Finite Group Databasebeta

← #30744087

Subgroups — #30744087

Order 17152.

556 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22559971C211yesmaximal, characteristic
2#22559903C211yesmaximal, characteristic
3#22558035C212yesmaximal, complemented
4#22558032C212yesmaximal, complemented
5C4×C22⋊Q8×C67C211yesmaximal, characteristic
6C22×#11761612—21no—
7C2×C22⋊(C67⋊C8)—21no—
8C2×#14523954—21no—
9#16863151—21no—
10#16863151C2212yes—
11#16863151C2212yes—
12C2×C22⋊Q8×C67C411yescharacteristic
13C2×C4×Q8×C67C411yescharacteristic
14C22×C42×C67C2211yescharacteristic
15#391×C67C2211yescharacteristic
16#389×C67C2211yescharacteristic
17#387×C67—21no—
18#385×C67—21no—
19#380×C67C411yescharacteristic
20C4×C4⋊C4×C67—21no—
21C4×C4⋊C4×C67C411yescharacteristic
22C4×C22⋊C4×C67—21no—
23C2×#11761612—24no—
24C2×#11761612—42no—
25C2×#11761612—42no—
26C22×C67⋊C8—22no—
27C22×C67⋊C8—22no—
28C22⋊(C67⋊C8)—24no—
29#14523954—42no—
30C22×Q8×C67C2×C411yescharacteristic
31C23×C4×C67C2×C411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C23×C4×C67D411yescharacteristic
34C22⋊Q8×C67—22no—
35C22⋊Q8×C67—22no—
36C4×Q8×C67—42no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67—21no—
45C2×C42×C67—21no—
46C2×C42×C67—21no—
47C2×C42×C67—42no—
48C2×C42×C67C2×C411yescharacteristic
49C2×C42×C67D411yescharacteristic
50C2×C42×C67D411yescharacteristic
51C2×C42×C67C2311yescharacteristic
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56#95×C67—21no—
57#11761612—48no—
58C2×C67⋊C8—24no—
59C2×C67⋊C8—24no—
60C2×C67⋊C8—42no—
61C2×C67⋊C8—42no—
62C2×C67⋊C8—42no—
63C2×C67⋊C8—42no—
64C24×C67C22⋊C411yescharacteristic
65C2×Q8×C67—21no—
66C2×Q8×C67—41no—
67C2×Q8×C67—41no—
68C2×Q8×C67M1612yescomplemented
69C22×C4×C67—22no—
70C22×C4×C67—22no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—22no—
76C22×C4×C67—21no—
77C22×C4×C67—21no—
78C22×C4×C67—22no—
79C22×C4×C67—21no—
80C22×C4×C67—21no—
81C22×C4×C67—41no—
82C22×C4×C67—41no—
83C22×C4×C67—41no—
84C22×C4×C67C22⋊C411yescharacteristic
85C22×C4×C67C22⋊C411yescharacteristic
86C22×C4×C67C22⋊C411yescharacteristic
87C22×C4×C67C22×C411yescharacteristic
88C22×C4×C67C2×D411yescharacteristic
89C22×C4×C67C2×D411yescharacteristic
90C4⋊C4×C67—24no—
91C4⋊C4×C67—21no—
92C4⋊C4×C67—41no—
93C4⋊C4×C67—41no—
94C4⋊C4×C67—42no—
95C4⋊C4×C67—41no—
96C4⋊C4×C67—41no—
97C4⋊C4×C67M1612yescomplemented
98C22⋊C4×C67—24no—
99C22⋊C4×C67—42no—
100C42×C67—22no—
101C42×C67—22no—
102C42×C67—41no—
103C42×C67—41no—
104C42×C67—42no—
105C42×C67—42no—
106C23×C67—21no—
107C23×C67—21no—
108C23×C67—21no—
109C23×C67—21no—
110C23×C67—21no—
111C23×C67—21no—
112C23×C67#10412yes—
113C23×C67C2×C22⋊C411yescharacteristic
114Q8×C67—42no—
115Q8×C67—42no—
116C2×C4×C67—24no—
117C2×C4×C67—24no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—22no—
129C2×C4×C67—22no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—44no—
134C2×C4×C67—42no—
135C2×C4×C67—42no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67—41no—
143C2×C4×C67—41no—
144C2×C4×C67—41no—
145C2×C4×C67—41no—
146C2×C4×C67#9912yes—
147C2×C4×C67#10412yes—
148C2×C4×C67C2×M1611yescharacteristic
149C2×C4×C67C2×M1611yescharacteristic
150C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C67#13711yescharacteristic
152C2×C4×C67#13712yes—
153C67⋊C8—44no—
154C67⋊C8—44no—
155C22×C67—22no—
156C22×C67—22no—
157C22×C67—22no—
158C22×C67—41no—
159C22×C67—41no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67#40711yescharacteristic
163C22×C67C2×#9911yescharacteristic
164C22×C67#41311yescharacteristic
165C22×C67#41711yescharacteristic
166C22×C67#41911yescharacteristic
167C22×C67C2×#10411yescharacteristic
168C22×C67C2×#10411yescharacteristic
169C4×C67—21no—
170C4×C67—21no—
171C4×C67—21no—
172C4×C67—22no—
173C4×C67—21no—
174C4×C67—22no—
175C4×C67—22no—
176C4×C67—41no—
177C4×C67—42no—
178C4×C67—42no—
179C4×C67—81no—
180C4×C67—81no—
181#7634—671nomaximal, Sylow
182C2×C67—42no—
183C2×C67#146511yescharacteristic
184C2×C67#148911yescharacteristic
185C2×C67#149111yescharacteristic
186C2×C67#150411yescharacteristic
187C2×C67#150811yescharacteristic
188C2×C67#152411yescharacteristic
189C2×C67#155911yescharacteristic
190C4×C22⋊Q8D6711yescharacteristic
191#1749—671no—
192#1728—671no—
193#1272—672no—
194#1271—672no—
195C67#763411yescharacteristic, complemented, Sylow
196C22×M16—1341no—
197C2×C22⋊Q8Dic6711yescharacteristic
198C2×C4×Q8Dic6711yescharacteristic
199C22×C42C2×D6711yescharacteristic
200C2×C22⋊C8—1341no—
201C2×#97—1341no—
202#391C2×D6711yescharacteristic
203#389C2×D6711yescharacteristic
204#387—21no—
205#385—21no—
206#380Dic6711yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4Dic6711yescharacteristic
209C4×C22⋊C4—21no—
210#336—672no—
211#336—672no—
212#336—1341no—
213C22×Q8C2×Dic6711yescharacteristic
214C23×C4C2×Dic6711yescharacteristic
215C23×C4C22⋊D6711yescharacteristic
216C23×C4C22⋊D6711yescharacteristic
217C2×M16—2682no—
218C2×M16—2682no—
219C2×M16—1344no—
220C22×C8—1342no—
221C22×C8—1342no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C4×Q8—42no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic6711yescharacteristic
229C2×C4⋊C4C2×Dic6711yescharacteristic
230C2×C4⋊C4C2×Dic6711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—42no—
236C2×C42C2×Dic6711yescharacteristic
237C2×C42C22⋊D6711yescharacteristic
238C2×C42C22⋊D6711yescharacteristic
239C2×C42C22×D6711yescharacteristic
240C22⋊C8—1344no—
241#97—2682no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247C24C22⋊Dic6711yescharacteristic
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8#1176161212yescomplemented
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C22⋊Dic6711yescharacteristic
268C22×C4C22⋊Dic6711yescharacteristic
269C22×C4C22⋊Dic6711yescharacteristic
270C22×C4C22×Dic6711yescharacteristic
271C22×C4C2×C22⋊D6711yescharacteristic
272C22×C4C2×C22⋊D6711yescharacteristic
273M16—2688no—
274C2×C8—2682no—
275C2×C8—2682no—
276C2×C8—2682no—
277C2×C8—2682no—
278C2×C8—1344no—
279C2×C8—1344no—
280C4⋊C4—24no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4#1176161212yescomplemented
288C22⋊C4—24no—
289C22⋊C4—42no—
290C42—22no—
291C42—22no—
292C42—41no—
293C42—41no—
294C42—42no—
295C42—42no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#1452397012yes—
303C23C2×C22⋊Dic6711yescharacteristic
304Q8—42no—
305Q8—42no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4#1452395812yes—
337C2×C4#1452397012yes—
338C2×C4C2×#1176161211yescharacteristic
339C2×C4C2×#1176161211yescharacteristic
340C2×C4#1452406811yescharacteristic
341C2×C4#1452407311yescharacteristic
342C2×C4#1452407412yes—
343C8—2684no—
344C8—2684no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#1686336011yescharacteristic
353C22C2×#1452395811yescharacteristic
354C22#1686338011yescharacteristic
355C22#1686339711yescharacteristic
356C22#1686340511yescharacteristic
357C22C2×#1452397011yescharacteristic
358C22C2×#1452397011yescharacteristic
359C4—22no—
360C4—22no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—41no—
367C4—42no—
368C4—42no—
369C4—81no—
370C4—81no—
371C2—42no—
372C2#2255854711yescharacteristic
373C2#2255864811yescharacteristic
374C2#2255865611yescharacteristic
375C2#2255871311yescharacteristic
376C2#2255872911yescharacteristic
377C2#2255879711yescharacteristic
378C2#2255893911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC4×C22⋊Q8×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenterC4×C22⋊Q8D67
hypocenterC67#7634
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22558035C2semidirect
4. #22558032C2semidirect
68. C2×Q8×C67M16semidirect
97. C4⋊C4×C67M16semidirect
195. C67#7634semidirect
251. C2×Q8#11761612semidirect
287. C4⋊C4#11761612semidirect

Maximal subgroups

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

subgroupindex
#225599712
#225599032
#225580352
#225580322
C4×C22⋊Q8×C672
#763467