Finite Group Databasebeta

← #29736747

Subgroups — #29736747

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456201C212yesmaximal, complemented
2#21450811C211yesmaximal, characteristic
3#21446977C214yesmaximal, complemented
4#21446764C212yesmaximal, complemented
5#21446207C214yesmaximal
6C4×#14519694C211yesmaximal, characteristic
7C61×#3501C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9C2×#14519764C2212yes—
10#16840132—24no—
11#16840114—22no—
12#16840114—22no—
13#16839770C2218yes—
14C2×#14519694C2212yes—
15C4×#11758194C2212yes—
16#16839687—22no—
17#16839667C2212yes—
18#16839622—22no—
19C4×C4⋊Dic61C2211yescharacteristic
20#16839531C2212yes—
21#16839521C2218yes—
22C61×#554—22no—
23C61×#554—24no—
24D4×Q8×C61—22no—
25C61×#544C2212yes—
26C61×#543—22no—
27C61×#533C2212yes—
28C2×C22⋊Q8×C61—22no—
29C61×#518C2212yes—
30C2×C4×D4×C61C2211yescharacteristic
31C22×Dic122C2312yes—
32C2×#11758194C2314yes—
33#14519777—22no—
34#14519777—22no—
35#14519764—28no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C22⋊Dic61—24no—
40C2×C4×Dic61C2312yes—
41#14519700—28no—
42#14519700—28no—
43#14519694—24no—
44#14519688—22no—
45C4×#92418—22no—
46#14519684—21no—
47#14519684—24no—
48C22×Q8×C61—22no—
49C22×D4×C61C2311yescharacteristic
50C23×C4×C61D412yes—
51C4⋊Q8×C61—24no—
52C4⋊Q8×C61D412yes—
53C4⋊Q8×C61D412yes—
54#125×C61—24no—
55#125×C61D414yes—
56C22⋊Q8×C61—28no—
57C22⋊Q8×C61—28no—
58C22⋊Q8×C61—24no—
59C22⋊Q8×C61—24no—
60C22⋊Q8×C61—22no—
61C22⋊Q8×C61—22no—
62C22⋊Q8×C61—24no—
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C4×D4×C61—24no—
66C4×D4×C61—24no—
67C4×D4×C61—22no—
68C4×D4×C61—22no—
69#117×C61C2314yes—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—24no—
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61D412yes—
75C2×C42×C61C2311yescharacteristic
76C24×C61C2×D412yes—
77C2×Q8×C61—24no—
78C2×Q8×C61—22no—
79C2×Q8×C61—22no—
80C2×Q8×C61—84no—
81C2×D4×C61—22no—
82C2×D4×C61—22no—
83C2×D4×C61#4114yes—
84C22×C4×C61—24no—
85C22×C4×C61—24no—
86C22×C4×C61—22no—
87C22×C4×C61—22no—
88C22×C4×C61—44no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2411yescharacteristic
93C22×Dic61—22no—
94C22×Dic61—22no—
95C2×Dic122—24no—
96C2×Dic122—24no—
97C2×Dic122—24no—
98C4⋊C4×C61—24no—
99C4⋊C4×C61—28no—
100C4⋊C4×C61—24no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61C2×D414yes—
106C4⋊C4×C61C2×D414yes—
107C22⋊C4×C61—28no—
108C22⋊C4×C61—24no—
109C22⋊C4×C61—24no—
110C42×C61—24no—
111C42×C61C2×D412yes—
112C42×C61C2×D412yes—
113C22⋊Dic61—48no—
114C4⋊Dic61—42no—
115C4⋊Dic61—42no—
116C4⋊Dic61—41no—
117C4⋊Dic61—41no—
118C4×Dic61—24no—
119#11758194—24no—
120#11758194—24no—
121C2×#92418—24no—
122C23×C61—22no—
123C23×C61—22no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61#12414yes—
127C23×C61C22×D411yescharacteristic
128Q8×C61—44no—
129Q8×C61—82no—
130Q8×C61—84no—
131D4×C61—28no—
132C2×C4×C61—22no—
133C2×C4×C61—24no—
134C2×C4×C61—24no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—44no—
138C2×C4×C61—42no—
139C2×C4×C61—82no—
140C2×C4×C61—84no—
141C2×C4×C61—84no—
142C2×C4×C61#12012yes—
143C2×C4×C61#12011yescharacteristic
144C2×C4×C61#12011yescharacteristic
145C2×C4×C61C22×D412yes—
146C2×C4×C61C2×#4112yes—
147C2×C4×C612−1+412yes—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—42no—
154Dic122—48no—
155#92418—44no—
156#25917—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—24no—
160C22×C61—44no—
161C22×C61—42no—
162C22×C61—42no—
163C22×C61—82no—
164C22×C61C2×#12011yescharacteristic
165C22×C61C2×#12411yescharacteristic
166C22×C61#53611yescharacteristic
167C4×C61—21no—
168C4×C61—22no—
169C4×C61—42no—
170C4×C61—44no—
171C4×C61—44no—
172C4×C61#45512yes—
173C4×C61#54814yes—
174Dic61—82no—
175Dic61—82no—
176#3501D6111yescharacteristic
177#3139—612no—
178#2547—611no—
179#2028—614no—
180#1986—612no—
181#1885—614no—
182C4×#100—611no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—44no—
186C2×C61#243611yescharacteristic
187C2×C61C2×#45511yescharacteristic
188C2×C61#302211yescharacteristic
189#554—22no—
190#554—24no—
191D4×Q8—22no—
192#544C2×D6112yes—
193#543—22no—
194#533C2×D6112yes—
195C2×C4⋊Q8—611no—
196C2×#124—612no—
197C2×C22⋊Q8—22no—
198#518C2×D6112yes—
199C2×C4×D4C2×D6111yescharacteristic
200#491—1224no—
201#487—1222no—
202#487—1222no—
203#419—618no—
204C2×#100—612no—
205C4×M16—612no—
206#397—1222no—
207#391—612no—
208#380—1222no—
209C4×C4⋊C4—611no—
210#343—612no—
211#339—618no—
212C61#2591711yescharacteristic, complemented, Sylow
213C22×Q8—22no—
214C22×Q8—612no—
215C22×D4C22×D6111yescharacteristic
216C23×C4C22⋊D6112yes—
217C2×M16—614no—
218C4⋊Q8—24no—
219C4⋊Q8—1222no—
220C4⋊Q8—1222no—
221C4⋊Q8C22⋊D6112yes—
222C4⋊Q8C22⋊D6112yes—
223#125—24no—
224#125C22⋊D6114yes—
225#124—1228no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230C22⋊Q8—22no—
231C22⋊Q8—24no—
232C22⋊Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—22no—
238C4×D4—22no—
239#117C22×D6114yes—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4C22×D6111yescharacteristic
246C2×C22⋊C4—24no—
247C2×C22⋊C4—1224no—
248C2×C22⋊C4C22⋊D6112yes—
249C2×C42—612no—
250C2×C42C22×D6111yescharacteristic
251#103—1228no—
252#103—1228no—
253#100—1224no—
254#97—1222no—
255C4×C8—1222no—
256#95—1221no—
257#95—1224no—
258C24C2×C22⋊D6112yes—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—84no—
263C2×Q8—1224no—
264C2×Q8—1224no—
265C2×Q8—1224no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4#1175822614yes—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—44no—
274C22×C4—1222no—
275C22×C4—1222no—
276C22×C4C2×C22⋊D6111yescharacteristic
277C22×C4C2×C22⋊D6112yes—
278C22×C4C2×C22⋊D6111yescharacteristic
279C22×C4C23×D6111yescharacteristic
280M16—1224no—
281M16—1224no—
282C2×C8—1224no—
283C4⋊C4—24no—
284C4⋊C4—28no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—2442no—
291C4⋊C4—2442no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C4⋊C4C2×C22⋊D6114yes—
295C4⋊C4C2×C22⋊D6114yes—
296C22⋊C4—28no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—2448no—
300C42—24no—
301C42—1224no—
302C42C2×C22⋊D6112yes—
303C42C2×C22⋊D6112yes—
304C23—22no—
305C23—22no—
306C23—42no—
307C23—42no—
308C23#1451976614yes—
309C23C22×C22⋊D6111yescharacteristic
310Q8—44no—
311Q8—82no—
312Q8—84no—
313Q8—2448no—
314D4—28no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4—2442no—
331C2×C4#1451974612yes—
332C2×C4#1451974611yescharacteristic
333C2×C4#1451974611yescharacteristic
334C2×C4C22×C22⋊D6112yes—
335C2×C4C2×#1175822612yes—
336C2×C4#1451982512yes—
337C8—2444no—
338C22—21no—
339C22—21no—
340C22—24no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—82no—
345C22C2×#1451974611yescharacteristic
346C22C2×#1451976611yescharacteristic
347C22#1684033611yescharacteristic
348C4—21no—
349C4—22no—
350C4—42no—
351C4—44no—
352C4—44no—
353C4—4882no—
354C4—4882no—
355C4#1683994012yes—
356C4#1684042314yes—
357C2—21no—
358C2—21no—
359C2—44no—
360C2#2144947311yescharacteristic
361C2C2×#1683994011yescharacteristic
362C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#3501D61
hypocenterC61#25917
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21456201C2semidirect
3. #21446977C2semidirect
4. #21446764C2semidirect
212. C61#25917semidirect

Maximal subgroups

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

subgroupindex
#214562012
#214508112
#214469772
#214467642
#214462072
C4×#145196942
C61×#35012
#2591761