Finite Group Databasebeta

← #30863730

Subgroups — #30863730

Order 17152.

898 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863527C211yesmaximal, characteristic, complemented
2C2×D4×C67⋊C8C211yesmaximal, characteristic
3#22561146C211yesmaximal, characteristic, complemented
4#22561004C211yesmaximal, characteristic
5#22560824C214yesmaximal, complemented
6#22560326C211yesmaximal, characteristic, complemented
7#22559987C214yesmaximal, complemented
8C4×#14523954C211yesmaximal, characteristic, complemented
9C4×#124×C67C211yesmaximal, characteristic
10C23×C67⋊C8C2212yes—
11#16863527—24no—
12D4×C67⋊C8—24no—
13C2×C4⋊(C67⋊C8)C2212yes—
14C2×C4⋊(C67⋊C8)C2211yescharacteristic
15C2×C4⋊(C67⋊C8)C2211yescharacteristic
16C2×C22⋊(C67⋊C8)C2212yes—
17C2×C22⋊(C67⋊C8)C2214yes—
18C2×#14523954C2212yes—
19C2×#14523954C2212yes—
20C2×#14523954C2212yes—
21C2×C4×C67⋊C8C2211yescharacteristic
22C2×C4×C67⋊C8C2211yescharacteristic
23#16863151C2214yes—
24#16863151C2214yes—
25C2×#124×C67C411yescharacteristic
26C2×C4×Q8×C67C2211yescharacteristic
27C2×C4×D4×C67C2211yescharacteristic
28#391×C67C411yescharacteristic
29#390×C67C411yescharacteristic
30#388×C67C414yes—
31#382×C67C411yescharacteristic
32C4×C22⋊C4×C67C2214yes—
33C43×C67C2211yescharacteristic
34C22×C67⋊C8—24no—
35C22×C67⋊C8—28no—
36C22×C67⋊C8C2312yes—
37C22×C67⋊C8C2312yes—
38C22×C67⋊C8C2311yescharacteristic
39C22×C67⋊C8C2312yes—
40C22×C67⋊C8C2311yescharacteristic
41C4⋊(C67⋊C8)—24no—
42C4⋊(C67⋊C8)—22no—
43C4⋊(C67⋊C8)—22no—
44C22⋊(C67⋊C8)—24no—
45C22⋊(C67⋊C8)—28no—
46#14523954—24no—
47#14523954—24no—
48#14523954—24no—
49C4×C67⋊C8—22no—
50C4×C67⋊C8D414yes—
51C22×Q8×C67C2×C411yescharacteristic
52C22×D4×C67C2×C411yescharacteristic
53C23×C4×C67C2312yes—
54#124×C67—24no—
55C4×Q8×C67—22no—
56C4×Q8×C67—22no—
57C4×D4×C67—22no—
58C4×D4×C67—22no—
59C2×C4⋊C4×C67C2×C411yescharacteristic
60C2×C4⋊C4×C67C2×C411yescharacteristic
61C2×C4⋊C4×C67C2×C412yes—
62C2×C22⋊C4×C67C2×C412yes—
63C2×C22⋊C4×C67C2×C414yes—
64C2×C22⋊C4×C67C2×C414yes—
65C2×C42×C67C2×C411yescharacteristic
66C2×C42×C67C2×C411yescharacteristic
67C2×C42×C67C2×C412yes—
68C2×C42×C67C2312yes—
69C2×C42×C67C2312yes—
70C2×C42×C67C2311yescharacteristic
71#95×C67C2×C414yes—
72#95×C67C2×C414yes—
73C2×C67⋊C8—22no—
74C2×C67⋊C8—22no—
75C2×C67⋊C8—22no—
76C2×C67⋊C8—22no—
77C2×C67⋊C8—22no—
78C2×C67⋊C8—24no—
79C2×C67⋊C8—24no—
80C2×C67⋊C8—48no—
81C2×C67⋊C8C2×D412yes—
82C2×C67⋊C8C2×D412yes—
83C2×C67⋊C8#4114yes—
84C2×C67⋊C8#4114yes—
85C24×C67C22×C412yes—
86C2×Q8×C67—22no—
87C2×Q8×C67—22no—
88C2×Q8×C67—22no—
89C2×D4×C67—22no—
90C2×D4×C67—22no—
91C2×D4×C67M1614yes—
92C22×C4×C67—22no—
93C22×C4×C67—22no—
94C22×C4×C67—24no—
95C22×C4×C67—24no—
96C22×C4×C67C22×C412yes—
97C22×C4×C67C22×C412yes—
98C22×C4×C67C22×C412yes—
99C22×C4×C67C22×C412yes—
100C22×C4×C67C22×C412yes—
101C22×C4×C67C22×C411yescharacteristic
102C22×C4×C67C22×C411yescharacteristic
103C22×C4×C67C2411yescharacteristic
104C4⋊C4×C67—22no—
105C4⋊C4×C67—22no—
106C4⋊C4×C67—24no—
107C22⋊C4×C67—28no—
108C22⋊C4×C67—24no—
109C22⋊C4×C67—44no—
110C42×C67—22no—
111C42×C67—22no—
112C42×C67—22no—
113C42×C67—22no—
114C42×C67—22no—
115C42×C67—22no—
116C42×C67—42no—
117C42×C67C2×D412yes—
118C42×C67#4112yes—
119C23×C67—22no—
120C23×C67—24no—
121C23×C67—22no—
122C23×C67—24no—
123C23×C67C2×M1614yes—
124C23×C67C23×C411yescharacteristic
125Q8×C67—44no—
126D4×C67—28no—
127C2×C4×C67—22no—
128C2×C4×C67—22no—
129C2×C4×C67—22no—
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—22no—
133C2×C4×C67—24no—
134C2×C4×C67—22no—
135C2×C4×C67—22no—
136C2×C4×C67—22no—
137C2×C4×C67—22no—
138C2×C4×C67—22no—
139C2×C4×C67—22no—
140C2×C4×C67—24no—
141C2×C4×C67—42no—
142C2×C4×C67—44no—
143C2×C4×C67—42no—
144C2×C4×C67C4×D412yes—
145C2×C4×C67C4×D412yes—
146C2×C4×C67C2×M1612yes—
147C2×C4×C67#13112yes—
148C2×C4×C67C22×D411yescharacteristic
149C2×C4×C67C2×#4112yes—
150C2×C4×C67C2×#4112yes—
151C2×C4×C67C2×#4111yescharacteristic
152C67⋊C8—24no—
153C67⋊C8—22no—
154C67⋊C8—42no—
155C67⋊C8—42no—
156C67⋊C8#12414yescomplemented
157C22×C67—24no—
158C22×C67—28no—
159C22×C67—42no—
160C22×C67—44no—
161C22×C67—42no—
162C22×C67C2×C4×D411yescharacteristic
163C22×C67C4×#4112yes—
164C22×C67C22×M1611yescharacteristic
165C22×C67C2×#13111yescharacteristic
166C22×C67#56812yes—
167C4×C67—22no—
168C4×C67—42no—
169C4×C67—42no—
170C4×C67—42no—
171C4×C67—42no—
172C4×C67—42no—
173C4×C67#43514yes—
174C4×C67C2×#12411yescharacteristic
175C4×C67#52912yes—
176C4×C67#53211yescharacteristic
177#21633—671nomaximal, Sylow
178C2×C67—44no—
179C2×C67C4×#12411yescharacteristic
180C2×C67C2×#43511yescharacteristic
181C2×C67#293211yescharacteristic
182C2×C67#296912yes—
183C2×C67#297012yes—
184C2×#446—671no—
185C2×C8×D4—671no—
186C4×#124D6711yescharacteristic
187#1984—671no—
188#1954—671no—
189#1924—674no—
190#1831—671no—
191#1752—674no—
192C4×#97—671no—
193C67#2163311yescharacteristic, complemented, Sylow
194C23×C8—672no—
195C2×#124Dic6711yescharacteristic
196C2×C4×Q8C2×D6711yescharacteristic
197C2×C4×D4C2×D6711yescharacteristic
198#446—1344no—
199C8×D4—1344no—
200C2×C4⋊C8—671no—
201C2×C4⋊C8—671no—
202C2×C4⋊C8—672no—
203C2×C22⋊C8—672no—
204C2×C22⋊C8—674no—
205C2×#97—672no—
206C2×#97—672no—
207C2×#97—672no—
208C2×C4×C8—671no—
209C2×C4×C8—671no—
210#391Dic6711yescharacteristic
211#390Dic6711yescharacteristic
212#388Dic6714yes—
213#382Dic6711yescharacteristic
214C4×C22⋊C4C2×D6714yes—
215C43C2×D6711yescharacteristic
216#336—674no—
217#336—674no—
218C22×Q8C2×Dic6711yescharacteristic
219C22×D4C2×Dic6711yescharacteristic
220C23×C4C22×D6712yes—
221C22×C8—671no—
222C22×C8—672no—
223C22×C8—671no—
224C22×C8—672no—
225C22×C8—672no—
226C22×C8—1348no—
227C22×C8—1344no—
228#124—24no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4C2×Dic6711yescharacteristic
234C2×C4⋊C4C2×Dic6711yescharacteristic
235C2×C4⋊C4C2×Dic6712yes—
236C2×C22⋊C4C2×Dic6712yes—
237C2×C22⋊C4C2×Dic6714yes—
238C2×C22⋊C4C2×Dic6714yes—
239C2×C42C2×Dic6711yescharacteristic
240C2×C42C2×Dic6711yescharacteristic
241C2×C42C2×Dic6712yes—
242C2×C42C22×D6712yes—
243C2×C42C22×D6712yes—
244C2×C42C22×D6711yescharacteristic
245C4⋊C8—1344no—
246C4⋊C8—1342no—
247C4⋊C8—1342no—
248C22⋊C8—1348no—
249C22⋊C8—1344no—
250#97—1344no—
251#97—1344no—
252#97—1344no—
253C4×C8—674no—
254C4×C8—1342no—
255#95C2×Dic6714yes—
256#95C2×Dic6714yes—
257C24C22×Dic6712yes—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4#1176161214yes—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4C22×Dic6712yes—
269C22×C4C22×Dic6712yes—
270C22×C4C22×Dic6712yes—
271C22×C4C22×Dic6712yes—
272C22×C4C22×Dic6712yes—
273C22×C4C22×Dic6711yescharacteristic
274C22×C4C22×Dic6711yescharacteristic
275C22×C4C23×D6711yescharacteristic
276C2×C8—2688no—
277C2×C8—672no—
278C2×C8—674no—
279C2×C8—672no—
280C2×C8—674no—
281C2×C8—1342no—
282C2×C8—1342no—
283C2×C8—1344no—
284C2×C8—1344no—
285C2×C8—1342no—
286C2×C8—1342no—
287C2×C8—1342no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C22⋊C4—28no—
292C22⋊C4—24no—
293C22⋊C4—44no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—42no—
301C42D4×D6712yes—
302C42#1176162712yes—
303C23—22no—
304C23—24no—
305C23—22no—
306C23—24no—
307C23C2×#1176161214yes—
308C23C23×Dic6711yescharacteristic
309Q8—44no—
310D4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4D4×Dic6712yes—
329C2×C4D4×Dic6712yes—
330C2×C4C2×#1176161212yes—
331C2×C4#1452405012yes—
332C2×C4C2×D4×D6711yescharacteristic
333C2×C4#41×D6712yes—
334C2×C4#41×D6712yes—
335C2×C4C2×#1176162711yescharacteristic
336C8—2682no—
337C8—2682no—
338C8—674no—
339C8—1344no—
340C8—1342no—
341C22—28no—
342C22—24no—
343C22—42no—
344C22—42no—
345C22—44no—
346C22C2×D4×Dic6711yescharacteristic
347C22#41×Dic6712yes—
348C22C22×#1176161211yescharacteristic
349C22C2×#1452405011yescharacteristic
350C22#1686416312yes—
351C4—22no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4#1686346814yes—
358C4#124×D6711yescharacteristic
359C4#1686392512yes—
360C4#1686394511yescharacteristic
361C2—44no—
362C2#124×Dic6711yescharacteristic
363C2C2×#1686346811yescharacteristic
364C2#2256892711yescharacteristic
365C2#2256921412yes—
366C2#2256922312yes—
subgroupG/H
center, FrattiniC22×C4C23×D67
commutator, socleC23×C67C23×C4
FittingC4×#124×C67C2
prime radicalC22×D4×C67C2×C4
cosocleC22×C4×C67C24
hypercenterC4×#124D67
hypocenterC67#21633
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16863527C2semidirect
3. #22561146C2semidirect
5. #22560824C2semidirect
6. #22560326C2semidirect
7. #22559987C2semidirect
8. C4×#14523954C2semidirect
156. C67⋊C8#124semidirect
193. C67#21633semidirect

Maximal subgroups

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

subgroupindex
C2×#168635272
C2×D4×C67⋊C82
#225611462
#225610042
#225608242
#225603262
#225599872
C4×#145239542
C4×#124×C672
#2163367