Finite Group Databasebeta

← #37582235

Subgroups — #37582235

Order 24448.

452 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133462C211yesmaximal, characteristic, complemented
2#27133459C211yesmaximal, characteristic, complemented
3C2×#20259343C211yesmaximal, characteristic, complemented
4#131×D191C211yesmaximal, characteristic, complemented
5#27132937C211yesmaximal, characteristic, complemented
6#27132936C211yesmaximal, characteristic, complemented
7#27132932C211yesmaximal, characteristic, complemented
8#27132931C211yesmaximal, characteristic, complemented
9#27132927C211yesmaximal, characteristic, complemented
10#27132926C211yesmaximal, characteristic, complemented
11#27132923C211yesmaximal, characteristic, complemented
12#108×D191C211yesmaximal, characteristic, complemented
13#27132646C211yesmaximal, characteristic, complemented
14#27132644C211yesmaximal, characteristic, complemented
15#471×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18#41×D191C2211yescharacteristic
19#20259355C2211yescharacteristic
20#20259354—21no—
21#20259353C2211yescharacteristic
22#20259352C2211yescharacteristic
23#20259351—21no—
24#20259350C2211yescharacteristic
25#20259349—21no—
26#20259348C2211yescharacteristic
27#20259347C2211yescharacteristic
28#20259346—21no—
29#20259345—21no—
30#20259343—21no—
31#20259343—21no—
32#20259343—21no—
33#20259343—21no—
34#20259343—21no—
35#20259343—21no—
36#20259341—21no—
37C2×#15690277C2211yescharacteristic, complemented
38C2×#15690276C2211yescharacteristic, complemented
39C2×#15690272C2211yescharacteristic, complemented
40#20259330—21no—
41#20259330C2211yescharacteristic
42#20259329C2211yescharacteristic
43#20259328—21no—
44#20259328C2211yescharacteristic
45M16×D191—21no—
46M16×D191C2211yescharacteristic
47M16×D191C2211yescharacteristic
48M16×D191C2211yescharacteristic
49C2×C8×D191—21no—
50C2×C8×D191C2211yescharacteristic
51#20259258C2211yescharacteristic, complemented
52#20259257C2211yescharacteristic, complemented
53#20259257C2211yescharacteristic, complemented
54#20259241C2211yescharacteristic, complemented
55#20259240C2211yescharacteristic, complemented
56#20259240C2211yescharacteristic, complemented
57#20259239C2211yescharacteristic, complemented
58#20259238C2211yescharacteristic, complemented
59#20259238C2211yescharacteristic, complemented
60#137×C191C2211yescharacteristic
61AGL(1,ℤ/8)×C191C2211yescharacteristic
62C2×SD16×C191C2211yescharacteristic, complemented
63#131×C191C2211yescharacteristic
64#108×C191C2211yescharacteristic, complemented
65#101×C191C2211yescharacteristic, complemented
66#100×C191C2211yescharacteristic, complemented
67#15690288—21no—
68#15690288—21no—
69#15690288—21no—
70#15690288—21no—
71#15690288D411yescharacteristic
72#15690287—21no—
73#15690287C2311yescharacteristic
74#15690286—21no—
75#15690286—21no—
76#15690286—21no—
77#15690286—21no—
78#15690286—21no—
79#15690286D411yescharacteristic
80C2×Dic382C2311yescharacteristic
81Q8×D191D411yescharacteristic
82C2×D764—21no—
83C2×D764C2311yescharacteristic
84C2×C22⋊D191—21no—
85D4×D191—21no—
86D4×D191D411yescharacteristic
87C22×Dic191—21no—
88C2×C4×D191—21no—
89C2×C4×D191—21no—
90C2×C4×D191C2311yescharacteristic
91Dic764—21no—
92#15690277—21no—
93#15690277—21no—
94#15690277—21no—
95#15690277—21no—
96#15690276—21no—
97#15690276—21no—
98#15690276—21no—
99#15690276—21no—
100#15690275—21no—
101#15690275—21no—
102#15690274—21no—
103#15690274—21no—
104D1528—21no—
105#15690272—21no—
106#15690272—21no—
107#15690272—21no—
108#15690272—21no—
109#15690271—21no—
110#15690271C2311yescharacteristic
111#15690271C2311yescharacteristic
112#15690271C2311yescharacteristic
113#15690270—21no—
114#15690270—21no—
115#15690270—21no—
116#15690270—21no—
117#15690270—21no—
118C2×C191⋊C8—21no—
119C2×C191⋊C8C2311yescharacteristic
120C8×D191—21no—
121C8×D191—21no—
122C8×D191—21no—
123C8×D191—21no—
124C8×D191—21no—
125C8×D191D411yescharacteristic
126C8×D191D411yescharacteristic
127C8×D191D411yescharacteristic
128C8×D191D411yescharacteristic
129#41×C191C2311yescharacteristic
130C2×Q8×C191C2311yescharacteristic
131C2×D4×C191C2311yescharacteristic
132Q16×C191—21no—
133SD16×C191—21no—
134SD16×C191—21no—
135SD16×C191—21no—
136SD16×C191—21no—
137D8×C191—21no—
138M16×C191—21no—
139M16×C191C2311yescharacteristic
140M16×C191C2311yescharacteristic
141M16×C191C2311yescharacteristic
142C2×C8×C191—21no—
143C2×C8×C191C2311yescharacteristic
144C22×D191—21no—
145C22×D191—21no—
146C22×D191#4111yescharacteristic
147Dic382—21no—
148Dic382—21no—
149Dic382—21no—
150Dic382C2×D411yescharacteristic
151D764—21no—
152D764—21no—
153D764—21no—
154D764—41no—
155D764—41no—
156D764C2×D411yescharacteristic
157C22⋊D191—21no—
158C22⋊D191—21no—
159C22⋊D191—21no—
160C22⋊D191—41no—
161C22⋊D191—41no—
162C2×Dic191—21no—
163C2×Dic191—21no—
164C2×Dic191—41no—
165C2×Dic191—41no—
166C2×Dic191#4111yescharacteristic
167C4×D191—21no—
168C4×D191—21no—
169C4×D191—21no—
170C4×D191—41no—
171C4×D191—41no—
172C4×D191C2×D411yescharacteristic
173C4×D191C2×D411yescharacteristic
174C4×D191C2×D411yescharacteristic
175C4×D191C2×D411yescharacteristic
176C191⋊C8—21no—
177C191⋊C8—21no—
178C191⋊C8—21no—
179C191⋊C8C2×D411yescharacteristic
180C191⋊C8C2×D411yescharacteristic
181C23×C191—21no—
182Q8×C191—21no—
183Q8×C191—21no—
184Q8×C191C2×D411yescharacteristic
185D4×C191—21no—
186D4×C191—21no—
187D4×C191—21no—
188D4×C191C2×D411yescharacteristic
189C2×C4×C191—21no—
190C2×C4×C191—21no—
191C2×C4×C191C2411yescharacteristic
192C8×C191—21no—
193C8×C191—21no—
194C8×C191—21no—
195C8×C191C2×D411yescharacteristic
196C8×C191C2×D411yescharacteristic
197C22×C191—21no—
198C22×C191—41no—
199C22×C191—81no—
200C22×C191C2×#4111yescharacteristic
201C2×D191—21no—
202C2×D191—41no—
203C2×D191—42no—
204C2×D191—41no—
205C2×D191—81no—
206C2×D191#12111yescharacteristic
207C2×D191#12111yescharacteristic
208C4×C191—21no—
209C4×C191—41no—
210C4×C191C22×D411yescharacteristic
211C4×C191C22×D411yescharacteristic
212Dic191—21no—
213Dic191—41no—
214Dic191#12111yescharacteristic
215Dic191#12111yescharacteristic
216C2×C191—21no—
217C2×C191—41no—
218C2×C191—81no—
219C2×C191C2×#12111yescharacteristic
220D191—21no—
221D191—21no—
222D191—41no—
223D191—81no—
224C191#306711yescharacteristic, complemented, Sylow
225#3067—1911nomaximal, Sylow
226#575—1911no—
227#575—1911no—
228C2×#135—1911no—
229C2×#131—1911no—
230#473—1911no—
231#473—1911no—
232#472—1911no—
233#472—1911no—
234#471—1911no—
235#471—1911no—
236#471—1911no—
237#471D19111yescharacteristic, complemented
238C2×#108—1911no—
239#413—1911no—
240#413—1911no—
241C2×#41—1911no—
242C2×#41—1911no—
243C2×#41—1911no—
244#137—3821no—
245#137—3821no—
246#137—1911no—
247#137—1911no—
248#137—1911no—
249#137C2×D19111yescharacteristic
250AGL(1,ℤ/8)—3821no—
251AGL(1,ℤ/8)—3821no—
252AGL(1,ℤ/8)—1911no—
253AGL(1,ℤ/8)—1911no—
254AGL(1,ℤ/8)—1911no—
255AGL(1,ℤ/8)C2×D19111yescharacteristic
256#135—3821no—
257#135—3821no—
258#135—3821no—
259#135—3821no—
260#135—3821no—
261#135—3821no—
262#135—3821no—
263#135—3821no—
264C2×Q16—1911no—
265C2×SD16—1911no—
266C2×SD16C2×D19111yescharacteristic, complemented
267C2×D8—1911no—
268#131—3821no—
269#131—3821no—
270#131—1911no—
271#131—1911no—
272#131—1911no—
273#131C2×D19111yescharacteristic
274C2×M16—3821no—
275C2×M16—1911no—
276C2×M16—1911no—
277C2×M16—1911no—
278C22×C8—3821no—
279C22×C8—1911no—
280#108—1911no—
281#108—1911no—
282#108—1911no—
283#108C2×D19111yescharacteristic, complemented
284#101—1911no—
285#101—1911no—
286#101—1911no—
287#101C2×D19111yescharacteristic, complemented
288#100—1911no—
289#100—1911no—
290#100—1911no—
291#100C2×D19111yescharacteristic, complemented
292#41—3821no—
293#41—3821no—
294#41—3821no—
295#41—3821no—
296#41—3821no—
297#41—3821no—
298#41—3821no—
299#41—3821no—
300#41—3821no—
301#41—3821no—
302#41—1911no—
303#41—1911no—
304#41—1911no—
305#41C22×D19111yescharacteristic
306C2×Q8—1911no—
307C2×Q8—1911no—
308C2×Q8C22×D19111yescharacteristic
309C2×D4—3821no—
310C2×D4—3821no—
311C2×D4—3821no—
312C2×D4—1911no—
313C2×D4—1911no—
314C2×D4C22×D19111yescharacteristic
315C22×C4—3821no—
316C22×C4—3821no—
317C22×C4—3821no—
318C22×C4—1911no—
319Q16—21no—
320Q16—3821no—
321Q16—3821no—
322Q16—3821no—
323Q16—3821no—
324Q16—3821no—
325SD16—21no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—3821no—
330SD16—3821no—
331SD16—3821no—
332SD16—3821no—
333SD16—3821no—
334SD16—3821no—
335SD16—3821no—
336SD16—3821no—
337D8—21no—
338D8—3821no—
339D8—3821no—
340D8—3821no—
341D8—3821no—
342D8—3821no—
343M16—21no—
344M16—3821no—
345M16—3821no—
346M16—3821no—
347M16—3821no—
348M16—3821no—
349M16—3821no—
350M16—1911no—
351M16—1911no—
352M16—1911no—
353M16C22×D19111yescharacteristic
354M16C22×D19111yescharacteristic
355M16C22×D19111yescharacteristic
356C2×C8—21no—
357C2×C8—3821no—
358C2×C8—3821no—
359C2×C8—3821no—
360C2×C8—3821no—
361C2×C8—3821no—
362C2×C8—3821no—
363C2×C8—1911no—
364C2×C8—1911no—
365C2×C8—1911no—
366C2×C8—1911no—
367C2×C8—1911no—
368C2×C8C22×D19111yescharacteristic
369C23—21no—
370C23—3821no—
371C23—3821no—
372C23—1911no—
373Q8—21no—
374Q8—21no—
375Q8—3821no—
376Q8—3821no—
377Q8—3821no—
378Q8—1911no—
379Q8D4×D19111yescharacteristic
380D4—21no—
381D4—21no—
382D4—21no—
383D4—3821no—
384D4—3821no—
385D4—3821no—
386D4—3821no—
387D4—3821no—
388D4—3821no—
389D4—1911no—
390D4—7641no—
391D4—7641no—
392D4—7641no—
393D4—7641no—
394D4D4×D19111yescharacteristic
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—3821no—
398C2×C4—3821no—
399C2×C4—3821no—
400C2×C4—3821no—
401C2×C4—3821no—
402C2×C4—1911no—
403C2×C4—1911no—
404C2×C4—1911no—
405C2×C4—1911no—
406C2×C4—1911no—
407C2×C4—7641no—
408C2×C4—7641no—
409C2×C4—7641no—
410C2×C4—7641no—
411C2×C4C23×D19111yescharacteristic
412C8—21no—
413C8—21no—
414C8—21no—
415C8—3821no—
416C8—3821no—
417C8—3821no—
418C8—1911no—
419C8—1911no—
420C8D4×D19111yescharacteristic
421C8D4×D19111yescharacteristic
422C22—21no—
423C22—41no—
424C22—81no—
425C22—3821no—
426C22—1911no—
427C22—1911no—
428C22—15281no—
429C22—7641no—
430C22—7642no—
431C22—7641no—
432C22#41×D19111yescharacteristic
433C4—21no—
434C4—41no—
435C4—3821no—
436C4—1911no—
437C4—1911no—
438C4—7641no—
439C4C2×D4×D19111yescharacteristic
440C4C2×D4×D19111yescharacteristic
441C2—21no—
442C2—41no—
443C2—81no—
444C2—3821no—
445C2—3821no—
446C2—15281no—
447C2—7641no—
448C2#121×D19111yescharacteristic
subgroupG/H
centerC2#121×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#471×C191C2
socleC2×C191C2×#121
hypercenter#471D191
hypocenterC191#3067
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #27133462C2semidirect
2. #27133459C2semidirect
3. C2×#20259343C2semidirect
4. #131×D191C2semidirect
5. #27132937C2semidirect
6. #27132936C2semidirect
7. #27132932C2semidirect
8. #27132931C2semidirect
9. #27132927C2semidirect
10. #27132926C2semidirect
11. #27132923C2semidirect
12. #108×D191C2semidirect
13. #27132646C2semidirect
14. #27132644C2semidirect
15. #471×C191C2semidirect
37. C2×#15690277C22semidirect
38. C2×#15690276C22semidirect
39. C2×#15690272C22semidirect
51. #20259258C22semidirect
52. #20259257C22semidirect
53. #20259257C22semidirect
54. #20259241C22semidirect
55. #20259240C22semidirect
56. #20259240C22semidirect
57. #20259239C22semidirect
58. #20259238C22semidirect
59. #20259238C22semidirect
62. C2×SD16×C191C22semidirect
64. #108×C191C22semidirect
65. #101×C191C22semidirect
66. #100×C191C22semidirect
224. C191#3067semidirect
237. #471D191semidirect
266. C2×SD16C2×D191semidirect
283. #108C2×D191semidirect
287. #101C2×D191semidirect
291. #100C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334622
#271334592
C2×#202593432
#131×D1912
#271329372
#271329362
#271329322
#271329312
#271329272
#271329262
#271329232
#108×D1912
#271326462
#271326442
#471×C1912
#3067191