Finite Group Databasebeta

← #49170297

Subgroups — #49170297

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×#22627653C211yesmaximal, characteristic, complemented
2C2×SD16×D293C211yesmaximal, characteristic, complemented
3Q8×D1172C211yesmaximal, characteristic, complemented
4#34036774C211yesmaximal, characteristic
5#34036523C211yesmaximal, characteristic, complemented
6#34036376C211yesmaximal, characteristic, complemented
7#34036371C211yesmaximal, characteristic, complemented
8#34036351C211yesmaximal, characteristic, complemented
9#34036334C211yesmaximal, characteristic, complemented
10#34036292C211yesmaximal, characteristic, complemented
11#34036286C211yesmaximal, characteristic, complemented
12#34036281C211yesmaximal, characteristic, complemented
13#34036269C211yesmaximal, characteristic, complemented
14#34036202C211yesmaximal, characteristic, complemented
15#440×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C2×Q8×D293C2211yescharacteristic
18C2×Q8×D293C2211yescharacteristic, complemented
19C2×D4×D293C2211yescharacteristic
20#22627653—24no—
21C2×Dic1172C2211yescharacteristic, complemented
22C2×Dic1172C2211yescharacteristic, complemented
23C2×#17989792C2211yescharacteristic, complemented
24C2×#17989791C2211yescharacteristic
25C2×#17989791C2211yescharacteristic
26C2×#17989790C2211yescharacteristic
27C2×#17989790C2211yescharacteristic
28C2×#17989789C2211yescharacteristic
29SD16×D293—24no—
30C2×#17989785C2211yescharacteristic, complemented
31C2×C8×D293C2211yescharacteristic
32#22627624—21no—
33#22627624C2211yescharacteristic, complemented
34#22627612C2211yescharacteristic
35#22627608—21no—
36#22627604—21no—
37#22627604—21no—
38#22627604C2211yescharacteristic
39#22627604C2211yescharacteristic
40#22627604C2211yescharacteristic, complemented
41#22627599C2211yescharacteristic
42#22627598—21no—
43#22627594—21no—
44C4×D1172—21no—
45C4×D1172C2211yescharacteristic
46C2×#17989779—21no—
47C4⋊#14556873C2211yescharacteristic, complemented
48#22627547C2211yescharacteristic, complemented
49#22627547C2211yescharacteristic, complemented
50#22627547C2211yescharacteristic, complemented
51#22627544C2211yescharacteristic
52#22627536C2211yescharacteristic
53#22627536C2211yescharacteristic, complemented
54C2×SD16×C293C2211yescharacteristic
55C4×Q8×C293C2211yescharacteristic, complemented
56C4×D4×C293C2211yescharacteristic
57#107×C293C2211yescharacteristic, complemented
58#103×C293C2211yescharacteristic, complemented
59#102×C293C2211yescharacteristic
60#97×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989801—22no—
63#17989801—41no—
64C2×Dic586—21no—
65C2×Dic586—21no—
66C2×Dic586C2311yescharacteristic
67C2×Dic586C2311yescharacteristic
68C2×Dic586C2311yescharacteristic
69Q8×D293—22no—
70Q8×D293—22no—
71Q8×D293—41no—
72Q8×D293—41no—
73C2×D1172—21no—
74C2×D1172D411yescharacteristic
75C2×D1172C2311yescharacteristic
76C2×C22⋊D293—21no—
77C2×C22⋊D293—21no—
78D4×D293—22no—
79D4×D293—41no—
80C22×Dic293—21no—
81C2×C4×D293—21no—
82C2×C4×D293—21no—
83C2×C4×D293C2311yescharacteristic
84C2×C4×D293C2311yescharacteristic
85Dic1172—22no—
86Dic1172—41no—
87#17989792—22no—
88#17989791—22no—
89#17989791—41no—
90#17989790—22no—
91#17989790—22no—
92#17989789—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×Q8×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C22×C4×C293—21no—
114SD16×C293D414yescomplemented
115C2×C8×C293C2311yescharacteristic
116C2×C8×C293C2311yescharacteristic
117C4⋊C4×C293—21no—
118C4⋊C4×C293C2311yescharacteristic
119C4⋊C4×C293C2311yescharacteristic
120C22⋊C4×C293—21no—
121C42×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—
135Dic586—42no—
136Dic586—42no—
137D1172—22no—
138D1172—21no—
139D1172—21no—
140D1172—81no—
141C22⋊D293—42no—
142C22⋊D293—42no—
143C2×Dic293—21no—
144C2×Dic293—21no—
145C2×Dic293—21no—
146C2×Dic293—42no—
147C2×Dic293C2×D411yescharacteristic
148C2×Dic293C2×D411yescharacteristic
149C4×D293—21no—
150C4×D293—21no—
151C4×D293—21no—
152C4×D293—21no—
153C4×D293—42no—
154C4×D293—42no—
155#14556873—21no—
156#14556873—21no—
157C23×C293—21no—
158Q8×C293—21no—
159Q8×C293C2×D412yes—
160D4×C293—21no—
161D4×C293C2×D412yes—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—81no—
166C2×C4×C293C2×D411yescharacteristic
167C2×C4×C293C2×D411yescharacteristic
168C2×C4×C293C2411yescharacteristic
169C8×C293—21no—
170C8×C293C2×D412yes—
171C22×C293—22no—
172C22×C293—41no—
173C22×C293—41no—
174C22×C293C22×D411yescharacteristic
175C2×D293—21no—
176C2×D293—21no—
177C2×D293—21no—
178C2×D293—21no—
179C2×D293—21no—
180C2×D293—41no—
181C2×D293—41no—
182C2×D293—41no—
183C2×D293—84no—
184C2×D293#13712yescomplemented
185C4×C293—22no—
186C4×C293—21no—
187C4×C293—21no—
188C4×C293—41no—
189C4×C293—41no—
190C4×C293C22×D411yescharacteristic
191C4×C293H(5,𝔽2)11yescharacteristic
192Dic293—21no—
193Dic293—21no—
194Dic293—41no—
195Dic293—41no—
196Dic293—41no—
197C2×C293—42no—
198C2×C293D4211yescharacteristic
199C2×C293C2×#13711yescharacteristic
200C2×C293#57711yescharacteristic
201D293—22no—
202D293—41no—
203D293—81no—
204C293#326611yescharacteristic, complemented, Sylow
205#3266—2931nomaximal, Sylow
206C2×#137—2931no—
207C22×SD16—2931no—
208D4×Q8—2931no—
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#440D29311yescharacteristic, complemented
220#435—2931no—
221C2×#41—2931no—
222C22×Q8—2931no—
223C22×Q8—2931no—
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—5864no—
235C2×SD16C2×D29311yescharacteristic
236C2×M16—2931no—
237C22×C8—2931no—
238C4⋊Q8—2931no—
239C4⋊Q8—5861no—
240#124—2931no—
241#123—5861no—
242C22⋊Q8—2931no—
243C22⋊Q8—2931no—
244C22⋊Q8—2931no—
245C22⋊Q8—5861no—
246C22⋊Q8—5861no—
247#121—2931no—
248#121—5861no—
249#120—5861no—
250C4×Q8C2×D29311yescharacteristic, complemented
251C4×D4—2931no—
252C4×D4—5861no—
253C4×D4C2×D29311yescharacteristic
254C2×C22⋊C4—5861no—
255#107C2×D29311yescharacteristic, complemented
256C4⋊C8—2931no—
257#103—2931no—
258#103—2931no—
259#103—2931no—
260#103C2×D29311yescharacteristic, complemented
261#102—2931no—
262#102C2×D29311yescharacteristic
263C22⋊C8—2931no—
264C22⋊C8—2931no—
265#97C2×D29311yescharacteristic, complemented
266C24—5861no—
267#41—5862no—
268#41—11721no—
269C2×Q8—2931no—
270C2×Q8—2931no—
271C2×Q8—2931no—
272C2×Q8—5861no—
273C2×Q8—5861no—
274C2×Q8—5862no—
275C2×Q8—5862no—
276C2×Q8—11721no—
277C2×Q8—11721no—
278C2×Q8C22×D29311yescharacteristic
279C2×D4—2931no—
280C2×D4—2931no—
281C2×D4—5861no—
282C2×D4—5861no—
283C2×D4—5862no—
284C2×D4—5861no—
285C2×D4—11721no—
286C2×D4C22×D29311yescharacteristic
287C22×C4—21no—
288C22×C4—2931no—
289C22×C4—2931no—
290C22×C4—5861no—
291C22×C4—5861no—
292C22×C4—5861no—
293Q16—5862no—
294Q16—5862no—
295Q16—11721no—
296SD16—5862no—
297SD16—5862no—
298SD16—5862no—
299SD16—5862no—
300SD16—11721no—
301SD16D117214yescomplemented
302M16—5862no—
303C2×C8—2931no—
304C2×C8—2931no—
305C2×C8—5862no—
306C2×C8C22×D29311yescharacteristic
307C2×C8C22×D29311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2931no—
310C4⋊C4—5861no—
311C4⋊C4—5861no—
312C4⋊C4—5861no—
313C4⋊C4—5861no—
314C4⋊C4—5861no—
315C4⋊C4C22×D29311yescharacteristic
316C4⋊C4C22×D29311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2931no—
319C22⋊C4—2931no—
320C22⋊C4—5861no—
321C22⋊C4—5861no—
322C22⋊C4—5861no—
323C22⋊C4—5861no—
324C22⋊C4—5861no—
325C22⋊C4—5861no—
326C42—21no—
327C42C22×D29311yescharacteristic
328C23—21no—
329C23—2931no—
330C23—2931no—
331C23—23441no—
332C23—5861no—
333C23—11722no—
334C23—11722no—
335Q8—21no—
336Q8—5861no—
337Q8—5861no—
338Q8—5861no—
339Q8—5861no—
340Q8—5861no—
341Q8—5861no—
342Q8—11722no—
343Q8—11722no—
344Q8C2×D117212yes—
345D4—21no—
346D4—23441no—
347D4—5862no—
348D4—5861no—
349D4—5861no—
350D4—11722no—
351D4—11722no—
352D4C2×D117212yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—2931no—
358C2×C4—2931no—
359C2×C4—5861no—
360C2×C4—5861no—
361C2×C4—5861no—
362C2×C4—5861no—
363C2×C4—5861no—
364C2×C4—5861no—
365C2×C4—5861no—
366C2×C4—11722no—
367C2×C4—11722no—
368C2×C4—11722no—
369C2×C4D4×D29311yescharacteristic
370C2×C4D4×D29311yescharacteristic
371C2×C4C23×D29311yescharacteristic
372C8—21no—
373C8—5861no—
374C8—5861no—
375C8C2×D117212yes—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—2932no—
380C22—23444no—
381C22—5861no—
382C22—5861no—
383C22—5861no—
384C22—5861no—
385C22—5861no—
386C22—11721no—
387C22—11721no—
388C22—11721no—
389C22C2×D4×D29311yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—5861no—
396C4—5861no—
397C4—11721no—
398C4—11721no—
399C4—11721no—
400C4C22×D117211yescharacteristic
401C4#2262767011yescharacteristic
402C2—42no—
403C2—23441no—
404C2—5862no—
405C2—11721no—
406C2D4×D117211yescharacteristic
407C2#137×D29311yescharacteristic
408C2#3403693911yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#440×C293C2
socleC22×C293C22×D4
prime radical#34036774C2
hypercenter#440D293
hypocenterC293#3266
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#22627653C2semidirect
2. C2×SD16×D293C2semidirect
3. Q8×D1172C2semidirect
5. #34036523C2semidirect
6. #34036376C2semidirect
7. #34036371C2semidirect
8. #34036351C2semidirect
9. #34036334C2semidirect
10. #34036292C2semidirect
11. #34036286C2semidirect
12. #34036281C2semidirect
13. #34036269C2semidirect
14. #34036202C2semidirect
15. #440×C293C2semidirect
18. C2×Q8×D293C22semidirect
21. C2×Dic1172C22semidirect
22. C2×Dic1172C22semidirect
23. C2×#17989792C22semidirect
30. C2×#17989785C22semidirect
33. #22627624C22semidirect
40. #22627604C22semidirect
47. C4⋊#14556873C22semidirect
48. #22627547C22semidirect
49. #22627547C22semidirect
50. #22627547C22semidirect
53. #22627536C22semidirect
55. C4×Q8×C293C22semidirect
57. #107×C293C22semidirect
58. #103×C293C22semidirect
60. #97×C293C22semidirect
114. SD16×C293D4semidirect
184. C2×D293#137semidirect
204. C293#3266semidirect
219. #440D293semidirect
250. C4×Q8C2×D293semidirect
255. #107C2×D293semidirect
260. #103C2×D293semidirect
265. #97C2×D293semidirect
301. SD16D1172semidirect

Maximal subgroups

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

subgroupindex
C2×#226276532
C2×SD16×D2932
Q8×D11722
#340367742
#340365232
#340363762
#340363712
#340363512
#340363342
#340362922
#340362862
#340362812
#340362692
#340362022
#440×C2932
#3266293