Finite Group Databasebeta

← C2×#22626064 #34022204

Subgroups — C2×#22626064 #34022204

Order 18688.

1878 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×#17989591C211yesmaximal, characteristic, complemented
2C22×#14556656C214yesmaximal, complemented
3C2×C42×#64408C211yesmaximal, characteristic, complemented
4#22626064C218yesmaximal, direct product factor
5C2×#17989138C211yesmaximal, characteristic
6C22×#11924038C2212yes—
7C22×C4×#64408—22no—
8C22×C4×#64408C2212yes—
9#17989591—22no—
10#17989591C2214yescomplemented
11C2×C4⋊#10596833—22no—
12C2×#14556656—216no—
13C2×#14556656C22116yescomplemented
14C42×#64408—22no—
15C42×#64408C2214yescomplemented
16C22×#11924029C2212yes—
17#17989138—22no—
18#17989138—22no—
19#17989138C414yescomplemented
20#17989138C2214yes—
21C2×#14556588C411yescharacteristic, complemented
22C2×#14556580C412yescomplemented
23C2×#14556566—22no—
24C2×#14556553—22no—
25C2×C42×D73C2211yescharacteristic
26C2×C4⋊Q8×C73C411yescharacteristic, complemented
27C23×#64408—22no—
28C2×#11924038—22no—
29C2×#11924038—28no—
30C2×#11924038C2314yes—
31C22×#10596833—22no—
32C2×C4×#64408—216no—
33C2×C4×#64408—28no—
34C2×C4×#64408—22no—
35C2×C4×#64408—24no—
36C2×C4×#64408D418yescomplemented
37C2×C4×#64408C2314yes—
38C4⋊#10596833—28no—
39#14556656—232no—
40C2×#11924029—22no—
41C2×#11924029—22no—
42C2×#11924029—28no—
43C2×#11924029—28no—
44C2×#11924029C2×C414yes—
45C2×#11924029C2314yes—
46C22×D292—22no—
47C22×D292—22no—
48C22×D292C2×C412yes—
49C22×C4×D73—22no—
50C22×C4×D73D412yes—
51C22×C4×D73C2311yescharacteristic
52#14556588—22no—
53#14556588C2×C414yescomplemented
54#14556580—24no—
55#14556580C2×C418yescomplemented
56#14556566—28no—
57#14556566—28no—
58#14556553—28no—
59#14556553—28no—
60C2×#11924005—24no—
61C2×C4×Dic73C2×C411yescharacteristic
62C2×C4×Dic73D412yes—
63C42×D73C2×C412yes—
64C42×D73D412yes—
65C42×D73D412yes—
66C42×D73C2312yes—
67C22×Q8×C73C2×C412yes—
68C4⋊Q8×C73—22no—
69C4⋊Q8×C73C2×C414yescomplemented
70C2×C4⋊C4×C73—22no—
71C2×C42×C73C2×C411yescharacteristic
72C22×#64408—216no—
73C22×#64408—28no—
74C22×#64408—24no—
75#11924038—28no—
76#11924038—44no—
77C2×#10596833—24no—
78C2×#10596833—44no—
79C4×#64408—232no—
80C4×#64408—28no—
81C4×#64408—24no—
82C4×#64408C2×D418yescomplemented
83C23×D73—22no—
84C23×D73C2×D411yescharacteristic
85#11924029—216no—
86#11924029—28no—
87#11924029—28no—
88#11924029—48no—
89#11924029—44no—
90#11924029—44no—
91C2×D292—28no—
92C2×D292—24no—
93C2×D292—24no—
94C2×D292—28no—
95C2×D292—22no—
96C2×D292—48no—
97C2×D292—48no—
98C2×D292C22×C414yes—
99C22×Dic73C22⋊C412yes—
100C22×Dic73C2×D411yescharacteristic
101C2×C4×D73—28no—
102C2×C4×D73—24no—
103C2×C4×D73—24no—
104C2×C4×D73—44no—
105C2×C4×D73—44no—
106C2×C4×D73C22⋊C414yes—
107C2×C4×D73C22×C411yescharacteristic
108C2×C4×D73C2×D414yes—
109C2×C4×D73C2×D414yes—
110C2×C4×D73C2×D411yescharacteristic
111C2×C4×D73C2×D411yescharacteristic
112C2×C4×D73#4114yes—
113C2×C4×D73C2411yescharacteristic
114#11924005—216no—
115C4×Dic73C22⋊C414yes—
116C4×Dic73C22⋊C412yes—
117C4×Dic73C22×C412yes—
118C4×Dic73C2×D414yes—
119C2×Q8×C73—22no—
120C2×Q8×C73—28no—
121C2×Q8×C73C22×C414yes—
122C22×C4×C73—22no—
123C22×C4×C73C22⋊C412yes—
124C22×C4×C73C22×C411yescharacteristic
125C4⋊C4×C73—28no—
126C42×C73C22⋊C412yes—
127C42×C73C22×C412yes—
128C23×C73C2×C22⋊C411yescharacteristic
129C22×D73—24no—
130C22×D73—24no—
131C22×D73—44no—
132C22×D73—48no—
133C22×D73C2×C22⋊C411yescharacteristic
134C22×D73#12114yes—
135C22×D73C22×D411yescharacteristic
136C2×#64408—216no—
137C2×#64408—232no—
138C2×#64408—28no—
139Q8×C73—28no—
140Q8×C73—44no—
141C2×C4×C73—24no—
142C2×C4×C73—44no—
143C2×C4×C73#10418yescomplemented
144C2×C4×C73C2×C22⋊C411yescharacteristic
145C2×C4×C73C2×C22⋊C414yes—
146C2×C4×C73C4×D414yes—
147C2×C4×C73C23×C411yescharacteristic
148C2×Dic73—24no—
149C2×Dic73C2×C22⋊C414yes—
150C2×Dic73C2×C22⋊C411yescharacteristic
151C2×Dic73#12412yes—
152C2×Dic73#12712yes—
153C2×Dic73C22×D411yescharacteristic
154D292—28no—
155D292—48no—
156D292—48no—
157D292—44no—
158D292—416no—
159C4×D73—28no—
160C4×D73—28no—
161C4×D73—44no—
162C4×D73—44no—
163C4×D73—48no—
164C4×D73C4×D414yes—
165C4×D73#12114yes—
166C4×D73#12412yes—
167C4×D73#12712yes—
168C4×D73C22×D412yes—
169C4×D73C2×#4112yes—
170#10596833—44no—
171C22×C73#39012yes—
172C22×C73C2×#10414yes—
173C22×C73C22×C22⋊C411yescharacteristic
174C2×D73—22no—
175C2×D73—22no—
176C2×D73—28no—
177C2×D73—48no—
178C2×D73—44no—
179C2×D73—88no—
180C2×D73#39012yes—
181C2×D73C2×#12112yes—
182#64408—216no—
183C4×C73—44no—
184C4×C73#39012yes—
185C4×C73C2×#10418yescomplemented
186C4×C73C2×C4×D412yes—
187Dic73—24no—
188Dic73#39012yes—
189Dic73C2×#12411yescharacteristic
190Dic73C2×#12711yescharacteristic
191C2×#1962—731nomaximal, Sylow
192C2×C73#196214yescomplemented
193C2×C73C2×#39011yescharacteristic
194C2×C73C22×#10412yes—
195D73—22no—
196D73—22no—
197D73—84no—
198C2×#532—731no—
199C2×#423—731no—
200C22×#104—734no—
201C2×C43—731no—
202#1962—738no—
203C73C2×#196211yescharacteristic, complemented, Sylow
204C22×#41—732no—
205C22×M16—732no—
206#532—734no—
207#532—734no—
208#532—1462no—
209#532—1462no—
210C2×C4⋊Q8#6440811yescharacteristic, complemented
211C2×#127—731no—
212C2×#124—732no—
213C2×#121—1462no—
214C2×C4×D4—1462no—
215C22×C42—731no—
216C22×C42—732no—
217C22×C42—1462no—
218#423—734no—
219#423—1462no—
220C2×C4⋊C8—1462no—
221C2×#104—7316no—
222C2×#104—14616no—
223C43—734no—
224C43—1462no—
225C2×#41—734no—
226C2×#41—734no—
227C2×#41—1468no—
228C2×#41—1468no—
229C2×#41—1462no—
230C2×#41—1462no—
231C22×Q8C2×#6440812yes—
232C22×D4—732no—
233C22×D4—1462no—
234C22×D4—1462no—
235C23×C4—731no—
236C23×C4—732no—
237C23×C4—1462no—
238C23×C4—1462no—
239C2×M16—734no—
240C2×M16—1468no—
241C2×M16—1462no—
242C22×C8—1462no—
243C4⋊Q8—22no—
244C4⋊Q8C2×#6440814yescomplemented
245#127—734no—
246#127—1462no—
247#124—738no—
248#124—1464no—
249#121—1468no—
250#121—1468no—
251C4×D4—1468no—
252C4×D4—1468no—
253C2×C4⋊C4—22no—
254C2×C22⋊C4—1464no—
255C2×C42—732no—
256C2×C42—731no—
257C2×C42—732no—
258C2×C42—732no—
259C2×C42—732no—
260C2×C42—732no—
261C2×C42—734no—
262C2×C42—738no—
263C2×C42—1468no—
264C2×C42—1462no—
265C2×C42—1464no—
266C2×C42—14616no—
267C2×C42C2×#6440811yescharacteristic
268C4⋊C8—1468no—
269#104—14632no—
270C24—731no—
271C24—1462no—
272#41—2928no—
273#41—2924no—
274#41—2924no—
275#41—1468no—
276#41—1468no—
277#41—14616no—
278C2×Q8—22no—
279C2×Q8—28no—
280C2×Q8C22×#6440814yes—
281C2×D4—2928no—
282C2×D4—2928no—
283C2×D4—734no—
284C2×D4—1464no—
285C2×D4—1464no—
286C2×D4—1468no—
287C2×D4—1462no—
288C2×D4—1468no—
289C22×C4—22no—
290C22×C4—2924no—
291C22×C4—2924no—
292C22×C4—731no—
293C22×C4—731no—
294C22×C4—731no—
295C22×C4—731no—
296C22×C4—731no—
297C22×C4—734no—
298C22×C4—734no—
299C22×C4—732no—
300C22×C4—734no—
301C22×C4—734no—
302C22×C4—1464no—
303C22×C4—1464no—
304C22×C4—1464no—
305C22×C4—1468no—
306C22×C4—14616no—
307C22×C4—1468no—
308C22×C4C22⋊#6440812yes—
309C22×C4C22×#6440811yescharacteristic
310M16—2924no—
311M16—1468no—
312C2×C8—2924no—
313C2×C8—1464no—
314C4⋊C4—28no—
315C22⋊C4—14616no—
316C42—732no—
317C42—734no—
318C42—732no—
319C42—734no—
320C42—738no—
321C42—1464no—
322C42—1468no—
323C42—14632no—
324C42C22⋊#6440812yes—
325C42C22×#6440812yes—
326C23—2924no—
327C23—2928no—
328C23—731no—
329C23—731no—
330C23—734no—
331C23—1464no—
332C23—1464no—
333C23C2×C22⋊#6440811yescharacteristic
334Q8—28no—
335Q8—44no—
336D4—2928no—
337D4—2928no—
338D4—2924no—
339D4—29216no—
340D4—1468no—
341C2×C4—24no—
342C2×C4—44no—
343C2×C4—2924no—
344C2×C4—2924no—
345C2×C4—2928no—
346C2×C4—731no—
347C2×C4—731no—
348C2×C4—732no—
349C2×C4—732no—
350C2×C4—732no—
351C2×C4—732no—
352C2×C4—734no—
353C2×C4—734no—
354C2×C4—732no—
355C2×C4—732no—
356C2×C4—734no—
357C2×C4—1468no—
358C2×C4—14616no—
359C2×C4—14632no—
360C2×C4—1464no—
361C2×C4—1468no—
362C2×C4—1468no—
363C2×C4#1455665618yescomplemented
364C2×C4C2×C22⋊#6440811yescharacteristic
365C2×C4C2×C22⋊#6440814yes—
366C2×C4D4×#6440814yes—
367C2×C4C23×#6440811yescharacteristic
368C8—2924no—
369C22—2924no—
370C22—2928no—
371C22—5848no—
372C22—732no—
373C22—732no—
374C22—1468no—
375C22—1462no—
376C22—1462no—
377C22#1798953912yes—
378C22C2×#1455665614yes—
379C22C22×C22⋊#6440811yescharacteristic
380C4—44no—
381C4—731no—
382C4—731no—
383C4—732no—
384C4—14616no—
385C4—1464no—
386C4#1798953912yes—
387C4C2×#1455665618yescomplemented
388C4C2×D4×#6440812yes—
389C2—5844no—
390C2—1462no—
391C2—1462no—
392C2#2262606414yesdirect product factor
393C2C2×#1798953911yescharacteristic
394C2C22×#1455665612yes—
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
FittingC2×C4⋊Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC2×#17989138C2
cosocleC2×C4×D73C24
hypercenterC2×C4⋊Q8#64408
hypocenterC73C2×#1962
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17989591C2semidirect
2. C22×#14556656C2semidirect
3. C2×C42×#64408C2semidirect
4. #22626064C2direct (N × H)
10. #17989591C22semidirect
13. C2×#14556656C22semidirect
15. C42×#64408C22semidirect
19. #17989138C4semidirect
21. C2×#14556588C4semidirect
22. C2×#14556580C4semidirect
26. C2×C4⋊Q8×C73C4semidirect
36. C2×C4×#64408D4semidirect
53. #14556588C2×C4semidirect
55. #14556580C2×C4semidirect
69. C4⋊Q8×C73C2×C4semidirect
82. C4×#64408C2×D4semidirect
143. C2×C4×C73#104semidirect
185. C4×C73C2×#104semidirect
192. C2×C73#1962semidirect
203. C73C2×#1962semidirect
210. C2×C4⋊Q8#64408semidirect
244. C4⋊Q8C2×#64408semidirect
363. C2×C4#14556656semidirect
387. C4C2×#14556656semidirect
392. C2#22626064direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179895912
C22×#145566562
C2×C42×#644082
#226260642
C2×#179891382
C2×#196273