Finite Group Databasebeta

← #49170298

Subgroups — #49170298

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
2C2×SD16×D293C211yesmaximal, characteristic, complemented
3Q8×C22⋊D293C211yesmaximal, characteristic, complemented
4#34036776C211yesmaximal, characteristic
5#34036520C211yesmaximal, characteristic, complemented
6#34036373C211yesmaximal, characteristic, complemented
7#34036369C211yesmaximal, characteristic, complemented
8#34036349C211yesmaximal, characteristic, complemented
9#34036333C211yesmaximal, characteristic, complemented
10#34036293C211yesmaximal, characteristic, complemented
11#34036286C211yesmaximal, characteristic, complemented
12#34036278C211yesmaximal, characteristic, complemented
13#34036232C211yesmaximal, characteristic, complemented
14#34036203C211yesmaximal, characteristic, complemented
15#448×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C2×Q8×D293C2211yescharacteristic
18C2×D4×D293C2211yescharacteristic
19#22627655—24no—
20C2×Dic1172C2211yescharacteristic, complemented
21C2×#17989792C2211yescharacteristic, complemented
22C2×#17989792C2211yescharacteristic, complemented
23C2×#17989791C2211yescharacteristic
24C2×#17989790C2211yescharacteristic
25C2×#17989789C2211yescharacteristic
26C2×#17989789C2211yescharacteristic
27SD16×D293—24no—
28C2×#17989786C2211yescharacteristic, complemented
29C2×C8×D293C2211yescharacteristic
30#22627622—21no—
31#22627622C2211yescharacteristic, complemented
32#22627611C2211yescharacteristic
33#22627606—21no—
34#22627604C2211yescharacteristic
35#22627603—21no—
36#22627603C2211yescharacteristic, complemented
37#22627600—21no—
38#22627600C2211yescharacteristic
39#22627595—21no—
40#22627592—21no—
41Q8×Dic293C2211yescharacteristic, complemented
42#22627586C2211yescharacteristic
43C4×C22⋊D293—21no—
44C4×C22⋊D293C2211yescharacteristic
45C22⋊C4×D293—21no—
46#22627557C2211yescharacteristic, complemented
47#22627553C2211yescharacteristic, complemented
48#22627549C2211yescharacteristic, complemented
49#22627547C2211yescharacteristic, complemented
50#22627546C2211yescharacteristic
51#22627545C2211yescharacteristic
52#22627536C2211yescharacteristic
53#22627534C2211yescharacteristic, complemented
54C22×Q8×C293C2211yescharacteristic, complemented
55C2×SD16×C293C2211yescharacteristic
56C2×SD16×C293C2211yescharacteristic
57#121×C293C2211yescharacteristic
58#103×C293C2211yescharacteristic, complemented
59#103×C293C2211yescharacteristic, complemented
60C22⋊C8×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989802—22no—
63#17989802—41no—
64C2×Dic586—21no—
65C2×Dic586C2311yescharacteristic
66C2×Dic586C2311yescharacteristic
67Q8×D293—22no—
68Q8×D293—41no—
69C2×D1172C2311yescharacteristic
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C22⋊D293D411yescharacteristic
73D4×D293—22no—
74D4×D293—41no—
75C2×C4×D293—21no—
76C2×C4×D293—21no—
77C2×C4×D293—21no—
78C2×C4×D293C2311yescharacteristic
79Dic1172—41no—
80#17989792—22no—
81#17989792—22no—
82#17989791—22no—
83#17989790—22no—
84#17989789—22no—
85#17989789D414yescomplemented
86#17989786—22no—
87C2×#14556873C2311yescharacteristic
88C2×#14556873C2311yescharacteristic
89C8×D293—22no—
90C4⋊Dic293—21no—
91C4⋊Dic293—21no—
92C4⋊Dic293C2311yescharacteristic
93#17989781—21no—
94#17989781—21no—
95#17989781—21no—
96#17989781—21no—
97#17989781D411yescharacteristic
98C22⋊Dic293—21no—
99C22⋊Dic293—21no—
100C22⋊Dic293—21no—
101C22⋊Dic293D411yescharacteristic
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779D411yescharacteristic
106C4×Dic293—21no—
107C4×Dic293C2311yescharacteristic
108C2×Q8×C293—21no—
109C2×Q8×C293—22no—
110C2×Q8×C293—41no—
111C2×Q8×C293C2311yescharacteristic
112C2×Q8×C293C2311yescharacteristic
113C2×D4×C293—21no—
114C2×D4×C293C2311yescharacteristic
115C22×C4×C293—21no—
116C22×C4×C293C2311yescharacteristic
117SD16×C293—22no—
118SD16×C293—41no—
119C2×C8×C293C2311yescharacteristic
120C2×C8×C293C2311yescharacteristic
121C4⋊C4×C293C2311yescharacteristic
122C22⋊C4×C293—21no—
123C22×D293—21no—
124C22×D293—42no—
125C22×D293—42no—
126C22×D293—81no—
127C22×D293C2×D411yescharacteristic
128Dic586—21no—
129Dic586—21no—
130Dic586—21no—
131Dic586—21no—
132Dic586—42no—
133D1172—21no—
134D1172C2×D412yes—
135C22⋊D293—22no—
136C22⋊D293—42no—
137C22⋊D293—42no—
138C2×Dic293—21no—
139C2×Dic293—21no—
140C2×Dic293—21no—
141C2×Dic293C2×D411yescharacteristic
142C2×Dic293C2×D411yescharacteristic
143C2×Dic293C2×D411yescharacteristic
144C4×D293—21no—
145C4×D293—21no—
146C4×D293—42no—
147C4×D293—42no—
148C4×D293—81no—
149#14556873—21no—
150#14556873C2×D412yes—
151C23×C293—21no—
152C23×C293C2×D411yescharacteristic
153Q8×C293—21no—
154Q8×C293—21no—
155Q8×C293—21no—
156Q8×C293—42no—
157Q8×C293C2×D412yes—
158D4×C293—21no—
159D4×C293—21no—
160D4×C293—81no—
161C2×C4×C293—21no—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—42no—
167C2×C4×C293C2×D411yescharacteristic
168C2×C4×C293C2411yescharacteristic
169C8×C293—21no—
170C8×C293—21no—
171C22×C293—21no—
172C22×C293—21no—
173C22×C293—21no—
174C22×C293—41no—
175C22×C293—41no—
176C22×C293—41no—
177C22×C293C22×D411yescharacteristic
178C2×D293—21no—
179C2×D293—21no—
180C2×D293—22no—
181C2×D293—41no—
182C2×D293—41no—
183C2×D293—84no—
184C2×D293#13712yescomplemented
185C4×C293—21no—
186C4×C293—22no—
187C4×C293—41no—
188C4×C293—41no—
189C4×C293C22×D411yescharacteristic
190C4×C293H(5,𝔽2)11yescharacteristic
191Dic293—21no—
192Dic293—21no—
193Dic293—21no—
194Dic293—41no—
195Dic293—41no—
196Dic293—41no—
197C2×C293—41no—
198C2×C293—81no—
199C2×C293D4211yescharacteristic
200C2×C293C2×#13711yescharacteristic
201C2×C293#57711yescharacteristic
202D293—22no—
203D293—42no—
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#448D29311yescharacteristic, complemented
219#440—2931no—
220#435—2931no—
221C2×#41—2931no—
222C22×Q8—2931no—
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—5864no—
234C2×SD16C2×D29311yescharacteristic
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—5861no—
248#121C2×D29311yescharacteristic
249#120—5861no—
250C4×Q8—2931no—
251C4×D4—2931no—
252C4×D4—2931no—
253C4×D4—5861no—
254C2×C22⋊C4—5861no—
255#107—2931no—
256C4⋊C8—2931no—
257#103—2931no—
258#103—2931no—
259#103C2×D29311yescharacteristic, complemented
260#103C2×D29311yescharacteristic, complemented
261#102—2931no—
262#102—2931no—
263C22⋊C8—2931no—
264C22⋊C8C2×D29311yescharacteristic, complemented
265#97—2931no—
266C24—5861no—
267#41—5862no—
268#41—11721no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2931no—
273C2×Q8—2931no—
274C2×Q8—5861no—
275C2×Q8—5862no—
276C2×Q8—11721no—
277C2×Q8C22×D29311yescharacteristic
278C2×Q8C22×D29311yescharacteristic
279C2×D4—21no—
280C2×D4—2931no—
281C2×D4—2931no—
282C2×D4—5861no—
283C2×D4—5862no—
284C2×D4—5861no—
285C2×D4—11721no—
286C2×D4C22×D29311yescharacteristic
287C22×C4—21no—
288C22×C4—2931no—
289C22×C4—5861no—
290C22×C4—5861no—
291C22×C4—5861no—
292C22×C4C22×D29311yescharacteristic
293Q16—5862no—
294Q16—5862no—
295Q16—11721no—
296SD16—22no—
297SD16—41no—
298SD16—2934no—
299SD16—5862no—
300SD16—5862no—
301SD16—5862no—
302M16—5862no—
303C2×C8—2931no—
304C2×C8—2931no—
305C2×C8—5862no—
306C2×C8C22×D29311yescharacteristic
307C2×C8C22×D29311yescharacteristic
308C4⋊C4—2931no—
309C4⋊C4—2931no—
310C4⋊C4—5861no—
311C4⋊C4—5861no—
312C4⋊C4—5861no—
313C4⋊C4—5861no—
314C4⋊C4—5861no—
315C4⋊C4—5861no—
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—2931no—
327C42—5861no—
328C23—21no—
329C23—2931no—
330C23—23441no—
331C23—5861no—
332C23—11722no—
333C23—11722no—
334C23D4×D29311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—5861no—
340Q8—5861no—
341Q8—5861no—
342Q8—5861no—
343Q8—11722no—
344Q8D4×D29312yes—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—2932no—
349D4—5862no—
350D4—5861no—
351D4—11722no—
352D4—11722no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—2931no—
360C2×C4—2931no—
361C2×C4—2931no—
362C2×C4—23441no—
363C2×C4—5861no—
364C2×C4—5861no—
365C2×C4—5861no—
366C2×C4—5861no—
367C2×C4—5861no—
368C2×C4—11722no—
369C2×C4—11722no—
370C2×C4D4×D29311yescharacteristic
371C2×C4C23×D29311yescharacteristic
372C8—21no—
373C8—21no—
374C8—2932no—
375C8—5861no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—2932no—
383C22—23444no—
384C22—5861no—
385C22—5861no—
386C22—5862no—
387C22—11721no—
388C22—11721no—
389C22C2×D4×D29311yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—5861no—
395C4—5861no—
396C4—5861no—
397C4—11721no—
398C4—11721no—
399C4—11721no—
400C4C2×D4×D29311yescharacteristic
401C4#2262766911yescharacteristic
402C2—41no—
403C2—81no—
404C2—5862no—
405C2—11722no—
406C2#3403676111yescharacteristic
407C2#137×D29311yescharacteristic
408C2#3403693511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#448×C293C2
socleC22×C293C22×D4
prime radical#34036776C2
hypercenter#448D293
hypocenterC293#3266
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#22627655C2semidirect
2. C2×SD16×D293C2semidirect
3. Q8×C22⋊D293C2semidirect
5. #34036520C2semidirect
6. #34036373C2semidirect
7. #34036369C2semidirect
8. #34036349C2semidirect
9. #34036333C2semidirect
10. #34036293C2semidirect
11. #34036286C2semidirect
12. #34036278C2semidirect
13. #34036232C2semidirect
14. #34036203C2semidirect
15. #448×C293C2semidirect
20. C2×Dic1172C22semidirect
21. C2×#17989792C22semidirect
22. C2×#17989792C22semidirect
28. C2×#17989786C22semidirect
31. #22627622C22semidirect
36. #22627603C22semidirect
41. Q8×Dic293C22semidirect
46. #22627557C22semidirect
47. #22627553C22semidirect
48. #22627549C22semidirect
49. #22627547C22semidirect
53. #22627534C22semidirect
54. C22×Q8×C293C22semidirect
58. #103×C293C22semidirect
59. #103×C293C22semidirect
60. C22⋊C8×C293C22semidirect
85. #17989789D4semidirect
184. C2×D293#137semidirect
204. C293#3266semidirect
218. #448D293semidirect
223. C22×Q8C2×D293semidirect
259. #103C2×D293semidirect
260. #103C2×D293semidirect
264. C22⋊C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276552
C2×SD16×D2932
Q8×C22⋊D2932
#340367762
#340365202
#340363732
#340363692
#340363492
#340363332
#340362932
#340362862
#340362782
#340362322
#340362032
#448×C2932
#3266293