Finite Group Databasebeta

← #30856334

Subgroups — #30856334

Order 17152.

854 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863457C212yesmaximal
2#22560361C212yesmaximal, complemented
3#22560329C212yesmaximal, complemented
4#22560193C212yesmaximal
5#22560189C212yesmaximal
6#22560066C212yesmaximal, complemented
7#22560062C211yesmaximal, characteristic, complemented
8C4×C4⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
9C4×#117×C67C211yesmaximal, characteristic
10#16863457—28no—
11C2×C4⋊(C67⋊C8)C2212yes—
12C2×C4⋊(C67⋊C8)C2212yes—
13C2×C4⋊(C67⋊C8)C2214yes—
14C2×C4⋊(C67⋊C8)C2214yes—
15C2×C22⋊(C67⋊C8)C2214yes—
16C2×#14523954C2212yes—
17C2×C4×C67⋊C8C2212yes—
18#16863151C2212yes—
19#16863151C2212yes—
20#16863151C2214yes—
21C2×#117×C67C2211yescharacteristic
22C2×#117×C67C2211yescharacteristic
23C22×C42×C67C2211yescharacteristic
24C4×C4⋊C4×C67C412yes—
25C4×C4⋊C4×C67C412yes—
26C4×C22⋊C4×C67C412yes—
27C4×C22⋊C4×C67C412yes—
28#376×C67C2211yescharacteristic
29#376×C67C2211yescharacteristic
30C43×C67C2211yescharacteristic
31C43×C67C2211yescharacteristic
32C22×C67⋊C8C2314yes—
33C22×C67⋊C8C2314yes—
34C4⋊(C67⋊C8)—28no—
35C4⋊(C67⋊C8)—24no—
36C4⋊(C67⋊C8)—24no—
37C4⋊(C67⋊C8)—28no—
38C22⋊(C67⋊C8)—28no—
39#14523954—24no—
40C4×C67⋊C8—24no—
41C23×C4×C67C2×C411yescharacteristic
42C23×C4×C67C2×C411yescharacteristic
43C23×C4×C67C2311yescharacteristic
44#117×C67—24no—
45#117×C67D414yes—
46#117×C67Q814yes—
47C2×C4⋊C4×C67C2×C412yes—
48C2×C4⋊C4×C67C2×C412yes—
49C2×C22⋊C4×C67C2×C412yes—
50C2×C22⋊C4×C67C2×C412yes—
51C2×C42×C67—24no—
52C2×C42×C67C2×C411yescharacteristic
53C2×C42×C67C2×C411yescharacteristic
54C2×C42×C67C2×C412yes—
55C2×C42×C67C2×C412yes—
56C2×C42×C67C2×C412yes—
57C2×C42×C67C2×C412yes—
58C2×C42×C67C2311yescharacteristic
59C2×C42×C67C2311yescharacteristic
60C2×C42×C67C2311yescharacteristic
61C2×C42×C67C2311yescharacteristic
62C2×C42×C67C2311yescharacteristic
63C2×C42×C67C2311yescharacteristic
64#95×C67C2×C412yes—
65#95×C67C2×C412yes—
66#95×C67C2×C412yes—
67#95×C67C2×C412yes—
68C2×C67⋊C8—24no—
69C2×C67⋊C8—24no—
70C2×C67⋊C8—24no—
71C2×C67⋊C8—24no—
72C2×C67⋊C8—24no—
73C2×C67⋊C8—24no—
74C24×C67C22×C411yescharacteristic
75C22×C4×C67—22no—
76C22×C4×C67—22no—
77C22×C4×C67—22no—
78C22×C4×C67—22no—
79C22×C4×C67—24no—
80C22×C4×C67—42no—
81C22×C4×C67C22×C412yes—
82C22×C4×C67C22×C411yescharacteristic
83C22×C4×C67C22×C411yescharacteristic
84C22×C4×C67C22×C412yes—
85C22×C4×C67C22×C412yes—
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C22×C411yescharacteristic
88C22×C4×C67C22×C412yes—
89C22×C4×C67C22×C411yescharacteristic
90C22×C4×C67C2×D412yes—
91C22×C4×C67C2×Q812yes—
92C22×C4×C67C2411yescharacteristic
93C4⋊C4×C67—24no—
94C4⋊C4×C67C4⋊C418yes—
95C22⋊C4×C67—24no—
96C22⋊C4×C67C4⋊C418yes—
97C42×C67—22no—
98C42×C67—22no—
99C42×C67—24no—
100C42×C67—24no—
101C42×C67—22no—
102C42×C67—22no—
103C42×C67—42no—
104C42×C67—42no—
105C42×C67C2×D412yes—
106C42×C67C2×D412yes—
107C42×C67C2×Q812yes—
108C42×C67C2×Q812yes—
109C42×C67#4114yes—
110C42×C67#4114yes—
111C23×C67—21no—
112C23×C67—21no—
113C23×C67—22no—
114C23×C67—22no—
115C23×C67C2×C4⋊C412yes—
116C23×C67C23×C411yescharacteristic
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—22no—
120C2×C4×C67—22no—
121C2×C4×C67—22no—
122C2×C4×C67—22no—
123C2×C4×C67—22no—
124C2×C4×C67—22no—
125C2×C4×C67—22no—
126C2×C4×C67—24no—
127C2×C4×C67—24no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—22no—
133C2×C4×C67—22no—
134C2×C4×C67—22no—
135C2×C4×C67—44no—
136C2×C4×C67—44no—
137C2×C4×C67C2×C4⋊C414yes—
138C2×C4×C67C2×C4⋊C414yes—
139C2×C4×C67C2×C4⋊C412yes—
140C2×C4×C67#11712yes—
141C2×C4×C67#11714yes—
142C2×C4×C67#11712yes—
143C2×C4×C67#13114yes—
144C2×C4×C67C22×D411yescharacteristic
145C2×C4×C67C22×Q811yescharacteristic
146C2×C4×C67C2×#4112yes—
147C2×C4×C67C2×#4112yes—
148C67⋊C8—44no—
149C67⋊C8—44no—
150C22×C67—24no—
151C22×C67—22no—
152C22×C67—41no—
153C22×C67—41no—
154C22×C67—42no—
155C22×C67C22×C4⋊C411yescharacteristic
156C22×C67C2×#11711yescharacteristic
157C22×C67C2×#11711yescharacteristic
158C22×C67C2×#13112yes—
159C22×C67#56812yes—
160C4×C67—24no—
161C4×C67—24no—
162C4×C67—22no—
163C4×C67—24no—
164C4×C67—22no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67—42no—
168C4×C67#43314yescomplemented
169C4×C67#53212yes—
170C4×C67#53312yes—
171#20812—671nomaximal, Sylow
172C2×C67—42no—
173C2×C67#228511yescharacteristic
174C2×C67#290612yes—
175C2×C67C2×#43311yescharacteristic
176C2×C67#292111yescharacteristic
177C2×C67#292512yes—
178C2×#433—672no—
179C4×#117D6711yescharacteristic
180#1842—672no—
181#1832—672no—
182#1798—672no—
183#1797—672no—
184#1768—672no—
185#1767—671no—
186C4×C4⋊C8—671no—
187C67#2081211yescharacteristic, complemented, Sylow
188C2×#117C2×D6711yescharacteristic
189C2×#117C2×D6711yescharacteristic
190C22×C42C2×D6711yescharacteristic
191#433—1348no—
192C2×C4⋊C8—672no—
193C2×C4⋊C8—672no—
194C2×C4⋊C8—674no—
195C2×C4⋊C8—674no—
196C2×C22⋊C8—674no—
197C2×#97—672no—
198C2×C4×C8—672no—
199C4×C4⋊C4Dic6712yes—
200C4×C4⋊C4Dic6712yes—
201C4×C22⋊C4Dic6712yes—
202C4×C22⋊C4Dic6712yes—
203#376C2×D6711yescharacteristic
204#376C2×D6711yescharacteristic
205C43C2×D6711yescharacteristic
206C43C2×D6711yescharacteristic
207#336—674no—
208#336—672no—
209#336—672no—
210C23×C4C2×Dic6711yescharacteristic
211C23×C4C2×Dic6711yescharacteristic
212C23×C4C22×D6711yescharacteristic
213C22×C8—674no—
214C22×C8—674no—
215#117—24no—
216#117Dic13414yes—
217#117D26814yes—
218C2×C4⋊C4C2×Dic6712yes—
219C2×C4⋊C4C2×Dic6712yes—
220C2×C22⋊C4C2×Dic6712yes—
221C2×C22⋊C4C2×Dic6712yes—
222C2×C42—24no—
223C2×C42C2×Dic6712yes—
224C2×C42C2×Dic6712yes—
225C2×C42C2×Dic6711yescharacteristic
226C2×C42C2×Dic6711yescharacteristic
227C2×C42C2×Dic6712yes—
228C2×C42C2×Dic6712yes—
229C2×C42C22×D6711yescharacteristic
230C2×C42C22×D6711yescharacteristic
231C2×C42C22×D6711yescharacteristic
232C2×C42C22×D6711yescharacteristic
233C2×C42C22×D6711yescharacteristic
234C2×C42C22×D6711yescharacteristic
235C4⋊C8—1348no—
236C4⋊C8—1344no—
237C4⋊C8—1344no—
238C4⋊C8—1348no—
239C22⋊C8—1348no—
240#97—1344no—
241C4×C8—1344no—
242#95C2×Dic6712yes—
243#95C2×Dic6712yes—
244#95C2×Dic6712yes—
245#95C2×Dic6712yes—
246C24C22×Dic6711yescharacteristic
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4C22×Dic6711yescharacteristic
254C22×C4C22×Dic6711yescharacteristic
255C22×C4C22×Dic6712yes—
256C22×C4C22×Dic6712yes—
257C22×C4C22×Dic6712yes—
258C22×C4C22×Dic6711yescharacteristic
259C22×C4C22×Dic6712yes—
260C22×C4C22×Dic6711yescharacteristic
261C22×C4C22×Dic6711yescharacteristic
262C22×C4C2×D26812yes—
263C22×C4C2×Dic13412yes—
264C22×C4C23×D6711yescharacteristic
265C2×C8—1344no—
266C2×C8—1344no—
267C2×C8—1344no—
268C2×C8—1344no—
269C2×C8—1344no—
270C2×C8—1344no—
271C4⋊C4—24no—
272C4⋊C4C4⋊Dic6718yes—
273C22⋊C4—24no—
274C22⋊C4C4⋊Dic6718yes—
275C42—22no—
276C42—22no—
277C42—24no—
278C42—24no—
279C42—22no—
280C42—22no—
281C42—42no—
282C42—42no—
283C42C2×D26812yes—
284C42C2×D26812yes—
285C42C2×Dic13412yes—
286C42C2×Dic13412yes—
287C42#1176162814yes—
288C42#1176162814yes—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23C2×C4⋊Dic6712yes—
294C23C23×Dic6711yescharacteristic
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4C2×C4⋊Dic6714yes—
316C2×C4C2×C4⋊Dic6714yes—
317C2×C4C2×C4⋊Dic6712yes—
318C2×C4#1452400212yes—
319C2×C4#1452400214yes—
320C2×C4#1452400212yes—
321C2×C4#1452405014yes—
322C2×C4C22×D26811yescharacteristic
323C2×C4C22×Dic13411yescharacteristic
324C2×C4C2×#1176162812yes—
325C2×C4C2×#1176162812yes—
326C8—2684no—
327C8—2684no—
328C22—22no—
329C22—24no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22C22×C4⋊Dic6711yescharacteristic
334C22C2×#1452400211yescharacteristic
335C22C2×#1452400211yescharacteristic
336C22C2×#1452405012yes—
337C22#1686416312yes—
338C4—22no—
339C4—24no—
340C4—24no—
341C4—22no—
342C4—24no—
343C4—41no—
344C4—41no—
345C4—42no—
346C4#1686345714yescomplemented
347C4#1686394612yes—
348C4#1686395312yes—
349C2—42no—
350C2#2256261611yescharacteristic
351C2#2256877712yes—
352C2C2×#1686345711yescharacteristic
353C2#2256884711yescharacteristic
354C2#2256887512yes—
subgroupG/H
center, FrattiniC22×C4C23×D67
commutator, socleC23×C67C23×C4
FittingC4×#117×C67C2
prime radicalC24×C67C22×C4
cosocleC22×C4×C67C24
hypercenterC4×#117D67
hypocenterC67#20812
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. #22560361C2semidirect
3. #22560329C2semidirect
6. #22560066C2semidirect
7. #22560062C2semidirect
8. C4×C4⋊(C67⋊C8)C2semidirect
168. C4×C67#433semidirect
187. C67#20812semidirect
346. C4#16863457semidirect

Maximal subgroups

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

subgroupindex
C2×#168634572
#225603612
#225603292
#225601932
#225601892
#225600662
#225600622
C4×C4⋊(C67⋊C8)2
C4×#117×C672
#2081267