Finite Group Databasebeta

← #49170312

Subgroups — #49170312

Order 37504.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627648C211yesmaximal, characteristic, complemented
2C2×#22627643C211yesmaximal, characteristic, complemented
3#34036783C211yesmaximal, characteristic, complemented
4#34036774C211yesmaximal, characteristic, complemented
5#34036523C211yesmaximal, characteristic, complemented
6#34036388C211yesmaximal, characteristic, complemented
7#34036359C211yesmaximal, characteristic, complemented
8#34036344C211yesmaximal, characteristic, complemented
9#34036334C211yesmaximal, characteristic, complemented
10#34036292C211yesmaximal, characteristic, complemented
11#34036292C211yesmaximal, characteristic, complemented
12#34036281C211yesmaximal, characteristic, complemented
13#34036275C211yesmaximal, characteristic, complemented
14#34036202C211yesmaximal, characteristic, complemented
15#442×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C2×#17989801C2211yescharacteristic
18C2×#17989801C2211yescharacteristic, complemented
19C2×D4×D293C2211yescharacteristic
20#22627648—24no—
21#22627643—24no—
22C2×Dic1172C2211yescharacteristic, complemented
23C2×Dic1172C2211yescharacteristic, complemented
24C2×#17989791C2211yescharacteristic
25C2×#17989791C2211yescharacteristic
26C2×#17989790C2211yescharacteristic
27C2×#17989790C2211yescharacteristic
28C2×#17989790C2211yescharacteristic, complemented
29C2×#17989787C2211yescharacteristic
30C2×#17989785C2211yescharacteristic
31C2×C8×D293C2211yescharacteristic, complemented
32#22627624C2211yescharacteristic, complemented
33#22627612C2211yescharacteristic
34#22627608—21no—
35#22627608—21no—
36#22627608—21no—
37#22627604C2211yescharacteristic
38#22627604C2211yescharacteristic
39#22627604C2211yescharacteristic, complemented
40#22627599C2211yescharacteristic
41#22627598—21no—
42#22627598—21no—
43#22627594—21no—
44C4×D1172C2211yescharacteristic
45C2×C4⋊Dic293—21no—
46C2×#17989779—21no—
47C4⋊#14556873C2211yescharacteristic, complemented
48#22627547C2211yescharacteristic, complemented
49#22627547C2211yescharacteristic, complemented
50#22627547C2211yescharacteristic, complemented
51#22627544C2211yescharacteristic
52#22627536C2211yescharacteristic
53#22627536C2211yescharacteristic, complemented
54C2×D8×C293C2211yescharacteristic
55C4×D4×C293C2211yescharacteristic
56C4×D4×C293C2211yescharacteristic, complemented
57#106×C293C2211yescharacteristic, complemented
58#102×C293C2211yescharacteristic
59#102×C293C2211yescharacteristic, complemented
60#97×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989801—22no—
63#17989801—22no—
64#17989801—22no—
65#17989801—41no—
66#17989801—41no—
67#17989801—41no—
68C2×Dic586C2311yescharacteristic
69C2×Dic586C2311yescharacteristic
70C2×Dic586C2311yescharacteristic
71C2×D1172—21no—
72C2×D1172D411yescharacteristic
73C2×D1172C2311yescharacteristic
74C2×C22⋊D293—21no—
75C2×C22⋊D293—21no—
76C2×C22⋊D293—21no—
77C2×C22⋊D293—21no—
78D4×D293—22no—
79D4×D293—41no—
80C22×Dic293—21no—
81C22×Dic293—21no—
82C22×Dic293—21no—
83C2×C4×D293C2311yescharacteristic
84C2×C4×D293C2311yescharacteristic
85Dic1172—22no—
86Dic1172—41no—
87#17989791—22no—
88#17989791—41no—
89#17989790—22no—
90#17989790—22no—
91#17989790—22no—
92#17989787—22no—
93#17989785—22no—
94C2×#14556873C2311yescharacteristic
95C2×#14556873C2311yescharacteristic
96C8×D293—22no—
97C4⋊Dic293—21no—
98C4⋊Dic293—21no—
99C4⋊Dic293—21no—
100C4⋊Dic293—21no—
101C4⋊Dic293—21no—
102C4⋊Dic293D411yescharacteristic
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779—21no—
109#17989779D411yescharacteristic
110#17989779D411yescharacteristic
111C2×D4×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C22×C4×C293—21no—
114C22×C4×C293—21no—
115D8×C293D414yescomplemented
116C2×C8×C293C2311yescharacteristic
117C2×C8×C293C2311yescharacteristic
118C4⋊C4×C293C2311yescharacteristic
119C4⋊C4×C293C2311yescharacteristic
120C22⋊C4×C293—21no—
121C22⋊C4×C293—21no—
122C42×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—42no—
125C22×D293—42no—
126C22×D293—81no—
127C22×D293C2×D411yescharacteristic
128C22×D293C2×D411yescharacteristic
129Dic586—21no—
130Dic586—21no—
131Dic586—21no—
132Dic586—21no—
133Dic586—21no—
134Dic586—21no—
135D1172—21no—
136D1172—21no—
137D1172—41no—
138D1172—81no—
139C22⋊D293—42no—
140C22⋊D293—42no—
141C22⋊D293—42no—
142C22⋊D293—42no—
143C2×Dic293—21no—
144C2×Dic293—21no—
145C2×Dic293—21no—
146C2×Dic293—42no—
147C2×Dic293—42no—
148C2×Dic293—42no—
149C2×Dic293C2×D411yescharacteristic
150C2×Dic293C2×D411yescharacteristic
151C4×D293—21no—
152C4×D293—21no—
153C4×D293—21no—
154C4×D293—21no—
155#14556873—21no—
156#14556873—21no—
157C23×C293—21no—
158C23×C293—21no—
159D4×C293—21no—
160D4×C293—21no—
161D4×C293C2×D412yes—
162D4×C293C2×D412yes—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—42no—
166C2×C4×C293—81no—
167C2×C4×C293C2×D411yescharacteristic
168C2×C4×C293C2×D411yescharacteristic
169C2×C4×C293C2411yescharacteristic
170C8×C293—21no—
171C8×C293C2×D412yes—
172C22×C293—22no—
173C22×C293—22no—
174C22×C293—41no—
175C22×C293—41no—
176C22×C293—41no—
177C22×C293—41no—
178C22×C293C22×D411yescharacteristic
179C2×D293—21no—
180C2×D293—21no—
181C2×D293—21no—
182C2×D293—21no—
183C2×D293—21no—
184C2×D293—21no—
185C2×D293—41no—
186C2×D293—41no—
187C2×D293—41no—
188C2×D293—84no—
189C4×C293—21no—
190C4×C293—21no—
191C4×C293—41no—
192C4×C293—41no—
193C4×C293C22×D411yescharacteristic
194C4×C293H(5,𝔽2)11yescharacteristic
195Dic293—21no—
196Dic293—21no—
197Dic293—41no—
198Dic293—41no—
199Dic293—41no—
200C2×C293—42no—
201C2×C293—42no—
202C2×C293D4211yescharacteristic
203C2×C293#57511yescharacteristic
204C2×C293#57711yescharacteristic
205D293—41no—
206D293—41no—
207D293—81no—
208C293#326711yescharacteristic, complemented, Sylow
209#3267—2931nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2931no—
211C2×#135—2931no—
212#547—2931no—
213#546—2931no—
214#497—2931no—
215#470—2931no—
216#465—2931no—
217#463—2931no—
218#461—2931no—
219#452—2931no—
220#452—2931no—
221#450—2931no—
222#449—2931no—
223#442D29311yescharacteristic, complemented
224#435—2931no—
225C2×#41—2931no—
226C2×#41—2931no—
227C2×#41—2931no—
228C22×D4—2931no—
229AGL(1,ℤ/8)—5864no—
230#135—5864no—
231C2×Q16—2931no—
232C2×Q16—2931no—
233C2×SD16—2931no—
234C2×SD16—2931no—
235C2×SD16—2931no—
236C2×SD16—2931no—
237C2×SD16—2931no—
238C2×D8—2931no—
239C2×D8C2×D29311yescharacteristic
240C2×M16—2931no—
241C22×C8—2931no—
242C4⋊Q8—2931no—
243#124—2931no—
244#123—5861no—
245#123—5861no—
246#123—5861no—
247C22⋊Q8—2931no—
248C22⋊Q8—2931no—
249C22⋊Q8—2931no—
250#121—2931no—
251#121—5861no—
252#121—5861no—
253#120—5861no—
254C4×D4—2931no—
255C4×D4C2×D29311yescharacteristic
256C4×D4C2×D29311yescharacteristic, complemented
257C2×C4⋊C4—5861no—
258C2×C22⋊C4—5861no—
259#106C2×D29311yescharacteristic, complemented
260C4⋊C8—2931no—
261#103—2931no—
262#103—2931no—
263#103—2931no—
264#102—2931no—
265#102C2×D29311yescharacteristic
266#102C2×D29311yescharacteristic, complemented
267C22⋊C8—2931no—
268C22⋊C8—2931no—
269#97C2×D29311yescharacteristic, complemented
270C24—5861no—
271#41—5862no—
272#41—5862no—
273#41—5862no—
274#41—11721no—
275#41—11721no—
276#41—11721no—
277C2×Q8—2931no—
278C2×Q8—2931no—
279C2×Q8—2931no—
280C2×D4—2931no—
281C2×D4—2931no—
282C2×D4—5861no—
283C2×D4—5861no—
284C2×D4—5862no—
285C2×D4—5861no—
286C2×D4—5861no—
287C2×D4—5861no—
288C2×D4—11721no—
289C2×D4C22×D29311yescharacteristic
290C2×D4C22×D29311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—2931no—
294C22×C4—2931no—
295C22×C4—5861no—
296C22×C4—5861no—
297C22×C4—5861no—
298Q16—5862no—
299Q16—11721no—
300SD16—5862no—
301SD16—5862no—
302SD16—5862no—
303SD16—5862no—
304SD16—11721no—
305D8—5862no—
306D8D117214yescomplemented
307M16—5862no—
308C2×C8—2931no—
309C2×C8—2931no—
310C2×C8—5862no—
311C2×C8C22×D29311yescharacteristic
312C2×C8C22×D29311yescharacteristic
313C4⋊C4—2931no—
314C4⋊C4—5861no—
315C4⋊C4—5861no—
316C4⋊C4—5861no—
317C4⋊C4—5861no—
318C4⋊C4—5861no—
319C4⋊C4C22×D29311yescharacteristic
320C4⋊C4C22×D29311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—2931no—
324C22⋊C4—2931no—
325C22⋊C4—5861no—
326C22⋊C4—5861no—
327C22⋊C4—5861no—
328C22⋊C4—5861no—
329C22⋊C4—5861no—
330C22⋊C4—5861no—
331C42C22×D29311yescharacteristic
332C23—21no—
333C23—21no—
334C23—2931no—
335C23—2931no—
336C23—23441no—
337C23—5861no—
338C23—11722no—
339C23—11722no—
340Q8—5861no—
341Q8—5861no—
342Q8—5861no—
343Q8—5861no—
344Q8—5861no—
345Q8—5861no—
346D4—21no—
347D4—21no—
348D4—23441no—
349D4—5861no—
350D4—5861no—
351D4—11721no—
352D4—11722no—
353D4—11722no—
354D4—11722no—
355D4—11722no—
356D4C2×D117212yes—
357D4C2×D117212yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—2931no—
363C2×C4—2931no—
364C2×C4—5861no—
365C2×C4—5861no—
366C2×C4—5861no—
367C2×C4—5861no—
368C2×C4—5861no—
369C2×C4—5861no—
370C2×C4—5861no—
371C2×C4—11722no—
372C2×C4—11722no—
373C2×C4—11722no—
374C2×C4D4×D29311yescharacteristic
375C2×C4D4×D29311yescharacteristic
376C2×C4C23×D29311yescharacteristic
377C8—21no—
378C8—5861no—
379C8—5861no—
380C8C2×D117212yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—23444no—
388C22—5861no—
389C22—5861no—
390C22—5861no—
391C22—5861no—
392C22—5861no—
393C22—5861no—
394C22—11721no—
395C22—11721no—
396C22—11721no—
397C22C2×D4×D29311yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—5861no—
403C4—5861no—
404C4—11721no—
405C4—11721no—
406C4—11721no—
407C4C22×D117211yescharacteristic
408C4#2262767011yescharacteristic
409C2—42no—
410C2—42no—
411C2—23441no—
412C2—11721no—
413C2—11721no—
414C2D4×D117211yescharacteristic
415C2#3403692511yescharacteristic
416C2#3403693911yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#442×C293C2
socleC22×C293C22×D4
hypercenter#442D293
hypocenterC293#3267
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#22627648C2semidirect
2. C2×#22627643C2semidirect
3. #34036783C2semidirect
4. #34036774C2semidirect
5. #34036523C2semidirect
6. #34036388C2semidirect
7. #34036359C2semidirect
8. #34036344C2semidirect
9. #34036334C2semidirect
10. #34036292C2semidirect
11. #34036292C2semidirect
12. #34036281C2semidirect
13. #34036275C2semidirect
14. #34036202C2semidirect
15. #442×C293C2semidirect
18. C2×#17989801C22semidirect
22. C2×Dic1172C22semidirect
23. C2×Dic1172C22semidirect
28. C2×#17989790C22semidirect
31. C2×C8×D293C22semidirect
32. #22627624C22semidirect
39. #22627604C22semidirect
47. C4⋊#14556873C22semidirect
48. #22627547C22semidirect
49. #22627547C22semidirect
50. #22627547C22semidirect
53. #22627536C22semidirect
56. C4×D4×C293C22semidirect
57. #106×C293C22semidirect
59. #102×C293C22semidirect
60. #97×C293C22semidirect
115. D8×C293D4semidirect
208. C293#3267semidirect
223. #442D293semidirect
256. C4×D4C2×D293semidirect
259. #106C2×D293semidirect
266. #102C2×D293semidirect
269. #97C2×D293semidirect
306. D8D1172semidirect

Maximal subgroups

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

subgroupindex
C2×#226276482
C2×#226276432
#340367832
#340367742
#340365232
#340363882
#340363592
#340363442
#340363342
#340362922
#340362812
#340362752
#340362022
#442×C2932
#3267293