Finite Group Databasebeta

← #49170314

Subgroups — #49170314

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×#22627651C211yesmaximal, characteristic, complemented
2C2×#22627644C211yesmaximal, characteristic, complemented
3#34036781C211yesmaximal, characteristic, complemented
4#34036767C211yesmaximal, characteristic, complemented
5#34036522C211yesmaximal, characteristic, complemented
6#34036386C211yesmaximal, characteristic, complemented
7#34036360C211yesmaximal, characteristic, complemented
8#34036347C211yesmaximal, characteristic, complemented
9#34036332C211yesmaximal, characteristic, complemented
10#34036290C211yesmaximal, characteristic, complemented
11#34036281C211yesmaximal, characteristic, complemented
12#34036276C211yesmaximal, characteristic, complemented
13#34036238C211yesmaximal, characteristic, complemented
14#34036197C211yesmaximal, characteristic, complemented
15#452×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989802C2211yescharacteristic, complemented
18C2×D4×D293C2211yescharacteristic
19#22627651—24no—
20#22627644—24no—
21C2×#17989792C2211yescharacteristic, complemented
22C2×#17989791C2211yescharacteristic
23C2×#17989790C2211yescharacteristic
24C2×#17989789C2211yescharacteristic
25C2×#17989789C2211yescharacteristic, complemented
26C2×D2344C2211yescharacteristic
27C2×#17989787C2211yescharacteristic
28C2×#17989785C2211yescharacteristic
29C22×#14556873C2211yescharacteristic, complemented
30#22627622C2211yescharacteristic, complemented
31#22627613C2211yescharacteristic
32#22627609—21no—
33#22627607—21no—
34#22627606—21no—
35#22627604C2211yescharacteristic
36#22627600C2211yescharacteristic
37#22627598C2211yescharacteristic
38#22627597—21no—
39#22627596—21no—
40#22627592—21no—
41D4×Dic293C2211yescharacteristic
42#22627586C2211yescharacteristic, complemented
43C4×C22⋊D293C2211yescharacteristic
44C4⋊C4×D293—21no—
45C2×#17989779—21no—
46#22627555C2211yescharacteristic, complemented
47#22627553C2211yescharacteristic, complemented
48#22627549C2211yescharacteristic, complemented
49#22627548C2211yescharacteristic, complemented
50#22627546C2211yescharacteristic, complemented
51#22627544C2211yescharacteristic
52#22627536C2211yescharacteristic
53#22627534C2211yescharacteristic, complemented
54C2×#41×C293C2211yescharacteristic
55C2×Q16×C293C2211yescharacteristic, complemented
56C2×SD16×C293C2211yescharacteristic
57C22⋊Q8×C293C2211yescharacteristic, complemented
58#103×C293C2211yescharacteristic, complemented
59#102×C293C2211yescharacteristic
60C22⋊C8×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989803—22no—
63#17989803—41no—
64#17989802—22no—
65#17989802—41no—
66C2×Dic586C2311yescharacteristic
67C2×D1172—21no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293D411yescharacteristic
74D4×D293—22no—
75D4×D293—41no—
76C22×Dic293—21no—
77C2×C4×D293—21no—
78C2×C4×D293—21no—
79C2×C4×D293—21no—
80C2×C4×D293C2311yescharacteristic
81#17989792—22no—
82#17989791—41no—
83#17989790—22no—
84#17989789—22no—
85#17989789—22no—
86D2344—22no—
87#17989787D414yescomplemented
88#17989785—22no—
89C2×#14556873—22no—
90C2×#14556873C2311yescharacteristic
91C2×#14556873C2311yescharacteristic
92C4⋊Dic293—21no—
93C4⋊Dic293C2311yescharacteristic
94#17989781—21no—
95#17989781—21no—
96#17989781—21no—
97#17989781D411yescharacteristic
98C22⋊Dic293—21no—
99C22⋊Dic293D411yescharacteristic
100#17989779—21no—
101#17989779—21no—
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779D411yescharacteristic
107C4×Dic293C2311yescharacteristic
108#41×C293—22no—
109#41×C293—41no—
110C2×Q8×C293C2311yescharacteristic
111C2×Q8×C293C2311yescharacteristic
112C2×D4×C293—21no—
113C2×D4×C293C2311yescharacteristic
114C22×C4×C293—21no—
115C22×C4×C293C2311yescharacteristic
116Q16×C293—41no—
117SD16×C293—22no—
118C2×C8×C293C2311yescharacteristic
119C2×C8×C293C2311yescharacteristic
120C4⋊C4×C293—21no—
121C4⋊C4×C293C2311yescharacteristic
122C22⋊C4×C293—21no—
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—42no—
126C22×D293—42no—
127C22×D293—81no—
128C22×D293C2×D411yescharacteristic
129Dic586—21no—
130Dic586—21no—
131D1172—21no—
132D1172—21no—
133D1172—21no—
134D1172—42no—
135D1172C2×D412yes—
136C22⋊D293—41no—
137C22⋊D293—42no—
138C22⋊D293—42no—
139C22⋊D293—81no—
140C2×Dic293—21no—
141C2×Dic293—21no—
142C2×Dic293—81no—
143C2×Dic293C2×D411yescharacteristic
144C2×Dic293C2×D411yescharacteristic
145C2×Dic293C2×D411yescharacteristic
146C4×D293—21no—
147C4×D293—21no—
148C4×D293—42no—
149C4×D293—42no—
150C4×D293—42no—
151#14556873—21no—
152#14556873C2×D412yes—
153C23×C293—21no—
154C23×C293C2×D411yescharacteristic
155Q8×C293—21no—
156Q8×C293—21no—
157Q8×C293—21no—
158Q8×C293—21no—
159D4×C293—21no—
160D4×C293—42no—
161D4×C293C2×D412yes—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293—42no—
168C2×C4×C293C2×D411yescharacteristic
169C2×C4×C293C2411yescharacteristic
170C8×C293—21no—
171C8×C293—21no—
172C22×C293—21no—
173C22×C293—22no—
174C22×C293—21no—
175C22×C293—21no—
176C22×C293—41no—
177C22×C293—41no—
178C22×C293C22×D411yescharacteristic
179C2×D293—21no—
180C2×D293—22no—
181C2×D293—21no—
182C2×D293—21no—
183C2×D293—41no—
184C2×D293—41no—
185C2×D293—41no—
186C2×D293—41no—
187C2×D293—41no—
188C2×D293—84no—
189C4×C293—21no—
190C4×C293—41no—
191C4×C293—41no—
192C4×C293—41no—
193C4×C293C22×D411yescharacteristic
194C4×C293H(5,𝔽2)11yescharacteristic
195Dic293—21no—
196Dic293—21no—
197Dic293—21no—
198Dic293—41no—
199Dic293—41no—
200C2×C293—42no—
201C2×C293—41no—
202C2×C293D4211yescharacteristic
203C2×C293#57511yescharacteristic
204C2×C293#57711yescharacteristic
205D293—41no—
206D293—42no—
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#452D29311yescharacteristic, complemented
221#450—2931no—
222#449—2931no—
223#442—2931no—
224#435—2931no—
225C2×#41—2931no—
226C2×#41—2931no—
227C2×#41C2×D29311yescharacteristic
228C22×D4—2931no—
229AGL(1,ℤ/8)—5864no—
230#135—5864no—
231C2×Q16—2931no—
232C2×Q16C2×D29311yescharacteristic, complemented
233C2×SD16—2931no—
234C2×SD16—2931no—
235C2×SD16—2931no—
236C2×SD16—2931no—
237C2×SD16C2×D29311yescharacteristic
238C2×D8—2931no—
239C2×D8—2931no—
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⋊Q8C2×D29311yescharacteristic, complemented
250#121—2931no—
251#121—5861no—
252#121—5861no—
253#120—5861no—
254C4×D4—2931no—
255C4×D4—2931no—
256C4×D4—2931no—
257C2×C4⋊C4—5861no—
258C2×C22⋊C4—5861no—
259#106—2931no—
260C4⋊C8—2931no—
261#103—2931no—
262#103—2931no—
263#103C2×D29311yescharacteristic, complemented
264#102—2931no—
265#102—2931no—
266#102C2×D29311yescharacteristic
267C22⋊C8—2931no—
268C22⋊C8C2×D29311yescharacteristic, complemented
269#97—2931no—
270C24—5861no—
271#41—22no—
272#41—41no—
273#41—5862no—
274#41—5862no—
275#41—11721no—
276#41—11721no—
277C2×Q8—2931no—
278C2×Q8C22×D29311yescharacteristic
279C2×Q8C22×D29311yescharacteristic
280C2×D4—21no—
281C2×D4—2931no—
282C2×D4—2931no—
283C2×D4—2931no—
284C2×D4—5861no—
285C2×D4—5861no—
286C2×D4—5861no—
287C2×D4—5862no—
288C2×D4—5861no—
289C2×D4—11721no—
290C2×D4C22×D29311yescharacteristic
291C22×C4—21no—
292C22×C4—2931no—
293C22×C4—5861no—
294C22×C4—5861no—
295C22×C4—5861no—
296C22×C4—5861no—
297C22×C4C22×D29311yescharacteristic
298Q16—41no—
299Q16—5862no—
300SD16—22no—
301SD16—5862no—
302SD16—5862no—
303SD16—5862no—
304SD16—11721no—
305D8—2934no—
306D8—5862no—
307M16—5862no—
308C2×C8—2931no—
309C2×C8—2931no—
310C2×C8—5862no—
311C2×C8C22×D29311yescharacteristic
312C2×C8C22×D29311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2931no—
315C4⋊C4—2931no—
316C4⋊C4—5861no—
317C4⋊C4—5861no—
318C4⋊C4—5861no—
319C4⋊C4—5861no—
320C4⋊C4C22×D29311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—2931no—
323C22⋊C4—2931no—
324C22⋊C4—5861no—
325C22⋊C4—5861no—
326C22⋊C4—5861no—
327C22⋊C4—5861no—
328C22⋊C4—5861no—
329C22⋊C4—5861no—
330C22⋊C4—5861no—
331C42—2931no—
332C23—21no—
333C23—2931no—
334C23—23441no—
335C23—5861no—
336C23—5861no—
337C23—11722no—
338C23—11722no—
339C23D4×D29311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5861no—
345Q8—5861no—
346D4—21no—
347D4—42no—
348D4—2932no—
349D4—23441no—
350D4—5861no—
351D4—5861no—
352D4—5861no—
353D4—11722no—
354D4—11721no—
355D4—11722no—
356D4—11722no—
357D4D4×D29312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—2931no—
365C2×C4—2931no—
366C2×C4—2931no—
367C2×C4—23441no—
368C2×C4—5861no—
369C2×C4—5861no—
370C2×C4—5861no—
371C2×C4—5861no—
372C2×C4—11722no—
373C2×C4—11722no—
374C2×C4—11722no—
375C2×C4D4×D29311yescharacteristic
376C2×C4C23×D29311yescharacteristic
377C8—21no—
378C8—21no—
379C8—2932no—
380C8—5861no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—23444no—
388C22—5861no—
389C22—5862no—
390C22—5861no—
391C22—5861no—
392C22—11721no—
393C22—11721no—
394C22—11721no—
395C22—11721no—
396C22—11721no—
397C22C2×D4×D29311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—5861no—
403C4—5861no—
404C4—5861no—
405C4—11721no—
406C4—11721no—
407C4C2×D4×D29311yescharacteristic
408C4#2262766911yescharacteristic
409C2—41no—
410C2—42no—
411C2—23441no—
412C2—11721no—
413C2—11722no—
414C2#3403676111yescharacteristic
415C2#3403692611yescharacteristic
416C2#3403693811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#452×C293C2
socleC22×C293C22×D4
hypercenter#452D293
hypocenterC293#3267
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. C2×#22627644C2semidirect
3. #34036781C2semidirect
4. #34036767C2semidirect
5. #34036522C2semidirect
6. #34036386C2semidirect
7. #34036360C2semidirect
8. #34036347C2semidirect
9. #34036332C2semidirect
10. #34036290C2semidirect
11. #34036281C2semidirect
12. #34036276C2semidirect
13. #34036238C2semidirect
14. #34036197C2semidirect
15. #452×C293C2semidirect
17. C2×#17989802C22semidirect
21. C2×#17989792C22semidirect
25. C2×#17989789C22semidirect
29. C22×#14556873C22semidirect
30. #22627622C22semidirect
42. #22627586C22semidirect
46. #22627555C22semidirect
47. #22627553C22semidirect
48. #22627549C22semidirect
49. #22627548C22semidirect
50. #22627546C22semidirect
53. #22627534C22semidirect
55. C2×Q16×C293C22semidirect
57. C22⋊Q8×C293C22semidirect
58. #103×C293C22semidirect
60. C22⋊C8×C293C22semidirect
87. #17989787D4semidirect
208. C293#3267semidirect
220. #452D293semidirect
232. C2×Q16C2×D293semidirect
249. C22⋊Q8C2×D293semidirect
263. #103C2×D293semidirect
268. C22⋊C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C2×#226276442
#340367812
#340367672
#340365222
#340363862
#340363602
#340363472
#340363322
#340362902
#340362812
#340362762
#340362382
#340361972
#452×C2932
#3267293