Finite Group Databasebeta

← #49170348

Subgroups — #49170348

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×#22627653C211yesmaximal, characteristic, complemented
2C22×#17989792C211yesmaximal, characteristic, complemented
3Q8×C22⋊D293C211yesmaximal, characteristic, complemented
4#34036785C211yesmaximal, characteristic
5#34036520C211yesmaximal, characteristic, complemented
6#34036386C211yesmaximal, characteristic, complemented
7#34036356C211yesmaximal, characteristic, complemented
8#34036349C211yesmaximal, characteristic, complemented
9#34036338C211yesmaximal, characteristic, complemented
10#34036288C211yesmaximal, characteristic, complemented
11#34036287C211yesmaximal, characteristic, complemented
12#34036286C211yesmaximal, characteristic, complemented
13#34036233C211yesmaximal, characteristic, complemented
14#34036197C211yesmaximal, characteristic, complemented
15#448×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C22×Dic586C2211yescharacteristic
18C2×Q8×D293C2211yescharacteristic
19#22627653—24no—
20C2×Dic1172C2211yescharacteristic
21C2×Dic1172C2211yescharacteristic
22C2×#17989792—24no—
23C2×#17989792C2211yescharacteristic
24C2×#17989792C2211yescharacteristic
25C2×#17989792C2211yescharacteristic
26C2×#17989792C2211yescharacteristic
27C2×#17989790C2211yescharacteristic
28C2×#17989785C2211yescharacteristic
29C22×#14556873C2211yescharacteristic
30#22627622—21no—
31#22627622C2211yescharacteristic
32#22627611C2211yescharacteristic
33#22627610—21no—
34#22627604C2211yescharacteristic
35#22627603—21no—
36#22627603C2211yescharacteristic
37#22627601—21no—
38#22627600—21no—
39#22627600C2211yescharacteristic
40#22627596—21no—
41Q8×Dic293C2211yescharacteristic
42#22627589C2211yescharacteristic
43D4×Dic293—21no—
44C4×C22⋊D293—21no—
45C4×C22⋊D293C2211yescharacteristic
46#22627555C2211yescharacteristic
47#22627553C2211yescharacteristic
48#22627549C2211yescharacteristic
49#22627549C2211yescharacteristic
50#22627547C2211yescharacteristic
51#22627545C2211yescharacteristic
52#22627536C2211yescharacteristic
53#22627534C2211yescharacteristic
54C22×Q8×C293C2211yescharacteristic
55C2×SD16×C293C2211yescharacteristic
56C2×SD16×C293C2211yescharacteristic
57#121×C293C2211yescharacteristic
58#103×C293C2211yescharacteristic
59#103×C293C2211yescharacteristic
60C22⋊C8×C293C2211yescharacteristic
61#17989801—22no—
62#17989801—41no—
63C2×Dic586—21no—
64C2×Dic586—22no—
65C2×Dic586—21no—
66C2×Dic586—41no—
67C2×Dic586C2311yescharacteristic
68C2×Dic586C2311yescharacteristic
69C2×Dic586C2311yescharacteristic
70Q8×D293—22no—
71Q8×D293—41no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293D411yescharacteristic
74C22×Dic293—21no—
75C22×Dic293—21no—
76C2×C4×D293—21no—
77C2×C4×D293C2311yescharacteristic
78Dic1172—22no—
79Dic1172—41no—
80#17989792—22no—
81#17989792—22no—
82#17989792—22no—
83#17989792D414yescomplemented
84#17989790—22no—
85#17989785—22no—
86C2×#14556873—22no—
87C2×#14556873C2311yescharacteristic
88C2×#14556873C2311yescharacteristic
89C4⋊Dic293—21no—
90C4⋊Dic293—21no—
91C4⋊Dic293C2311yescharacteristic
92#17989781—21no—
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#17989779D411yescharacteristic
105C4×Dic293—21no—
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×D293C2×D411yescharacteristic
124Dic586—21no—
125Dic586—21no—
126Dic586—21no—
127Dic586—21no—
128Dic586—21no—
129Dic586—42no—
130Dic586—42no—
131Dic586C2×D412yes—
132C22⋊D293—22no—
133C22⋊D293—42no—
134C2×Dic293—21no—
135C2×Dic293—21no—
136C2×Dic293—21no—
137C2×Dic293—21no—
138C2×Dic293—42no—
139C2×Dic293—42no—
140C2×Dic293C2×D411yescharacteristic
141C2×Dic293C2×D411yescharacteristic
142C2×Dic293C2×D411yescharacteristic
143C4×D293—21no—
144C4×D293—21no—
145C4×D293—42no—
146#14556873—21no—
147#14556873C2×D412yes—
148C23×C293—21no—
149C23×C293C2×D411yescharacteristic
150Q8×C293—21no—
151Q8×C293—21no—
152Q8×C293—21no—
153Q8×C293—42no—
154Q8×C293C2×D412yes—
155D4×C293—21no—
156D4×C293—21no—
157D4×C293—42no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—21no—
162C2×C4×C293—21no—
163C2×C4×C293—42no—
164C2×C4×C293C2×D411yescharacteristic
165C2×C4×C293C2411yescharacteristic
166C8×C293—21no—
167C8×C293—21no—
168C22×C293—21no—
169C22×C293—21no—
170C22×C293—41no—
171C22×C293—41no—
172C22×C293—41no—
173C22×C293#13712yescomplemented
174C22×C293C22×D411yescharacteristic
175C2×D293—21no—
176C2×D293—21no—
177C2×D293—21no—
178C4×C293—21no—
179C4×C293—22no—
180C4×C293—41no—
181C4×C293—41no—
182C4×C293C22×D411yescharacteristic
183C4×C293H(5,𝔽2)11yescharacteristic
184Dic293—21no—
185Dic293—22no—
186Dic293—21no—
187Dic293—21no—
188Dic293—41no—
189Dic293—41no—
190Dic293—41no—
191C2×C293—22no—
192C2×C293—81no—
193C2×C293D4211yescharacteristic
194C2×C293C2×#13711yescharacteristic
195C2×C293#57811yescharacteristic
196D293—41no—
197C293#327011yescharacteristic, complemented, Sylow
198#3270—2931nomaximal, Sylow
199C2×#137—2931no—
200C22×Q16—2931no—
201D4×Q8—2931no—
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#448D29311yescharacteristic, complemented
212#441—2931no—
213#435—2931no—
214C2×#41—2931no—
215C22×Q8—2931no—
216C22×Q8—2931no—
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×SD16C2×D29311yescharacteristic
228C2×SD16C2×D29311yescharacteristic
229C2×M16—2931no—
230C22×C8—2931no—
231C4⋊Q8—2931no—
232C4⋊Q8—5861no—
233#124—2931no—
234#124—5861no—
235C22⋊Q8—2931no—
236C22⋊Q8—2931no—
237C22⋊Q8—2931no—
238C22⋊Q8—5861no—
239C22⋊Q8—5861no—
240C22⋊Q8—5861no—
241#121—5861no—
242#121C2×D29311yescharacteristic
243C4×Q8—2931no—
244C4×Q8—2931no—
245C4×D4—2931no—
246C4×D4—5861no—
247C4×D4—5861no—
248#106—2931no—
249C4⋊C8—2931no—
250#103—2931no—
251#103—2931no—
252#103—2931no—
253#103C2×D29311yescharacteristic
254#103C2×D29311yescharacteristic
255#102—2931no—
256C22⋊C8—2931no—
257C22⋊C8C2×D29311yescharacteristic
258#97—2931no—
259#41—5862no—
260#41—11721no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—2931no—
265C2×Q8—2931no—
266C2×Q8—2931no—
267C2×Q8—5861no—
268C2×Q8—5862no—
269C2×Q8—5861no—
270C2×Q8—5862no—
271C2×Q8—11721no—
272C2×Q8—11721no—
273C2×Q8C22×D29311yescharacteristic
274C2×Q8C22×D29311yescharacteristic
275C2×D4—21no—
276C2×D4—2931no—
277C2×D4—5861no—
278C2×D4C22×D29311yescharacteristic
279C22×C4—21no—
280C22×C4—2931no—
281C22×C4—5861no—
282C22×C4—5861no—
283C22×C4—5861no—
284C22×C4C22×D29311yescharacteristic
285Q16—2934no—
286Q16—5862no—
287Q16—5862no—
288Q16—5862no—
289Q16—5862no—
290Q16—11721no—
291SD16—22no—
292SD16—41no—
293SD16—5862no—
294M16—5862no—
295C2×C8—2931no—
296C2×C8—2931no—
297C2×C8—5862no—
298C2×C8C22×D29311yescharacteristic
299C2×C8C22×D29311yescharacteristic
300C4⋊C4—2931no—
301C4⋊C4—2931no—
302C4⋊C4—5861no—
303C4⋊C4—5861no—
304C4⋊C4—5861no—
305C4⋊C4—5861no—
306C4⋊C4—5861no—
307C4⋊C4—5861no—
308C4⋊C4—5861no—
309C4⋊C4C22×D29311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—2931no—
312C22⋊C4—2931no—
313C22⋊C4—5861no—
314C22⋊C4—5861no—
315C22⋊C4—5861no—
316C22⋊C4—5861no—
317C22⋊C4—5861no—
318C42—2931no—
319C42—5861no—
320C42—5861no—
321C23—21no—
322C23—2931no—
323C23D4×D29311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—2932no—
329Q8—5861no—
330Q8—5861no—
331Q8—5861no—
332Q8—5861no—
333Q8—5861no—
334Q8—11722no—
335Q8—11722no—
336Q8D4×D29312yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—5862no—
341D4—11722no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—2931no—
349C2×C4—2931no—
350C2×C4—2931no—
351C2×C4—5861no—
352C2×C4—5861no—
353C2×C4—5861no—
354C2×C4—5861no—
355C2×C4—5861no—
356C2×C4—5861no—
357C2×C4—11722no—
358C2×C4—11722no—
359C2×C4—11722no—
360C2×C4D4×D29311yescharacteristic
361C2×C4C23×D29311yescharacteristic
362C8—21no—
363C8—21no—
364C8—2932no—
365C8—5861no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—5861no—
372C22—5861no—
373C22—5861no—
374C22#2262765312yescomplemented
375C22C2×D4×D29311yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—5861no—
381C4—5862no—
382C4—5861no—
383C4—5861no—
384C4—11721no—
385C4—11721no—
386C4—11721no—
387C4C2×D4×D29311yescharacteristic
388C4#2262766911yescharacteristic
389C2—22no—
390C2—81no—
391C2—11721no—
392C2#3403676111yescharacteristic
393C2C2×#2262765311yescharacteristic
394C2#3403694511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#448×C293C2
socleC22×C293C22×D4
prime radical#34036785C2
hypercenter#448D293
hypocenterC293#3270
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×#22627653C2semidirect
2. C22×#17989792C2semidirect
3. Q8×C22⋊D293C2semidirect
5. #34036520C2semidirect
6. #34036386C2semidirect
7. #34036356C2semidirect
8. #34036349C2semidirect
9. #34036338C2semidirect
10. #34036288C2semidirect
11. #34036287C2semidirect
12. #34036286C2semidirect
13. #34036233C2semidirect
14. #34036197C2semidirect
15. #448×C293C2semidirect
83. #17989792D4semidirect
173. C22×C293#137semidirect
197. C293#3270semidirect
211. #448D293semidirect
374. C22#22627653semidirect

Maximal subgroups

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

subgroupindex
C2×#226276532
C22×#179897922
Q8×C22⋊D2932
#340367852
#340365202
#340363862
#340363562
#340363492
#340363382
#340362882
#340362872
#340362862
#340362332
#340361972
#448×C2932
#3270293