Finite Group Databasebeta

← #30744141

Subgroups — #30744141

Order 17152.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560277C211yesmaximal, characteristic
2#22560189C211yesmaximal, characteristic
3#22558035C212yesmaximal, complemented
4#22558032C212yesmaximal, complemented
5C4×C22⋊Q8×C67C211yesmaximal, characteristic
6C2×C4⋊(C67⋊C8)—21no—
7C2×C22⋊(C67⋊C8)—21no—
8C2×C22⋊(C67⋊C8)—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—
23C22×C67⋊C8—22no—
24C22×C67⋊C8—22no—
25C4⋊(C67⋊C8)—42no—
26C22⋊(C67⋊C8)—42no—
27C22⋊(C67⋊C8)—42no—
28C22×Q8×C67C2×C411yescharacteristic
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C23×C4×C67D411yescharacteristic
32C22⋊Q8×C67—22no—
33C22⋊Q8×C67—22no—
34C4×Q8×C67—42no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C22⋊C4×C67—21no—
42C2×C22⋊C4×C67—21no—
43C2×C42×C67—21no—
44C2×C42×C67—21no—
45C2×C42×C67—42no—
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67D411yescharacteristic
48C2×C42×C67D411yescharacteristic
49C2×C42×C67C2311yescharacteristic
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55C2×C67⋊C8—42no—
56C2×C67⋊C8—42no—
57C2×C67⋊C8—42no—
58C2×C67⋊C8—42no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—42no—
61C24×C67C22⋊C411yescharacteristic
62C2×Q8×C67—21no—
63C2×Q8×C67—41no—
64C2×Q8×C67—41no—
65C2×Q8×C67M1612yes—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—22no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—22no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—41no—
75C22×C4×C67—41no—
76C22×C4×C67—41no—
77C22×C4×C67—41no—
78C22×C4×C67—41no—
79C22×C4×C67C22⋊C411yescharacteristic
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67SD1612yes—
83C22×C4×C67Q1612yes—
84C22×C4×C67C22×C411yescharacteristic
85C22×C4×C67C2×D411yescharacteristic
86C22×C4×C67C2×D411yescharacteristic
87C4⋊C4×C67—24no—
88C4⋊C4×C67—21no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—42no—
92C4⋊C4×C67—42no—
93C4⋊C4×C67M1612yes—
94C22⋊C4×C67—24no—
95C22⋊C4×C67—42no—
96C42×C67—22no—
97C42×C67—22no—
98C42×C67—41no—
99C42×C67—41no—
100C42×C67—42no—
101C42×C67—42no—
102C23×C67—21no—
103C23×C67—21no—
104C23×C67—21no—
105C23×C67—21no—
106C23×C67—21no—
107C23×C67—21no—
108C23×C67#10312yes—
109C23×C67C2×C22⋊C411yescharacteristic
110Q8×C67—42no—
111Q8×C67—42no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—22no—
123C2×C4×C67—22no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—42no—
128C2×C4×C67—42no—
129C2×C4×C67—44no—
130C2×C4×C67—42no—
131C2×C4×C67—42no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
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#9912yes—
143C2×C4×C67#10312yes—
144C2×C4×C67C2×M1611yescharacteristic
145C2×C4×C67C2×M1611yescharacteristic
146C2×C4×C67C2×SD1611yescharacteristic
147C2×C4×C67C2×Q1611yescharacteristic
148C2×C4×C67#13512yes—
149C67⋊C8—82no—
150C67⋊C8—82no—
151C22×C67—22no—
152C22×C67—22no—
153C22×C67—22no—
154C22×C67—41no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67#40711yescharacteristic
159C22×C67C2×#9911yescharacteristic
160C22×C67#41311yescharacteristic
161C22×C67C2×#10311yescharacteristic
162C22×C67#41611yescharacteristic
163C22×C67#42111yescharacteristic
164C22×C67#42111yescharacteristic
165C4×C67—21no—
166C4×C67—21no—
167C4×C67—22no—
168C4×C67—22no—
169C4×C67—21no—
170C4×C67—22no—
171C4×C67—21no—
172C4×C67—41no—
173C4×C67—42no—
174C4×C67—42no—
175C4×C67—81no—
176C4×C67—81no—
177#7648—671nomaximal, Sylow
178C2×C67—42no—
179C2×C67#146511yescharacteristic
180C2×C67#148711yescharacteristic
181C2×C67#148811yescharacteristic
182C2×C67#151611yescharacteristic
183C2×C67#152011yescharacteristic
184C2×C67#153511yescharacteristic
185C2×C67#154211yescharacteristic
186C4×C22⋊Q8D6711yescharacteristic
187#1817—671no—
188#1797—671no—
189#1272—672no—
190#1271—672no—
191C67#764811yescharacteristic, complemented, Sylow
192C2×C22⋊Q8Dic6711yescharacteristic
193C2×C4×Q8Dic6711yescharacteristic
194C22×C42C2×D6711yescharacteristic
195C2×C4⋊C8—1341no—
196C2×C22⋊C8—1341no—
197C2×C22⋊C8—1341no—
198#391C2×D6711yescharacteristic
199#389C2×D6711yescharacteristic
200#387—21no—
201#385—21no—
202#380Dic6711yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4Dic6711yescharacteristic
205C4×C22⋊C4—21no—
206#336—672no—
207#336—672no—
208#336—1341no—
209C22×Q8C2×Dic6711yescharacteristic
210C23×C4C2×Dic6711yescharacteristic
211C23×C4C22⋊D6711yescharacteristic
212C23×C4C22⋊D6711yescharacteristic
213C22×C8—1342no—
214C22×C8—1342no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C4×Q8—42no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×Dic6711yescharacteristic
222C2×C4⋊C4C2×Dic6711yescharacteristic
223C2×C4⋊C4C2×Dic6711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42C2×Dic6711yescharacteristic
230C2×C42C22⋊D6711yescharacteristic
231C2×C42C22⋊D6711yescharacteristic
232C2×C42C22×D6711yescharacteristic
233C4⋊C8—2682no—
234C22⋊C8—2682no—
235C22⋊C8—2682no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241C24C22⋊Dic6711yescharacteristic
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8#1176161212yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C22⋊Dic6711yescharacteristic
260C22×C4C22⋊Dic6711yescharacteristic
261C22×C4C22⋊Dic6711yescharacteristic
262C22×C4#1176161512yes—
263C22×C4#1176161812yes—
264C22×C4C22×Dic6711yescharacteristic
265C22×C4C2×C22⋊D6711yescharacteristic
266C22×C4C2×C22⋊D6711yescharacteristic
267C2×C8—2682no—
268C2×C8—2682no—
269C2×C8—2682no—
270C2×C8—2682no—
271C2×C8—2682no—
272C2×C8—2682no—
273C4⋊C4—24no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4#1176161212yes—
280C22⋊C4—24no—
281C22⋊C4—42no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—42no—
287C42—42no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1452396712yes—
295C23C2×C22⋊Dic6711yescharacteristic
296Q8—42no—
297Q8—42no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4#1452395812yes—
329C2×C4#1452396712yes—
330C2×C4C2×#1176161211yescharacteristic
331C2×C4C2×#1176161211yescharacteristic
332C2×C4C2×#1176161511yescharacteristic
333C2×C4C2×#1176161811yescharacteristic
334C2×C4#1452406312yes—
335C8—5362no—
336C8—5362no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#1686336011yescharacteristic
345C22C2×#1452395811yescharacteristic
346C22#1686338011yescharacteristic
347C22C2×#1452396711yescharacteristic
348C22#1686339211yescharacteristic
349C22#1686341311yescharacteristic
350C22#1686341311yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—42no—
360C4—42no—
361C4—81no—
362C4—81no—
363C2—42no—
364C2#2255854711yescharacteristic
365C2#2255863811yescharacteristic
366C2#2255864311yescharacteristic
367C2#2255876511yescharacteristic
368C2#2255878111yescharacteristic
369C2#2255884011yescharacteristic
370C2#2255886611yescharacteristic
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#7648
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
191. C67#7648semidirect

Maximal subgroups

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

subgroupindex
#225602772
#225601892
#225580352
#225580322
C4×C22⋊Q8×C672
#764867