Finite Group Databasebeta

← #30745260

Subgroups — #30745260

Order 17152.

552 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863453C211yesmaximal, characteristic
2C2×#16863449C211yesmaximal, characteristic
3#22558039C214yesmaximal, complemented
4C67×#2920C211yesmaximal, characteristic
5#16863453—21no—
6#16863453—21no—
7#16863453—21no—
8#16863453—21no—
9#16863449—21no—
10#16863449—21no—
11#16863449C412yes—
12#16863449C2212yes—
13C2×C4⋊(C67⋊C8)—21no—
14C2×C4⋊(C67⋊C8)C412yes—
15C2×C22⋊(C67⋊C8)—21no—
16C2×C22⋊(C67⋊C8)C412yes—
17C2×#14523954C2212yes—
18C2×C4×C67⋊C8C412yes—
19C22×C42×C67C2211yescharacteristic
20#432×C67—21no—
21#432×C67—21no—
22#431×C67C412yes—
23#431×C67C2212yes—
24#423×C67—21no—
25#407×C67—21no—
26C4×M16×C67C412yes—
27C22×C67⋊C8—22no—
28C22×C67⋊C8C2×C414yes—
29C4⋊(C67⋊C8)—22no—
30C4⋊(C67⋊C8)—22no—
31C4⋊(C67⋊C8)—22no—
32C4⋊(C67⋊C8)—22no—
33C22⋊(C67⋊C8)—22no—
34C22⋊(C67⋊C8)—22no—
35C22⋊(C67⋊C8)—22no—
36C22⋊(C67⋊C8)—22no—
37#14523954—22no—
38#14523954—22no—
39C4×C67⋊C8—22no—
40C4×C67⋊C8C2×C414yes—
41C23×C4×C67C2×C411yescharacteristic
42C23×C4×C67D411yescharacteristic
43C23×C4×C67Q811yescharacteristic
44C2×M16×C67—22no—
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67C2×C411yescharacteristic
48C2×C42×C67C2×C411yescharacteristic
49C2×C42×C67C2×C411yescharacteristic
50C2×C42×C67D411yescharacteristic
51C2×C42×C67D411yescharacteristic
52C2×C42×C67D411yescharacteristic
53C2×C42×C67D411yescharacteristic
54C2×C42×C67D411yescharacteristic
55C2×C42×C67Q811yescharacteristic
56C2×C42×C67C2311yescharacteristic
57C4⋊C8×C67—22no—
58C4⋊C8×C67—22no—
59C22⋊C8×C67—22no—
60C22⋊C8×C67—22no—
61#97×C67—22no—
62C4×C8×C67C2×C414yes—
63C2×C67⋊C8—24no—
64C2×C67⋊C8—24no—
65C2×C67⋊C8—28no—
66C2×C67⋊C8—44no—
67C24×C67C4⋊C411yescharacteristic
68C22×C4×C67—24no—
69C22×C4×C67—42no—
70C22×C4×C67—42no—
71C22×C4×C67C4211yescharacteristic
72C22×C4×C67C4211yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C22⋊C411yescharacteristic
79C22×C4×C67C4⋊C411yescharacteristic
80C22×C4×C67C4⋊C411yescharacteristic
81C22×C4×C67C4⋊C411yescharacteristic
82C22×C4×C67C4⋊C411yescharacteristic
83C22×C4×C67C4⋊C411yescharacteristic
84C22×C4×C67C4⋊C411yescharacteristic
85C22×C4×C67C4⋊C411yescharacteristic
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C2×D411yescharacteristic
88C22×C4×C67C2×Q811yescharacteristic
89M16×C67—44no—
90C2×C8×C67—24no—
91C42×C67C4211yescharacteristic
92C42×C67C4211yescharacteristic
93C42×C67C22⋊C411yescharacteristic
94C42×C67C22⋊C411yescharacteristic
95C42×C67C22⋊C411yescharacteristic
96C42×C67C22⋊C411yescharacteristic
97C42×C67C22⋊C411yescharacteristic
98C42×C67C22⋊C411yescharacteristic
99C42×C67C4⋊C411yescharacteristic
100C42×C67C4⋊C411yescharacteristic
101C42×C67C4⋊C411yescharacteristic
102C42×C67C4⋊C411yescharacteristic
103C42×C67C22×C411yescharacteristic
104C42×C67C22×C411yescharacteristic
105C42×C67C2×D411yescharacteristic
106C42×C67C2×D411yescharacteristic
107C23×C67—22no—
108C23×C67—22no—
109C23×C67—41no—
110C23×C67#9511yescharacteristic
111C23×C67#9511yescharacteristic
112C23×C67C2×C4⋊C411yescharacteristic
113C2×C4×C67—22no—
114C2×C4×C67—22no—
115C2×C4×C67—22no—
116C2×C4×C67—22no—
117C2×C4×C67—22no—
118C2×C4×C67—22no—
119C2×C4×C67—22no—
120C2×C4×C67—22no—
121C2×C4×C67—22no—
122C2×C4×C67—22no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67#9511yescharacteristic
131C2×C4×C67#9511yescharacteristic
132C2×C4×C67#9511yescharacteristic
133C2×C4×C67#9511yescharacteristic
134C2×C4×C67#9511yescharacteristic
135C2×C4×C67#9511yescharacteristic
136C2×C4×C67C2×C4211yescharacteristic
137C2×C4×C67C2×C22⋊C411yescharacteristic
138C2×C4×C67C2×C22⋊C411yescharacteristic
139C2×C4×C67C2×C22⋊C411yescharacteristic
140C2×C4×C67C2×C4⋊C411yescharacteristic
141C2×C4×C67C2×C4⋊C411yescharacteristic
142C2×C4×C67#13114yes—
143C8×C67—44no—
144C67⋊C8—44no—
145C67⋊C8—44no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—44no—
154C22×C67H(3,ℤ/4)12yes—
155C22×C67C2×#9511yescharacteristic
156C22×C67#42412yes—
157C4×C67—21no—
158C4×C67—21no—
159C4×C67—21no—
160C4×C67—21no—
161C4×C67—21no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67#33812yes—
167C4×C67#40512yes—
168C4×C67#40812yes—
169#7863—671nomaximal, Sylow
170C2×C67—22no—
171C2×C67—22no—
172C2×C67—41no—
173C2×C67#172911yescharacteristic
174C2×C67C2×H(3,ℤ/4)11yescharacteristic
175C2×C67C2×#33811yescharacteristic
176#2920D6711yescharacteristic
177C2×#432—671no—
178C2×#431—671no—
179#1275—674no—
180C67#786311yescharacteristic, complemented, Sylow
181C22×C42C2×D6711yescharacteristic
182#432—21no—
183#432—21no—
184#432—1341no—
185#432—1341no—
186#432—1341no—
187#432—1341no—
188#431—672no—
189#431—672no—
190#431—1341no—
191#431—1341no—
192#431Dic6712yes—
193#431C2×D6712yes—
194#423—21no—
195C2×C4⋊C8—672no—
196C2×C4⋊C8—1341no—
197#407—21no—
198C2×C22⋊C8—672no—
199C2×C22⋊C8—1341no—
200C4×M16Dic6712yes—
201C2×#97—672no—
202C2×C4×C8—672no—
203C23×C4Dic13411yescharacteristic
204C23×C4C4×D6711yescharacteristic
205C23×C4C22⋊D6711yescharacteristic
206C2×M16—22no—
207C22×C8—674no—
208C22×C8—1342no—
209C2×C42Dic13411yescharacteristic
210C2×C42C4×D6711yescharacteristic
211C2×C42C4×D6711yescharacteristic
212C2×C42C4×D6711yescharacteristic
213C2×C42D26811yescharacteristic
214C2×C42D26811yescharacteristic
215C2×C42C2×Dic6711yescharacteristic
216C2×C42C2×Dic6711yescharacteristic
217C2×C42C22⋊D6711yescharacteristic
218C2×C42C22⋊D6711yescharacteristic
219C2×C42C22⋊D6711yescharacteristic
220C2×C42C22×D6711yescharacteristic
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—1342no—
224C4⋊C8—1342no—
225C4⋊C8—1342no—
226C4⋊C8—1342no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—1342no—
230C22⋊C8—1342no—
231C22⋊C8—1342no—
232C22⋊C8—1342no—
233#97—22no—
234#97—1342no—
235#97—1342no—
236C4×C8—674no—
237C4×C8—1342no—
238C4×C8C2×Dic6714yes—
239C24Dic67⋊C411yescharacteristic
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C4×Dic6711yescharacteristic
244C22×C4C4×Dic6711yescharacteristic
245C22×C4(C2×D67)⋊C411yescharacteristic
246C22×C4(C2×D67)⋊C411yescharacteristic
247C22×C4(C2×D67)⋊C411yescharacteristic
248C22×C4(C2×D67)⋊C411yescharacteristic
249C22×C4C22⋊Dic6711yescharacteristic
250C22×C4C22⋊Dic6711yescharacteristic
251C22×C4Dic67⋊C411yescharacteristic
252C22×C4Dic67⋊C411yescharacteristic
253C22×C4Dic67⋊C411yescharacteristic
254C22×C4Dic67⋊C411yescharacteristic
255C22×C4Dic67⋊C411yescharacteristic
256C22×C4C4⋊Dic6711yescharacteristic
257C22×C4C4⋊Dic6711yescharacteristic
258C22×C4C2×C4×D6711yescharacteristic
259C22×C4C2×C22⋊D6711yescharacteristic
260C22×C4C2×Dic13411yescharacteristic
261M16—44no—
262C2×C8—24no—
263C2×C8—2684no—
264C2×C8—1344no—
265C2×C8—1344no—
266C2×C8—1348no—
267C42C4×Dic6711yescharacteristic
268C42C4×Dic6711yescharacteristic
269C42(C2×D67)⋊C411yescharacteristic
270C42(C2×D67)⋊C411yescharacteristic
271C42(C2×D67)⋊C411yescharacteristic
272C42(C2×D67)⋊C411yescharacteristic
273C42C22⋊Dic6711yescharacteristic
274C42C22⋊Dic6711yescharacteristic
275C42Dic67⋊C411yescharacteristic
276C42Dic67⋊C411yescharacteristic
277C42C4⋊Dic6711yescharacteristic
278C42C4⋊Dic6711yescharacteristic
279C42C2×C4×D6711yescharacteristic
280C42C22×Dic6711yescharacteristic
281C42C2×C22⋊D6711yescharacteristic
282C42C2×D26811yescharacteristic
283C23—22no—
284C23—22no—
285C23—41no—
286C23C67⋊#9511yescharacteristic
287C23C67⋊#9511yescharacteristic
288C23C2×Dic67⋊C411yescharacteristic
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4C67⋊#9511yescharacteristic
307C2×C4C67⋊#9511yescharacteristic
308C2×C4C67⋊#9511yescharacteristic
309C2×C4C67⋊#9511yescharacteristic
310C2×C4C67⋊#9511yescharacteristic
311C2×C4C67⋊#9511yescharacteristic
312C2×C4C2×C4×Dic6711yescharacteristic
313C2×C4C2×(C2×D67)⋊C411yescharacteristic
314C2×C4C2×(C2×D67)⋊C411yescharacteristic
315C2×C4C2×C22⋊Dic6711yescharacteristic
316C2×C4C2×Dic67⋊C411yescharacteristic
317C2×C4C2×C4⋊Dic6711yescharacteristic
318C2×C4#1452405114yes—
319C8—44no—
320C8—2684no—
321C8—2684no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—44no—
330C22#1686315212yes—
331C22C2×C67⋊#9511yescharacteristic
332C22#1686342412yes—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4C67⋊#33812yes—
343C4#1686335412yes—
344C4#1686336412yes—
345C2—22no—
346C2—22no—
347C2—41no—
348C2#2255990711yescharacteristic
349C2C2×#1686315211yescharacteristic
350C2C2×C67⋊#33811yescharacteristic
subgroupG/H
centerC2×C4C2×C4×Dic67
commutatorC2×C4×C67C2×C42
FrattiniC2×C42C22×D67
FittingC67×#2920C2
socleC22×C67C2×#95
prime radicalC24×C67C4⋊C4
cosocleC2×C42×C67C23
hypercenter#2920D67
hypocenterC67#7863
LCSG ▷ C2×C4×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #2920

Complements

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

NH ≅ G/Ntype
3. #22558039C2semidirect
180. C67#7863semidirect

Maximal subgroups

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

subgroupindex
C2×#168634532
C2×#168634492
#225580392
C67×#29202
#786367