Finite Group Databasebeta

← #49170296

Subgroups — #49170296

Order 37504.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627655C211yesmaximal, characteristic, complemented
2C22×#17989789C211yesmaximal, characteristic, complemented
3#34036775C211yesmaximal, characteristic
4#34036521C211yesmaximal, characteristic, complemented
5#34036377C211yesmaximal, characteristic, complemented
6#34036368C211yesmaximal, characteristic, complemented
7#34036351C211yesmaximal, characteristic, complemented
8#34036331C211yesmaximal, characteristic, complemented
9#34036289C211yesmaximal, characteristic, complemented
10#34036285C211yesmaximal, characteristic, complemented
11#34036279C211yesmaximal, characteristic, complemented
12#34036268C211yesmaximal, characteristic, complemented
13#34036227C211yesmaximal, characteristic, complemented
14#34036201C211yesmaximal, characteristic, complemented
15D4×Q8×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18#22627655—24no—
19C2×#17989792C2211yescharacteristic, complemented
20C2×#17989792C2211yescharacteristic, complemented
21C2×#17989792C2211yescharacteristic, complemented
22C2×#17989789—24no—
23C2×#17989789C2211yescharacteristic
24C2×#17989789C2211yescharacteristic
25C2×#17989789C2211yescharacteristic
26C2×#17989789C2211yescharacteristic
27C2×#17989789C2211yescharacteristic
28C2×#17989789C2211yescharacteristic
29C2×#17989786C2211yescharacteristic, complemented
30C22×#14556873C2211yescharacteristic
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#22627548C2211yescharacteristic, complemented
43#22627548C2211yescharacteristic, complemented
44#22627547C2211yescharacteristic, complemented
45#22627544C2211yescharacteristic
46#22627544C2211yescharacteristic
47C22⋊#14556873C2211yescharacteristic
48C22⋊#14556873C2211yescharacteristic, complemented
49#22627535C2211yescharacteristic, complemented
50C22×Q8×C293C2211yescharacteristic
51C22×Q8×C293C2211yescharacteristic, complemented
52C4⋊Q8×C293—21no—
53C4⋊Q8×C293C2211yescharacteristic, complemented
54C22⋊Q8×C293—21no—
55C22⋊Q8×C293—21no—
56C22⋊Q8×C293C2211yescharacteristic
57C22⋊Q8×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#17989792—22no—
76#17989792—41no—
77#17989789—22no—
78#17989789—22no—
79#17989789—22no—
80#17989789—22no—
81#17989789—41no—
82#17989789D414yescomplemented
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—
96C2×Q8×C293—21no—
97C2×Q8×C293—21no—
98C2×Q8×C293—22no—
99C2×Q8×C293—22no—
100C2×Q8×C293—41no—
101C2×Q8×C293—41no—
102C2×Q8×C293C2311yescharacteristic
103C2×Q8×C293C2311yescharacteristic
104C2×Q8×C293C2311yescharacteristic
105C2×D4×C293D411yescharacteristic
106C22×C4×C293—21no—
107C22×C4×C293—21no—
108C22×C4×C293C2311yescharacteristic
109C22×C4×C293C2311yescharacteristic
110C4⋊C4×C293—21no—
111C4⋊C4×C293—21no—
112C4⋊C4×C293—21no—
113C4⋊C4×C293—21no—
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×C293C2×D411yescharacteristic
146C23×C293C2×D411yescharacteristic
147Q8×C293—21no—
148Q8×C293—21no—
149Q8×C293—21no—
150Q8×C293—21no—
151Q8×C293—21no—
152Q8×C293—42no—
153Q8×C293—42no—
154Q8×C293C2×D412yes—
155D4×C293—22no—
156C2×C4×C293—21no—
157C2×C4×C293—21no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—21no—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—42no—
165C2×C4×C293—42no—
166C2×C4×C293C2×D411yescharacteristic
167C2×C4×C293C2×D411yescharacteristic
168C2×C4×C293C2×D411yescharacteristic
169C2×C4×C293C2×D411yescharacteristic
170C2×C4×C293C2411yescharacteristic
171C22×C293—21no—
172C22×C293—21no—
173C22×C293—21no—
174C22×C293—21no—
175C22×C293—21no—
176C22×C293#13712yescomplemented
177C22×C293C22×D411yescharacteristic
178C2×D293—22no—
179C2×D293—41no—
180C2×D293—41no—
181C2×D293—41no—
182C2×D293—41no—
183C2×D293—41no—
184C2×D293—84no—
185C4×C293—21no—
186C4×C293—21no—
187C4×C293—22no—
188C4×C293—21no—
189C4×C293—21no—
190C4×C293—41no—
191C4×C293—41no—
192C4×C293—41no—
193C4×C293C22×D411yescharacteristic
194C4×C293H(5,𝔽2)11yescharacteristic
195Dic293—41no—
196Dic293—41no—
197C2×C293—22no—
198C2×C293—41no—
199C2×C293D4211yescharacteristic
200C2×C293C2×#13711yescharacteristic
201C2×C293#57711yescharacteristic
202D293—42no—
203D293—81no—
204C293#326611yescharacteristic, complemented, Sylow
205#3266—2931nomaximal, Sylow
206C2×#137—2931no—
207C22×SD16—2931no—
208D4×Q8D29311yescharacteristic, complemented
209#546—2931no—
210#497—2931no—
211#468—2931no—
212#467—2931no—
213#464—2931no—
214#461—2931no—
215#452—2931no—
216#451—2931no—
217#450—2931no—
218#448—2931no—
219#440—2931no—
220#435—2931no—
221C2×#41—2931no—
222C22×Q8C2×D29311yescharacteristic
223C22×Q8C2×D29311yescharacteristic, complemented
224C22×D4—2931no—
225#137—5864no—
226C2×Q16—2931no—
227C2×Q16—2931no—
228C2×Q16—2931no—
229C2×SD16—2931no—
230C2×SD16—2931no—
231C2×SD16—2931no—
232C2×SD16—2931no—
233C2×SD16—2931no—
234C2×SD16—2931no—
235C2×SD16—5864no—
236C2×M16—2931no—
237C22×C8—2931no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D29311yescharacteristic, complemented
240#124—2931no—
241#123—5861no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—2931no—
245C22⋊Q8C2×D29311yescharacteristic
246C22⋊Q8C2×D29311yescharacteristic, complemented
247#121—2931no—
248#121—5861no—
249#120—5861no—
250C4×Q8C2×D29311yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—2931no—
253C4×D4C2×D29311yescharacteristic
254C2×C22⋊C4—5861no—
255#107—2931no—
256C4⋊C8—2931no—
257#103—2931no—
258#103—2931no—
259#103—2931no—
260#103—2931no—
261#102—2931no—
262#102—2931no—
263C22⋊C8—2931no—
264C22⋊C8—2931no—
265#97—2931no—
266C24—5861no—
267#41—5862no—
268#41—11721no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—2931no—
276C2×Q8C22×D29311yescharacteristic
277C2×Q8C22×D29311yescharacteristic
278C2×Q8C22×D29311yescharacteristic
279C2×D4—2931no—
280C2×D4—2931no—
281C2×D4—5862no—
282C2×D4—5861no—
283C2×D4—5861no—
284C2×D4—5861no—
285C2×D4—11721no—
286C2×D4C22⋊D29311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—5861no—
290C22×C4—5861no—
291C22×C4C22×D29311yescharacteristic
292C22×C4C22×D29311yescharacteristic
293Q16—5862no—
294Q16—5862no—
295Q16—11721no—
296SD16—2934no—
297SD16—5862no—
298SD16—5862no—
299SD16—5862no—
300SD16—5862no—
301SD16—11721no—
302M16—5862no—
303C2×C8—2931no—
304C2×C8—2931no—
305C2×C8—2931no—
306C2×C8—2931no—
307C2×C8—5862no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—2931no—
314C4⋊C4—5861no—
315C4⋊C4C22⋊D29311yescharacteristic
316C4⋊C4C22×D29311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—5861no—
320C22⋊C4—5861no—
321C22⋊C4—5861no—
322C22⋊C4—5861no—
323C22⋊C4—5861no—
324C22⋊C4C22⋊D29311yescharacteristic
325C22⋊C4C22⋊D29311yescharacteristic
326C42—21no—
327C42C22×D29311yescharacteristic
328C23—23441no—
329C23—5861no—
330C23—5861no—
331C23—11722no—
332C23—11722no—
333C23C2×C22⋊D29311yescharacteristic
334C23C2×C22⋊D29311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—5861no—
343Q8—5861no—
344Q8D4×D29312yes—
345D4—22no—
346D4—2932no—
347D4—23441no—
348D4—5861no—
349D4—5861no—
350D4—5861no—
351D4—11722no—
352D4—11722no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—23441no—
364C2×C4—5861no—
365C2×C4—5861no—
366C2×C4—11722no—
367C2×C4C2×C22⋊D29311yescharacteristic
368C2×C4C2×C22⋊D29311yescharacteristic
369C2×C4C2×C22⋊D29311yescharacteristic
370C2×C4C2×C22⋊D29311yescharacteristic
371C2×C4C23×D29311yescharacteristic
372C8—2932no—
373C8—5861no—
374C8—5861no—
375C8—5861no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—23444no—
382C22—5862no—
383C22—11721no—
384C22—11721no—
385C22—11721no—
386C22—11721no—
387C22—11721no—
388C22#2262765512yescomplemented
389C22C22×C22⋊D29311yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—11721no—
399C4—11721no—
400C4C2×D4×D29311yescharacteristic
401C4#2262766911yescharacteristic
402C2—22no—
403C2—41no—
404C2—23441no—
405C2—11722no—
406C2D4×C22⋊D29311yescharacteristic
407C2C2×#2262765511yescharacteristic
408C2#3403693711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingD4×Q8×C293C2
socleC22×C293C22×D4
prime radical#34036775C2
hypercenterD4×Q8D293
hypocenterC293#3266
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22627655C2semidirect
2. C22×#17989789C2semidirect
4. #34036521C2semidirect
5. #34036377C2semidirect
6. #34036368C2semidirect
7. #34036351C2semidirect
8. #34036331C2semidirect
9. #34036289C2semidirect
10. #34036285C2semidirect
11. #34036279C2semidirect
12. #34036268C2semidirect
13. #34036227C2semidirect
14. #34036201C2semidirect
15. D4×Q8×C293C2semidirect
19. C2×#17989792C22semidirect
20. C2×#17989792C22semidirect
21. C2×#17989792C22semidirect
29. C2×#17989786C22semidirect
39. #22627557C22semidirect
40. C4⋊#14556873C22semidirect
41. #22627548C22semidirect
42. #22627548C22semidirect
43. #22627548C22semidirect
44. #22627547C22semidirect
48. C22⋊#14556873C22semidirect
49. #22627535C22semidirect
51. C22×Q8×C293C22semidirect
53. C4⋊Q8×C293C22semidirect
57. C22⋊Q8×C293C22semidirect
58. C4×Q8×C293C22semidirect
82. #17989789D4semidirect
176. C22×C293#137semidirect
204. C293#3266semidirect
208. D4×Q8D293semidirect
223. C22×Q8C2×D293semidirect
239. C4⋊Q8C2×D293semidirect
246. C22⋊Q8C2×D293semidirect
250. C4×Q8C2×D293semidirect
388. C22#22627655semidirect

Maximal subgroups

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

subgroupindex
C2×#226276552
C22×#179897892
#340367752
#340365212
#340363772
#340363682
#340363512
#340363312
#340362892
#340362852
#340362792
#340362682
#340362272
#340362012
D4×Q8×C2932
#3266293