacteristic
317C2×D4—21no—
318C2×D4—3821no—
319C2×D4—3821no—
320C2×D4—1911no—
321C2×D4—1911no—
322C2×D4—1911no—
323C2×D4C22×D19111yescharacteristic
324C22×C4—21no—
325C22×C4—3821no—
326C22×C4—3821no—
327C22×C4—3821no—
328C22×C4—3821no—
329C22×C4—3821no—
330C22×C4—3821no—
331C22×C4—1911no—
332C22×C4—1911no—
333C22×C4C22×D19111yescharacteristic
334Q16—22no—
335Q16—3822no—
336Q16—3822no—
337Q16—3822no—
338Q16—3822no—
339Q16—3822no—
340SD16—22no—
341SD16—3822no—
342M16—3822no—
343C2×C8—3822no—
344C2×C8—1911no—
345C2×C8—1911no—
346C2×C8C22×D19111yescharacteristic
347C2×C8C22×D19111yescharacteristic
348C4⋊C4—21no—
349C4⋊C4—3821no—
350C4⋊C4—3821no—
351C4⋊C4—3821no—
352C4⋊C4—1911no—
353C4⋊C4C22×D19111yescharacteristic
354C22⋊C4—21no—
355C22⋊C4—3821no—
356C22⋊C4—3821no—
357C22⋊C4—3821no—
358C22⋊C4—3821no—
359C42—3821no—
360C23—21no—
361C23—1911no—
362C23—1911no—
363C23D4×D19111yescharacteristic
364Q8—21no—
365Q8—21no—
366Q8—21no—
367Q8—3824no—
368Q8—3824no—
369Q8—3822no—
370Q8—3824no—
371Q8—3821no—
372Q8—3821no—
373Q8—3821no—
374Q8—3821no—
375Q8—1912no—
376Q8—1912no—
377Q8—7642no—
378Q8D4×D19112yes—
379D4—24no—
380D4—21no—
381D4—3822no—
382D4—3824no—
383D4—3824no—
384D4—3822no—
385D4—3822no—
386D4D4×D19112yes—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—3824no—
394C2×C4—3824no—
395C2×C4—3824no—
396C2×C4—3824no—
397C2×C4—3821no—
398C2×C4—3821no—
399C2×C4—3824no—
400C2×C4—3821no—
401C2×C4—3821no—
402C2×C4—3821no—
403C2×C4—3821no—
404C2×C4—1911no—
405C2×C4—1911no—
406C2×C4—1912no—
407C2×C4—1912no—
408C2×C4—7642no—
409C2×C4D4×D19111yescharacteristic
410C2×C4C23×D19111yescharacteristic
411C8—21no—
412C8—21no—
413C8—3821no—
414C8—3821no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—3821no—
422C22—3821no—
423C22—3821no—
424C22—3821no—
425C22—3821no—
426C22—1912no—
427C22C2×D4×D19111yescharacteristic
428C4—21no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—3822no—
433C4—3822no—
434C4—3821no—
435C4—1912no—
436C4—7641no—
437C4—7641no—
438C4C2×D4×D19111yescharacteristic
439C4C2×D4×D19111yescharacteristic
440C2—41no—
441C2—42no—
442C2—3822no—
443C2—7641no—
444C2#120×D19111yescharacteristic
445C2#2713347711yescharacteristic
446C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#452×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259369C2
hypercenter#452D191
hypocenterC191#3011
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
2. C2×#20259351C2semidirect
3. C2×Q16×D191C2semidirect
4. #27133236C2semidirect
5. #27132826C2semidirect
6. #27132825C2semidirect
7. #27132822C2semidirect
8. #27132821C2semidirect
9. #27132820C2semidirect
10. #27132818C2semidirect
11. #27132817C2semidirect
12. #27132670C2semidirect
13. #27132656C2semidirect
14. #27132628C2semidirect
15. #452×C191C2semidirect
223. C191#3011semidirect
232. #452D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593692
C2×#202593512
C2×Q16×D1912
#271332362
#271328262
#271328252
#271328222
#271328212
#271328202
#271328182
#271328172
#271326702
#271326562
#271326282
#452×C1912
#3011191