Finite Group Databasebeta

← #30748462

Subgroups — #30748462

Order 17152.

1134 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560133C212yesmaximal, complemented
2#22558539C212yesmaximal, complemented
3#22558536C212yesmaximal
4C22×C22⋊Q8×C67C211yesmaximal, characteristic
5C2×#14523962—24no—
6#16863360C2214yes—
7#16863122—24no—
8#16863120—24no—
9C23×Q8×C67C2211yescharacteristic
10C24×C4×C67C411yescharacteristic
11C2×C22⋊Q8×C67—24no—
12C2×C22⋊Q8×C67—24no—
13C2×C22⋊Q8×C67—24no—
14C22×C4⋊C4×C67C411yescharacteristic
15C22×C4⋊C4×C67C411yescharacteristic
16C22×C4⋊C4×C67C2211yescharacteristic
17C22×C22⋊C4×C67C411yescharacteristic
18C22×C22⋊C4×C67C2211yescharacteristic
19C2×#11761612—24no—
20#14523962—48no—
21C22⋊(C67⋊C8)—24no—
22C22⋊(C67⋊C8)—24no—
23C25×C67C2×C411yescharacteristic
24C22×Q8×C67—22no—
25C22×Q8×C67—22no—
26C22×Q8×C67—44no—
27C22×Q8×C67D414yes—
28C23×C4×C67—22no—
29C23×C4×C67—22no—
30C23×C4×C67—22no—
31C23×C4×C67—44no—
32C23×C4×C67C2×C411yescharacteristic
33C23×C4×C67C2×C411yescharacteristic
34C23×C4×C67C2×C411yescharacteristic
35C23×C4×C67C2×C411yescharacteristic
36C23×C4×C67C2×C411yescharacteristic
37C23×C4×C67C2311yescharacteristic
38C22⋊Q8×C67—232no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67—22no—
41C2×C4⋊C4×C67—22no—
42C2×C4⋊C4×C67—22no—
43C2×C4⋊C4×C67—22no—
44C2×C4⋊C4×C67—22no—
45C2×C4⋊C4×C67—22no—
46C2×C4⋊C4×C67—22no—
47C2×C4⋊C4×C67D414yes—
48C2×C22⋊C4×C67—22no—
49C2×C22⋊C4×C67—22no—
50C2×C22⋊C4×C67—22no—
51C2×C22⋊C4×C67—22no—
52C2×C22⋊C4×C67—22no—
53C2×C22⋊C4×C67D414yes—
54#11761612—48no—
55C2×C67⋊C8—44no—
56C24×C67—24no—
57C24×C67—21no—
58C24×C67—21no—
59C24×C67—42no—
60C24×C67—42no—
61C24×C67C22⋊C412yes—
62C24×C67C22×C411yescharacteristic
63C2×Q8×C67—28no—
64C2×Q8×C67—28no—
65C2×Q8×C67—44no—
66C2×Q8×C67—48no—
67C2×Q8×C67—48no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—24no—
71C22×C4×C67—28no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67—21no—
77C22×C4×C67—21no—
78C22×C4×C67—21no—
79C22×C4×C67—21no—
80C22×C4×C67—42no—
81C22×C4×C67—42no—
82C22×C4×C67—42no—
83C22×C4×C67—48no—
84C22×C4×C67—48no—
85C22×C4×C67—44no—
86C22×C4×C67—44no—
87C22×C4×C67—42no—
88C22×C4×C67—42no—
89C22×C4×C67C22⋊C412yes—
90C22×C4×C67C22⋊C412yes—
91C22×C4×C67C22⋊C412yes—
92C22×C4×C67C22⋊C412yes—
93C22×C4×C67C22⋊C412yes—
94C22×C4×C67C2×D412yes—
95C22×C4×C67C2×D412yes—
96C22×C4×C67C2×D412yes—
97C4⋊C4×C67—28no—
98C4⋊C4×C67—28no—
99C4⋊C4×C67—28no—
100C22⋊C4×C67—28no—
101C22⋊C4×C67—28no—
102C23×C67—22no—
103C23×C67—22no—
104C23×C67—22no—
105C23×C67—24no—
106C23×C67—24no—
107C23×C67—22no—
108C23×C67—22no—
109C23×C67—42no—
110C23×C67—42no—
111C23×C67—42no—
112C23×C67—416no—
113C23×C67—42no—
114C23×C67—42no—
115C23×C67—42no—
116C23×C67#10114yescomplemented
117C23×C67C2×C22⋊C411yescharacteristic
118C23×C67C2×C22⋊C411yescharacteristic
119C23×C67C2×C22⋊C411yescharacteristic
120Q8×C67—416no—
121C2×C4×C67—24no—
122C2×C4×C67—24no—
123C2×C4×C67—22no—
124C2×C4×C67—22no—
125C2×C4×C67—24no—
126C2×C4×C67—22no—
127C2×C4×C67—22no—
128C2×C4×C67—22no—
129C2×C4×C67—22no—
130C2×C4×C67—24no—
131C2×C4×C67—24no—
132C2×C4×C67—42no—
133C2×C4×C67—42no—
134C2×C4×C67—416no—
135C2×C4×C67—42no—
136C2×C4×C67—42no—
137C2×C4×C67—42no—
138C2×C4×C67—42no—
139C2×C4×C67—42no—
140C2×C4×C67—42no—
141C2×C4×C67—42no—
142C2×C4×C67—42no—
143C2×C4×C67—42no—
144C2×C4×C67—42no—
145C2×C4×C67#9914yes—
146C2×C4×C67#12014yes—
147C67⋊C8—84no—
148C22×C67—28no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—24no—
156C22×C67—44no—
157C22×C67—48no—
158C22×C67—44no—
159C22×C67—48no—
160C22×C67—42no—
161C22×C67—42no—
162C22×C67#37911yescharacteristic
163C22×C67C2×#9912yes—
164C22×C67C2×#10112yes—
165C22×C67#41312yes—
166C4×C67—24no—
167C4×C67—24no—
168C4×C67—42no—
169C4×C67—42no—
170C4×C67—42no—
171C4×C67—42no—
172#8550—671nomaximal, Sylow
173C2×C67—22no—
174C2×C67—22no—
175C2×C67—22no—
176C2×C67—44no—
177C2×C67#178111yescharacteristic
178C2×C67#178411yescharacteristic
179C2×C67#178511yescharacteristic
180C22×C22⋊Q8D6711yescharacteristic
181#1784—672no—
182#1463—672no—
183#1462—672no—
184C67#855011yescharacteristic, complemented, Sylow
185C23×Q8C2×D6711yescharacteristic
186C24×C4Dic6711yescharacteristic
187C2×C22⋊Q8—24no—
188C2×C22⋊Q8—24no—
189C2×C22⋊Q8—24no—
190C22×C4⋊C4Dic6711yescharacteristic
191C22×C4⋊C4Dic6711yescharacteristic
192C22×C4⋊C4C2×D6711yescharacteristic
193C22×C22⋊C4Dic6711yescharacteristic
194C22×C22⋊C4C2×D6711yescharacteristic
195C2×#101—1344no—
196#407—674no—
197#324—1344no—
198#323—1344no—
199C25C2×Dic6711yescharacteristic
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—44no—
203C22×Q8C22⋊D6714yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—44no—
208C23×C4C2×Dic6711yescharacteristic
209C23×C4C2×Dic6711yescharacteristic
210C23×C4C2×Dic6711yescharacteristic
211C23×C4C2×Dic6711yescharacteristic
212C23×C4C2×Dic6711yescharacteristic
213C23×C4C22×D6711yescharacteristic
214C2×M16—1344no—
215C22⋊Q8—232no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C22⋊D6714yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C22⋊D6714yes—
231#101—2688no—
232C22⋊C8—1344no—
233C22⋊C8—1344no—
234C24—24no—
235C24—21no—
236C24—21no—
237C24—42no—
238C24—42no—
239C24C22⋊Dic6712yes—
240C24C22×Dic6711yescharacteristic
241C2×Q8—28no—
242C2×Q8—28no—
243C2×Q8—44no—
244C2×Q8—48no—
245C2×Q8—48no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C22⋊Dic6712yes—
268C22×C4C22⋊Dic6712yes—
269C22×C4C22⋊Dic6712yes—
270C22×C4C22⋊Dic6712yes—
271C22×C4C22⋊Dic6712yes—
272C22×C4C2×C22⋊D6712yes—
273C22×C4C2×C22⋊D6712yes—
274C22×C4C2×C22⋊D6712yes—
275M16—2688no—
276C2×C8—2684no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—24no—
286C23—24no—
287C23—22no—
288C23—22no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—416no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23#1452396214yescomplemented
297C23C2×C22⋊Dic6711yescharacteristic
298C23C2×C22⋊Dic6711yescharacteristic
299C23C2×C22⋊Dic6711yescharacteristic
300Q8—416no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—416no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4#1452395814yes—
326C2×C4#1452401214yes—
327C8—5364no—
328C22—28no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—44no—
337C22—48no—
338C22—44no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22#1686325211yescharacteristic
343C22C2×#1452395812yes—
344C22C2×#1452396212yes—
345C22#1686338012yes—
346C4—24no—
347C4—24no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C2—22no—
353C2—22no—
354C2—22no—
355C2—44no—
356C2#2256012411yescharacteristic
357C2#2256013311yescharacteristic
358C2#2256013711yescharacteristic
subgroupG/H
centerC22#16863252
commutatorC24×C67C22×C4
FrattiniC23×C4C22×D67
FittingC22×C22⋊Q8×C67C2
socleC22×C67#379
prime radicalC25×C67C2×C4
cosocleC23×C4×C67C23
hypercenterC22×C22⋊Q8D67
hypocenterC67#8550
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22560133C2semidirect
2. #22558539C2semidirect
116. C23×C67#101semidirect
184. C67#8550semidirect
296. C23#14523962semidirect

Maximal subgroups

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

subgroupindex
#225601332
#225585392
#225585362
C22×C22⋊Q8×C672
#855067