Finite Group Databasebeta

← #49170326

Subgroups — #49170326

Order 37504.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627649C211yesmaximal, characteristic, complemented
2C2×#22627644C211yesmaximal, characteristic, complemented
3#34036775C211yesmaximal, characteristic, complemented
4#34036516C211yesmaximal, characteristic, complemented
5#34036373C211yesmaximal, characteristic, complemented
6#34036363C211yesmaximal, characteristic, complemented
7#34036351C211yesmaximal, characteristic, complemented
8#34036321C211yesmaximal, characteristic, complemented
9#34036291C211yesmaximal, characteristic, complemented
10#34036274C211yesmaximal, characteristic, complemented
11#34036272C211yesmaximal, characteristic, complemented
12#34036262C211yesmaximal, characteristic, complemented
13#34036227C211yesmaximal, characteristic, complemented
14#34036201C211yesmaximal, characteristic, complemented
15#550×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18#22627649—24no—
19#22627644—24no—
20C2×#17989792C2211yescharacteristic, complemented
21C2×#17989790C2211yescharacteristic, complemented
22C2×#17989790C2211yescharacteristic, complemented
23C2×#17989789C2211yescharacteristic
24C2×#17989789C2211yescharacteristic
25C2×#17989787C2211yescharacteristic
26C2×#17989787C2211yescharacteristic
27C2×#17989787C2211yescharacteristic
28C2×#17989787C2211yescharacteristic
29C2×#17989786C2211yescharacteristic
30C22×#14556873C2211yescharacteristic, complemented
31#22627612C2211yescharacteristic
32#22627609—21no—
33#22627603C2211yescharacteristic
34#22627598C2211yescharacteristic
35#22627597—21no—
36#22627592—21no—
37C4×D1172C2211yescharacteristic
38C22⋊C4×D293—21no—
39#22627557C2211yescharacteristic, complemented
40C4⋊#14556873C2211yescharacteristic, complemented
41#22627548C2211yescharacteristic, complemented
42#22627547C2211yescharacteristic, complemented
43#22627545C2211yescharacteristic, complemented
44#22627545C2211yescharacteristic, complemented
45#22627544C2211yescharacteristic
46#22627544C2211yescharacteristic
47C22⋊#14556873C2211yescharacteristic
48C22⋊#14556873C2211yescharacteristic, complemented
49#22627535C2211yescharacteristic, complemented
50C2×#41×C293C2211yescharacteristic
51C2×#41×C293C2211yescharacteristic, complemented
52#127×C293—21no—
53#127×C293C2211yescharacteristic, complemented
54#121×C293—21no—
55#121×C293—21no—
56#121×C293C2211yescharacteristic
57#121×C293C2211yescharacteristic, complemented
58C4×Q8×C293C2211yescharacteristic, complemented
59C4×D4×C293—21no—
60C4×D4×C293C2211yescharacteristic
61C23×D293—21no—
62#17989802—22no—
63#17989802—41no—
64C2×Dic586C2311yescharacteristic
65C2×D1172—22no—
66C2×D1172—21no—
67C2×D1172—41no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C4×D293—21no—
73C2×C4×D293—21no—
74#17989792—22no—
75#17989790—22no—
76#17989790—41no—
77#17989789—22no—
78#17989789D414yescomplemented
79#17989787—22no—
80#17989787—22no—
81#17989787—22no—
82#17989787—41no—
83#17989786—22no—
84C2×#14556873—22no—
85C2×#14556873C2311yescharacteristic
86C2×#14556873C2311yescharacteristic
87C2×#14556873C2311yescharacteristic
88C2×#14556873C2311yescharacteristic
89C4⋊Dic293C2311yescharacteristic
90#17989781—21no—
91C22⋊Dic293—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95#17989779—21no—
96#41×C293—22no—
97#41×C293—22no—
98#41×C293—41no—
99#41×C293—41no—
100C2×Q8×C293C2311yescharacteristic
101C2×D4×C293—21no—
102C2×D4×C293—21no—
103C2×D4×C293—21no—
104C2×D4×C293—21no—
105C2×D4×C293—21no—
106C2×D4×C293—21no—
107C2×D4×C293D411yescharacteristic
108C2×D4×C293C2311yescharacteristic
109C2×D4×C293C2311yescharacteristic
110C22×C4×C293—21no—
111C22×C4×C293—21no—
112C22×C4×C293C2311yescharacteristic
113C22×C4×C293C2311yescharacteristic
114C4⋊C4×C293—21no—
115C4⋊C4×C293D411yescharacteristic
116C4⋊C4×C293C2311yescharacteristic
117C22⋊C4×C293—21no—
118C22⋊C4×C293—21no—
119C22⋊C4×C293D411yescharacteristic
120C22⋊C4×C293D411yescharacteristic
121C42×C293—21no—
122C42×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—42no—
126C22×D293—42no—
127C22×D293—81no—
128Dic586—21no—
129Dic586—21no—
130D1172—21no—
131D1172—21no—
132D1172—21no—
133D1172—42no—
134D1172C2×D412yes—
135C22⋊D293—42no—
136C22⋊D293—81no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C4×D293—42no—
140C4×D293—81no—
141#14556873—21no—
142#14556873—21no—
143#14556873—21no—
144#14556873C2×D412yes—
145C23×C293—21no—
146C23×C293—21no—
147C23×C293C2×D411yescharacteristic
148C23×C293C2×D411yescharacteristic
149Q8×C293—21no—
150Q8×C293C2×D412yes—
151D4×C293—21no—
152D4×C293—21no—
153D4×C293—21no—
154D4×C293—21no—
155D4×C293—41no—
156D4×C293—42no—
157D4×C293—42no—
158D4×C293—42no—
159D4×C293—81no—
160D4×C293—81no—
161D4×C293—81no—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293—21no—
168C2×C4×C293—42no—
169C2×C4×C293—42no—
170C2×C4×C293C2×D411yescharacteristic
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2×D411yescharacteristic
173C2×C4×C293C2×D411yescharacteristic
174C2×C4×C293C2411yescharacteristic
175C22×C293—21no—
176C22×C293—21no—
177C22×C293—21no—
178C22×C293—21no—
179C22×C293—21no—
180C22×C293—21no—
181C22×C293—41no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293—41no—
185C22×C293—41no—
186C22×C293—41no—
187C22×C293C22×D411yescharacteristic
188C2×D293—22no—
189C2×D293—41no—
190C2×D293—41no—
191C2×D293—41no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—84no—
195C4×C293—21no—
196C4×C293—21no—
197C4×C293—22no—
198C4×C293—21no—
199C4×C293—21no—
200C4×C293—41no—
201C4×C293C22×D411yescharacteristic
202C4×C293H(5,𝔽2)11yescharacteristic
203Dic293—41no—
204Dic293—41no—
205C2×C293—41no—
206C2×C293—41no—
207C2×C293—81no—
208C2×C293—81no—
209C2×C293D4211yescharacteristic
210C2×C293#57511yescharacteristic
211C2×C293#57611yescharacteristic
212D293—42no—
213D293—81no—
214C293#326811yescharacteristic, complemented, Sylow
215#3268—2931nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2931no—
217C2×#135—2931no—
218#550D29311yescharacteristic, complemented
219#546—2931no—
220#496—2931no—
221#468—2931no—
222#466—2931no—
223#464—2931no—
224#459—2931no—
225#452—2931no—
226#449—2931no—
227#449—2931no—
228#447—2931no—
229#440—2931no—
230#435—2931no—
231C2×#41—2931no—
232C2×#41C2×D29311yescharacteristic
233C2×#41C2×D29311yescharacteristic, complemented
234C22×D4—2931no—
235AGL(1,ℤ/8)—5864no—
236#135—5864no—
237C2×Q16—2931no—
238C2×SD16—2931no—
239C2×SD16—2931no—
240C2×SD16—2931no—
241C2×SD16—2931no—
242C2×D8—2931no—
243C2×D8—2931no—
244C2×D8—2931no—
245C2×D8—2931no—
246C2×M16—2931no—
247C22×C8—2931no—
248#127—21no—
249#127C2×D29311yescharacteristic, complemented
250#124—2931no—
251#123—5861no—
252C22⋊Q8—2931no—
253#121—21no—
254#121—21no—
255#121—2931no—
256#121—5861no—
257#121C2×D29311yescharacteristic
258#121C2×D29311yescharacteristic, complemented
259#120—5861no—
260C4×Q8C2×D29311yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—2931no—
263C4×D4C2×D29311yescharacteristic
264C2×C22⋊C4—5861no—
265#107—2931no—
266C4⋊C8—2931no—
267#103—2931no—
268#103—2931no—
269#102—2931no—
270#102—2931no—
271#102—2931no—
272#102—2931no—
273C22⋊C8—2931no—
274C22⋊C8—2931no—
275#97—2931no—
276C24—5861no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—5862no—
282#41—11721no—
283C2×Q8—2931no—
284C2×Q8C22×D29311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2931no—
292C2×D4—2931no—
293C2×D4—5862no—
294C2×D4—5861no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—11721no—
298C2×D4C22⋊D29311yescharacteristic
299C2×D4C22×D29311yescharacteristic
300C2×D4C22×D29311yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—5861no—
304C22×C4—5861no—
305C22×C4C22×D29311yescharacteristic
306C22×C4C22×D29311yescharacteristic
307Q16—5862no—
308SD16—2934no—
309SD16—5862no—
310SD16—5862no—
311SD16—11721no—
312D8—5862no—
313D8—5862no—
314D8—5862no—
315D8—11721no—
316M16—5862no—
317C2×C8—2931no—
318C2×C8—2931no—
319C2×C8—2931no—
320C2×C8—2931no—
321C2×C8—5862no—
322C4⋊C4—21no—
323C4⋊C4—2931no—
324C4⋊C4—5861no—
325C4⋊C4C22⋊D29311yescharacteristic
326C4⋊C4C22×D29311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—5861no—
330C22⋊C4—5861no—
331C22⋊C4—5861no—
332C22⋊C4—5861no—
333C22⋊C4—5861no—
334C22⋊C4C22⋊D29311yescharacteristic
335C22⋊C4C22⋊D29311yescharacteristic
336C42—21no—
337C42C22×D29311yescharacteristic
338C23—21no—
339C23—21no—
340C23—23441no—
341C23—5861no—
342C23—5861no—
343C23—11722no—
344C23—11722no—
345C23C2×C22⋊D29311yescharacteristic
346C23C2×C22⋊D29311yescharacteristic
347Q8—21no—
348Q8—5861no—
349Q8—5861no—
350Q8D4×D29312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—2932no—
363D4—23441no—
364D4—5861no—
365D4—5861no—
366D4—5861no—
367D4—11722no—
368D4—11722no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—23441no—
378C2×C4—5861no—
379C2×C4—5861no—
380C2×C4—11722no—
381C2×C4C2×C22⋊D29311yescharacteristic
382C2×C4C2×C22⋊D29311yescharacteristic
383C2×C4C2×C22⋊D29311yescharacteristic
384C2×C4C2×C22⋊D29311yescharacteristic
385C2×C4C23×D29311yescharacteristic
386C8—2932no—
387C8—5861no—
388C8—5861no—
389C8—5861no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—23444no—
403C22—5862no—
404C22—11721no—
405C22—11721no—
406C22—11721no—
407C22—11721no—
408C22—11721no—
409C22C22×C22⋊D29311yescharacteristic
410C4—21no—
411C4—21no—
412C4—22no—
413C4—21no—
414C4—21no—
415C4—41no—
416C4—11721no—
417C4—11721no—
418C4C2×D4×D29311yescharacteristic
419C4#2262766911yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—23441no—
425C2—11722no—
426C2D4×C22⋊D29311yescharacteristic
427C2#3403692711yescharacteristic
428C2#3403693211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#550×C293C2
socleC22×C293C22×D4
hypercenter#550D293
hypocenterC293#3268
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22627649C2semidirect
2. C2×#22627644C2semidirect
3. #34036775C2semidirect
4. #34036516C2semidirect
5. #34036373C2semidirect
6. #34036363C2semidirect
7. #34036351C2semidirect
8. #34036321C2semidirect
9. #34036291C2semidirect
10. #34036274C2semidirect
11. #34036272C2semidirect
12. #34036262C2semidirect
13. #34036227C2semidirect
14. #34036201C2semidirect
15. #550×C293C2semidirect
20. C2×#17989792C22semidirect
21. C2×#17989790C22semidirect
22. C2×#17989790C22semidirect
30. C22×#14556873C22semidirect
39. #22627557C22semidirect
40. C4⋊#14556873C22semidirect
41. #22627548C22semidirect
42. #22627547C22semidirect
43. #22627545C22semidirect
44. #22627545C22semidirect
48. C22⋊#14556873C22semidirect
49. #22627535C22semidirect
51. C2×#41×C293C22semidirect
53. #127×C293C22semidirect
57. #121×C293C22semidirect
58. C4×Q8×C293C22semidirect
78. #17989789D4semidirect
214. C293#3268semidirect
218. #550D293semidirect
233. C2×#41C2×D293semidirect
249. #127C2×D293semidirect
258. #121C2×D293semidirect
260. C4×Q8C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276492
C2×#226276442
#340367752
#340365162
#340363732
#340363632
#340363512
#340363212
#340362912
#340362742
#340362722
#340362622
#340362272
#340362012
#550×C2932
#3268293