Finite Group Databasebeta

← #30826044

Subgroups — #30826044

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570213C211yesmaximal, characteristic
2#22569144C211yesmaximal, characteristic, complemented
3C2×#16863473C211yesmaximal, characteristic
4#22559096C214yesmaximal, complemented
5#22559049C214yesmaximal, complemented
6#22558979C211yesmaximal, characteristic, complemented
7#22558635C211yesmaximal, characteristic
8#22558493C211yesmaximal, characteristic, complemented
9C67×#1481C211yesmaximal, characteristic, complemented
10C22×#11761611—21no—
11C2×C4×D268C2211yescharacteristic
12#16863724C412yes—
13#16863683C412yes—
14#16863598C412yes—
15#16863546C412yes—
16#16863527—21no—
17#16863527C2212yescomplemented
18Q8×C67⋊C8—21no—
19Q8×C67⋊C8C2212yescomplemented
20C4×#11761615C2214yes—
21#16863473—21no—
22#16863473—21no—
23#16863473C2212yes—
24#16863473C2212yes—
25#16863449C2211yescharacteristic
26#16863423—21no—
27#16863420C2211yescharacteristic
28C2×(C2×D67)⋊C8—21no—
29#16863352—21no—
30C2×C4×C67⋊C8C2211yescharacteristic
31#16863146C2214yescomplemented
32#16863125C2214yes—
33#16863118C2212yescomplemented
34C8×C67⋊C8C2212yescomplemented
35#533×C67C2211yescharacteristic
36#431×C67C2211yescharacteristic
37C2×C4⋊C8×C67C2211yescharacteristic
38Q8⋊C8×C67C2214yescomplemented
39C22×D268C2×C411yescharacteristic
40C22×C4×D67—21no—
41C2×#11761615C2×C414yes—
42C2×#11761612—21no—
43C2×#11761611—22no—
44C2×#11761611—22no—
45C2×#11761611—44no—
46C22×C67⋊C8D411yescharacteristic
47C22×C67⋊C8D411yescharacteristic
48C4×D268—21no—
49C4×D268—21no—
50C4×D268C2×C412yes—
51C4×D268C2312yes—
52C2×C4⋊Dic67C2×C411yescharacteristic
53C2×(C2×D67)⋊C4—21no—
54#14523974C2×C414yes—
55C4⋊(C67⋊C8)—22no—
56C4⋊(C67⋊C8)—22no—
57C4⋊(C67⋊C8)—21no—
58C4⋊(C67⋊C8)C2×C412yes—
59C4⋊(C67⋊C8)C2312yes—
60#14523967C2×C414yes—
61#14523963C2×C414yes—
62C22⋊(C67⋊C8)C2×C412yes—
63(C2×D67)⋊C8—22no—
64(C2×D67)⋊C8—22no—
65#14523954—22no—
66C4×C67⋊C8—22no—
67C4×C67⋊C8D411yescharacteristic
68C4×C67⋊C8D411yescharacteristic
69C4×C67⋊C8C2312yes—
70C4×C67⋊C8C2311yescharacteristic
71C4×C67⋊C8C2311yescharacteristic
72C22×C8×C67—21no—
73C4⋊Q8×C67C2×C411yescharacteristic
74C4⋊Q8×C67C2×C411yescharacteristic
75#125×C67—21no—
76C22⋊Q8×C67—21no—
77C22⋊Q8×C67C2×C412yes—
78C4×Q8×C67—21no—
79C4×Q8×C67C2312yes—
80#117×C67—21no—
81C2×C42×C67C2311yescharacteristic
82C4⋊C8×C67—21no—
83C4⋊C8×C67—21no—
84C4⋊C8×C67C2312yes—
85C22⋊C8×C67—21no—
86C4×C8×C67C2312yes—
87C23×D67—21no—
88C2×D268—21no—
89C2×D268—42no—
90C2×D268C22×C412yes—
91C22×Dic67—21no—
92C2×C4×D67—22no—
93C2×C4×D67—22no—
94C2×C4×D67—42no—
95C2×C4×D67—42no—
96#11761615—44no—
97#11761612—42no—
98#11761611—44no—
99#11761611—44no—
100C2×C67⋊C8—22no—
101C2×C67⋊C8—22no—
102C2×C67⋊C8SD1614yescomplemented
103C2×C67⋊C8C22×C414yes—
104C2×C67⋊C8C2×D411yescharacteristic
105C2×C67⋊C8C2×D411yescharacteristic
106C2×C67⋊C8C2×D411yescharacteristic
107C2×C67⋊C8C2×D411yescharacteristic
108C4⋊Dic67—21no—
109C4⋊Dic67C22×C412yes—
110(C2×D67)⋊C4—22no—
111(C2×D67)⋊C4—22no—
112C2×Q8×C67—21no—
113C2×Q8×C67C22×C412yes—
114C22×C4×C67C22×C411yescharacteristic
115C22×C4×C67C2×D411yescharacteristic
116C22×C4×C67#4111yescharacteristic
117C2×C8×C67—22no—
118C2×C8×C67—22no—
119C2×C8×C67—42no—
120C4⋊C4×C67—21no—
121C4⋊C4×C67—21no—
122C4⋊C4×C67—21no—
123C4⋊C4×C67—22no—
124C4⋊C4×C67—22no—
125C4⋊C4×C67C22×C412yes—
126C22⋊C4×C67—22no—
127C42×C67—22no—
128C42×C67C22×C411yescharacteristic
129C42×C67C2×D411yescharacteristic
130C42×C67#4111yescharacteristic
131C42×C67C2411yescharacteristic
132C23×C67C4×D411yescharacteristic
133C22×D67—22no—
134C22×D67—42no—
135C22×D67—42no—
136C22×D67—42no—
137Q8×C67—24no—
138Q8×C67—42no—
139C2×C4×C67—22no—
140C2×C4×C67—22no—
141C2×C4×C67—22no—
142C2×C4×C67—41no—
143C2×C4×C67C4×D411yescharacteristic
144C2×C4×C67C4×D411yescharacteristic
145C2×C4×C67C4×D411yescharacteristic
146C2×C4×C67C2×SD1612yes—
147C2×C4×C67#13512yes—
148C2×C4×C67C23×C411yescharacteristic
149C2×C4×C67C22×D411yescharacteristic
150C2×C4×C67C2×#4111yescharacteristic
151C2×Dic67—22no—
152C2×Dic67—42no—
153D268—22no—
154D268—22no—
155D268—42no—
156C4×D67—42no—
157C4×D67—42no—
158C4×D67—44no—
159C8×C67—42no—
160C8×C67—42no—
161C67⋊C8—24no—
162C67⋊C8—21no—
163C67⋊C8—21no—
164C67⋊C8—42no—
165C67⋊C8C2×SD1612yes—
166C67⋊C8C2×SD1612yes—
167C22×C67—21no—
168C22×C67—21no—
169C22×C67C4×SD1612yes—
170C22×C67C2×C4×D411yescharacteristic
171C2×D67—42no—
172C2×D67—42no—
173C2×D67—42no—
174C2×D67—84no—
175C4×C67—21no—
176C4×C67—21no—
177C4×C67—21no—
178C4×C67—42no—
179C4×C67—42no—
180C4×C67C4×SD1612yes—
181C4×C67#56812yes—
182C4×C67C22×SD1611yescharacteristic
183C4×C67C2×#13511yescharacteristic
184Dic67—42no—
185#17930—671nomaximal, Sylow
186C2×C67—22no—
187C2×C67#293011yescharacteristic
188C2×C67C2×C4×SD1611yescharacteristic
189C2×C67#295511yescharacteristic
190D67—82no—
191#3101—671no—
192#2960—671no—
193C2×#436—671no—
194#1590—674no—
195#1583—674no—
196#1570—671no—
197#1486—671no—
198#1481D6711yescharacteristic, complemented
199#1450—671no—
200C67#1793011yescharacteristic, complemented, Sylow
201C22×M16—1341no—
202#533C2×D6711yescharacteristic
203C2×C4×D4—671no—
204#483—672no—
205#476—672no—
206#461—672no—
207#450—672no—
208#446—672no—
209#446—1341no—
210C8×Q8—672no—
211C8×Q8—1341no—
212C4×SD16—674no—
213#436—672no—
214#436—672no—
215#436—1341no—
216#436—1341no—
217#431—671no—
218#431C2×D6711yescharacteristic
219#424—1341no—
220#423—671no—
221C2×C4⋊C8C2×D6711yescharacteristic
222C2×C22⋊C8—1341no—
223#405—1341no—
224C2×C4×C8—671no—
225#334—674no—
226Q8⋊C8C2×D6714yescomplemented
227D4⋊C8—674no—
228#322—672no—
229C82—672no—
230C22×D4—671no—
231C23×C4—1341no—
232C2×SD16—674no—
233C2×M16—2684no—
234C2×M16—1342no—
235C2×M16—1342no—
236C2×M16—1341no—
237C22×C8—21no—
238C22×C8—671no—
239C22×C8—671no—
240C4⋊Q8C4×D6711yescharacteristic
241C4⋊Q8C4×D6711yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8C4×D6712yes—
245C4×Q8—21no—
246C4×Q8C22×D6712yes—
247C4×D4—672no—
248C4×D4—672no—
249C4×D4—1341no—
250C4×D4—1341no—
251#117—21no—
252C2×C4⋊C4—671no—
253C2×C22⋊C4—1341no—
254C2×C42C22×D6711yescharacteristic
255#106—674no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—672no—
259C4⋊C8—672no—
260C4⋊C8—1342no—
261C4⋊C8—1342no—
262C4⋊C8—1341no—
263C4⋊C8C22×D6712yes—
264#103—674no—
265#102—674no—
266C22⋊C8—21no—
267C22⋊C8—672no—
268C22⋊C8—1342no—
269C22⋊C8—1342no—
270#97—1342no—
271C4×C8—671no—
272C4×C8—671no—
273C4×C8—671no—
274C4×C8—672no—
275C4×C8—671no—
276C4×C8—1342no—
277C4×C8C22×D6712yes—
278C24—1341no—
279C2×Q8—21no—
280C2×Q8C2×C4×D6712yes—
281C2×D4—2682no—
282C2×D4—672no—
283C2×D4—1341no—
284C22×C4—2682no—
285C22×C4—2682no—
286C22×C4—1341no—
287C22×C4—1342no—
288C22×C4—1342no—
289C22×C4C2×C4×D6711yescharacteristic
290C22×C4D4×D6711yescharacteristic
291C22×C4#1176162711yescharacteristic
292SD16—2684no—
293M16—2684no—
294M16—2684no—
295M16—2682no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—674no—
300C2×C8—671no—
301C2×C8—671no—
302C2×C8—674no—
303C2×C8—671no—
304C2×C8—671no—
305C2×C8—1342no—
306C2×C8—1342no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—672no—
313C4⋊C4—1341no—
314C4⋊C4C2×C4×D6712yes—
315C22⋊C4—22no—
316C22⋊C4—1342no—
317C22⋊C4—1342no—
318C42—22no—
319C42C2×C4×D6711yescharacteristic
320C42D4×D6711yescharacteristic
321C42#1176162711yescharacteristic
322C42C23×D6711yescharacteristic
323C23—2682no—
324C23—2682no—
325C23—2682no—
326C23—1342no—
327C23(C22⋊D67)⋊C411yescharacteristic
328Q8—24no—
329Q8—42no—
330D4—2682no—
331D4—1342no—
332D4—1342no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—2682no—
338C2×C4—2682no—
339C2×C4—2682no—
340C2×C4—2684no—
341C2×C4—1342no—
342C2×C4(C22⋊D67)⋊C411yescharacteristic
343C2×C4(C22⋊D67)⋊C411yescharacteristic
344C2×C4(C22⋊D67)⋊C411yescharacteristic
345C2×C4SD16×D6712yes—
346C2×C4#1452406612yes—
347C2×C4C22×C4×D6711yescharacteristic
348C2×C4C2×D4×D6711yescharacteristic
349C2×C4C2×#1176162711yescharacteristic
350C8—42no—
351C8—42no—
352C8—2682no—
353C8—672no—
354C8—672no—
355C8—1344no—
356C8—1341no—
357C8—1341no—
358C22—21no—
359C22—21no—
360C22—2682no—
361C22—2682no—
362C22—2682no—
363C22—5364no—
364C22#1686348112yes—
365C22C2×(C22⋊D67)⋊C411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—42no—
371C4—2682no—
372C4#1686348112yes—
373C4#1686416112yes—
374C4C2×SD16×D6711yescharacteristic
375C4C2×#1452406611yescharacteristic
376C2—22no—
377C2—5362no—
378C2#2256891511yescharacteristic
379C2C2×#1686348111yescharacteristic
380C2#2256910811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1481C2
socleC22×C67C2×C4×D4
prime radicalC22×D268C2×C4
cosocleC42×C67C24
hypercenter#1481D67
hypocenterC67#17930
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
2. #22569144C2semidirect
4. #22559096C2semidirect
5. #22559049C2semidirect
6. #22558979C2semidirect
8. #22558493C2semidirect
9. C67×#1481C2semidirect
17. #16863527C22semidirect
19. Q8×C67⋊C8C22semidirect
31. #16863146C22semidirect
33. #16863118C22semidirect
34. C8×C67⋊C8C22semidirect
38. Q8⋊C8×C67C22semidirect
102. C2×C67⋊C8SD16semidirect
198. #1481D67semidirect
200. C67#17930semidirect
226. Q8⋊C8C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225702132
#225691442
C2×#168634732
#225590962
#225590492
#225589792
#225586352
#225584932
C67×#14812
#1793067