Finite Group Databasebeta

← #49170356

Subgroups — #49170356

Order 37504.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627655C211yesmaximal, characteristic, complemented
2C22×#17989792C211yesmaximal, characteristic, complemented
3#34036788C211yesmaximal, characteristic
4#34036521C211yesmaximal, characteristic, complemented
5#34036385C211yesmaximal, characteristic, complemented
6#34036360C211yesmaximal, characteristic, complemented
7#34036351C211yesmaximal, characteristic, complemented
8#34036336C211yesmaximal, characteristic, complemented
9#34036289C211yesmaximal, characteristic, complemented
10#34036285C211yesmaximal, characteristic, complemented
11#34036284C211yesmaximal, characteristic, complemented
12#34036268C211yesmaximal, characteristic, complemented
13#34036234C211yesmaximal, characteristic, complemented
14#34036201C211yesmaximal, characteristic, complemented
15D4×Q8×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C22×Dic586C2211yescharacteristic
18#22627655—24no—
19C2×#17989792—24no—
20C2×#17989792C2211yescharacteristic
21C2×#17989792C2211yescharacteristic
22C2×#17989792C2211yescharacteristic
23C2×#17989792C2211yescharacteristic
24C2×#17989792C2211yescharacteristic
25C2×#17989792C2211yescharacteristic
26C2×#17989789C2211yescharacteristic
27C2×#17989789C2211yescharacteristic
28C2×#17989789C2211yescharacteristic
29C2×#17989786C2211yescharacteristic
30C22×#14556873C2211yescharacteristic
31#22627612C2211yescharacteristic
32#22627610—21no—
33#22627603C2211yescharacteristic
34#22627602—21no—
35#22627598C2211yescharacteristic
36#22627596—21no—
37C4×Dic586C2211yescharacteristic
38#22627584—21no—
39#22627555C2211yescharacteristic
40C4⋊#14556873C2211yescharacteristic
41#22627548C2211yescharacteristic
42#22627548C2211yescharacteristic
43#22627548C2211yescharacteristic
44#22627547C2211yescharacteristic
45#22627547C2211yescharacteristic
46#22627544C2211yescharacteristic
47C22⋊#14556873C2211yescharacteristic
48C22⋊#14556873C2211yescharacteristic
49#22627535C2211yescharacteristic
50C22×Q8×C293C2211yescharacteristic
51C22×Q8×C293C2211yescharacteristic
52C4⋊Q8×C293—21no—
53C4⋊Q8×C293C2211yescharacteristic
54C22⋊Q8×C293—21no—
55C22⋊Q8×C293—21no—
56C22⋊Q8×C293C2211yescharacteristic
57C22⋊Q8×C293C2211yescharacteristic
58C4×Q8×C293C2211yescharacteristic
59C4×D4×C293—21no—
60C4×D4×C293C2211yescharacteristic
61#17989802—22no—
62#17989802—41no—
63C2×Dic586—22no—
64C2×Dic586—21no—
65C2×Dic586—41no—
66C2×Dic586C2311yescharacteristic
67C2×Dic586C2311yescharacteristic
68C2×D1172C2311yescharacteristic
69C2×C22⋊D293—21no—
70C2×C22⋊D293—21no—
71C22×Dic293—21no—
72C2×C4×D293—21no—
73#17989792—22no—
74#17989792—22no—
75#17989792—22no—
76#17989792—22no—
77#17989792—41no—
78#17989792D414yescomplemented
79#17989789—22no—
80#17989789—22no—
81#17989789—41no—
82#17989786—22no—
83C2×#14556873—22no—
84C2×#14556873C2311yescharacteristic
85C2×#14556873C2311yescharacteristic
86C2×#14556873C2311yescharacteristic
87C2×#14556873C2311yescharacteristic
88C4⋊Dic293C2311yescharacteristic
89#17989781—21no—
90#17989781—21no—
91C22⋊Dic293—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95C4×Dic293—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—
124Dic586—21no—
125Dic586—21no—
126Dic586—21no—
127Dic586—42no—
128Dic586C2×D412yes—
129D1172—21no—
130D1172—21no—
131C22⋊D293—42no—
132C22⋊D293—42no—
133C2×Dic293—21no—
134C2×Dic293—21no—
135C2×Dic293—21no—
136C2×Dic293—42no—
137C4×D293—42no—
138#14556873—21no—
139#14556873—21no—
140#14556873—21no—
141#14556873C2×D412yes—
142C23×C293C2×D411yescharacteristic
143C23×C293C2×D411yescharacteristic
144Q8×C293—21no—
145Q8×C293—21no—
146Q8×C293—21no—
147Q8×C293—21no—
148Q8×C293—21no—
149Q8×C293—42no—
150Q8×C293—42no—
151Q8×C293C2×D412yes—
152D4×C293—22no—
153C2×C4×C293—21no—
154C2×C4×C293—21no—
155C2×C4×C293—21no—
156C2×C4×C293—21no—
157C2×C4×C293—21no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—42no—
162C2×C4×C293—42no—
163C2×C4×C293C2×D411yescharacteristic
164C2×C4×C293C2×D411yescharacteristic
165C2×C4×C293C2×D411yescharacteristic
166C2×C4×C293C2×D411yescharacteristic
167C2×C4×C293C2411yescharacteristic
168C22×C293—21no—
169C22×C293—21no—
170C22×C293—21no—
171C22×C293—21no—
172C22×C293—21no—
173C22×C293#13712yescomplemented
174C22×C293C22×D411yescharacteristic
175C2×D293—41no—
176C2×D293—41no—
177C2×D293—41no—
178C4×C293—21no—
179C4×C293—21no—
180C4×C293—22no—
181C4×C293—21no—
182C4×C293—21no—
183C4×C293—41no—
184C4×C293—41no—
185C4×C293—41no—
186C4×C293C22×D411yescharacteristic
187C4×C293H(5,𝔽2)11yescharacteristic
188Dic293—22no—
189Dic293—41no—
190Dic293—41no—
191C2×C293—22no—
192C2×C293—41no—
193C2×C293D4211yescharacteristic
194C2×C293C2×#13711yescharacteristic
195C2×C293#57811yescharacteristic
196D293—81no—
197C293#327011yescharacteristic, complemented, Sylow
198#3270—2931nomaximal, Sylow
199C2×#137—2931no—
200C22×Q16—2931no—
201D4×Q8D29311yescharacteristic, complemented
202#548—2931no—
203#497—2931no—
204#470—2931no—
205#465—2931no—
206#464—2931no—
207#462—2931no—
208#452—2931no—
209#451—2931no—
210#451—2931no—
211#448—2931no—
212#441—2931no—
213#435—2931no—
214C2×#41—2931no—
215C22×Q8—2931no—
216C22×Q8C2×D29311yescharacteristic
217C22×Q8C2×D29311yescharacteristic
218#137—5864no—
219C2×Q16—2931no—
220C2×Q16—2931no—
221C2×Q16—2931no—
222C2×Q16—2931no—
223C2×Q16—2931no—
224C2×Q16—2931no—
225C2×Q16—5864no—
226C2×SD16—2931no—
227C2×SD16—2931no—
228C2×SD16—2931no—
229C2×M16—2931no—
230C22×C8—2931no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D29311yescharacteristic
233#124—2931no—
234#124—5861no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—2931no—
238C22⋊Q8—5861no—
239C22⋊Q8C2×D29311yescharacteristic
240C22⋊Q8C2×D29311yescharacteristic
241#121—2931no—
242#121—5861no—
243C4×Q8—2931no—
244C4×Q8C2×D29311yescharacteristic
245C4×D4—21no—
246C4×D4—5861no—
247C4×D4C2×D29311yescharacteristic
248#106—2931no—
249C4⋊C8—2931no—
250#103—2931no—
251#103—2931no—
252#103—2931no—
253#103—2931no—
254#103—2931no—
255#102—2931no—
256C22⋊C8—2931no—
257C22⋊C8—2931no—
258#97—2931no—
259#41—5862no—
260#41—11721no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—2931no—
268C2×Q8—2931no—
269C2×Q8—5862no—
270C2×Q8—5861no—
271C2×Q8—11721no—
272C2×Q8C22×D29311yescharacteristic
273C2×Q8C22×D29311yescharacteristic
274C2×Q8C22×D29311yescharacteristic
275C2×D4—2931no—
276C2×D4—5861no—
277C2×D4—5861no—
278C2×D4C22⋊D29311yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—5861no—
282C22×C4—5861no—
283C22×C4C22×D29311yescharacteristic
284C22×C4C22×D29311yescharacteristic
285Q16—2934no—
286Q16—5862no—
287Q16—5862no—
288Q16—5862no—
289Q16—5862no—
290Q16—11721no—
291SD16—5862no—
292SD16—5862no—
293SD16—11721no—
294M16—5862no—
295C2×C8—2931no—
296C2×C8—2931no—
297C2×C8—2931no—
298C2×C8—2931no—
299C2×C8—5862no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—2931no—
306C4⋊C4—5861no—
307C4⋊C4—5861no—
308C4⋊C4C22⋊D29311yescharacteristic
309C4⋊C4C22×D29311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—5861no—
313C22⋊C4—5861no—
314C22⋊C4—5861no—
315C22⋊C4—5861no—
316C22⋊C4C22⋊D29311yescharacteristic
317C22⋊C4C22⋊D29311yescharacteristic
318C42—21no—
319C42—5861no—
320C42C22×D29311yescharacteristic
321C23—5861no—
322C23C2×C22⋊D29311yescharacteristic
323C23C2×C22⋊D29311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—2932no—
332Q8—5861no—
333Q8—5861no—
334Q8—5861no—
335Q8—11722no—
336Q8D4×D29312yes—
337D4—22no—
338D4—5861no—
339D4—5861no—
340D4—11722no—
341D4—11722no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—5861no—
353C2×C4—5861no—
354C2×C4—5861no—
355C2×C4—11722no—
356C2×C4—11722no—
357C2×C4C2×C22⋊D29311yescharacteristic
358C2×C4C2×C22⋊D29311yescharacteristic
359C2×C4C2×C22⋊D29311yescharacteristic
360C2×C4C2×C22⋊D29311yescharacteristic
361C2×C4C23×D29311yescharacteristic
362C8—2932no—
363C8—5861no—
364C8—5861no—
365C8—5861no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—11721no—
372C22—11721no—
373C22—11721no—
374C22#2262765512yescomplemented
375C22C22×C22⋊D29311yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—5862no—
385C4—11721no—
386C4—11721no—
387C4C2×D4×D29311yescharacteristic
388C4#2262766911yescharacteristic
389C2—22no—
390C2—41no—
391C2—23441no—
392C2D4×C22⋊D29311yescharacteristic
393C2C2×#2262765511yescharacteristic
394C2#3403694411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingD4×Q8×C293C2
socleC22×C293C22×D4
prime radical#34036788C2
hypercenterD4×Q8D293
hypocenterC293#3270
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×#17989792C2semidirect
4. #34036521C2semidirect
5. #34036385C2semidirect
6. #34036360C2semidirect
7. #34036351C2semidirect
8. #34036336C2semidirect
9. #34036289C2semidirect
10. #34036285C2semidirect
11. #34036284C2semidirect
12. #34036268C2semidirect
13. #34036234C2semidirect
14. #34036201C2semidirect
15. D4×Q8×C293C2semidirect
78. #17989792D4semidirect
173. C22×C293#137semidirect
197. C293#3270semidirect
201. D4×Q8D293semidirect
374. 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×#179897922
#340367882
#340365212
#340363852
#340363602
#340363512
#340363362
#340362892
#340362852
#340362842
#340362682
#340362342
#340362012
D4×Q8×C2932
#3270293