Finite Group Databasebeta

← #29708139

Subgroups — #29708139

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459224C211yesmaximal, characteristic
2C2×#16839802C212yesmaximal
3#21447169C214yesmaximal, complemented
4#21447164C214yesmaximal, complemented
5#21445558C211yesmaximal, characteristic
6#21445529C212yesmaximal
7C61×#2317C211yesmaximal, characteristic
8C22×#11758194C2212yes—
9#16840393C2212yes—
10#16840393C2212yes—
11#16840390—22no—
12#16840390—22no—
13#16840388—22no—
14#16840319—22no—
15#16840315—24no—
16C2×#14519736C2212yes—
17#16839802—24no—
18#16839802C2218yes—
19C2×#14519710—22no—
20C2×#14519708—22no—
21#16839531C2212yes—
22#16839531C2212yes—
23#16839522C2218yes—
24C2×#117×C61C2211yescharacteristic
25C2×#117×C61C2211yescharacteristic
26C22×C4⋊C4×C61C2211yescharacteristic
27C61×#389—22no—
28C61×#385—22no—
29C61×#384C2212yes—
30C61×#382C2212yes—
31C2×#11758194—22no—
32C2×#11758194—24no—
33C2×#11758194Q818yescomplemented
34C2×#11758194C2314yes—
35C22×#92418—22no—
36#14519777—22no—
37#14519777—22no—
38#14519776—24no—
39#14519776—24no—
40#14519769—24no—
41#14519768—24no—
42#14519768—24no—
43#14519756—24no—
44#14519756—24no—
45#14519754—28no—
46#14519754—28no—
47C4×Dic122—24no—
48C4×Dic122—24no—
49#14519736—22no—
50#14519736—22no—
51#14519736D414yes—
52#14519736C2314yes—
53C2×C4⋊Dic61D412yes—
54C2×C4⋊Dic61D412yes—
55C2×C22⋊Dic61—22no—
56C2×C4×Dic61—22no—
57#14519710—28no—
58#14519708—28no—
59C23×C4×C61—21no—
60C23×C4×C61C2311yescharacteristic
61#117×C61—22no—
62#117×C61—22no—
63#117×C61—22no—
64#117×C61Q814yescomplemented
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—24no—
70C2×C4⋊C4×C61C2312yes—
71C2×C4⋊C4×C61C2311yescharacteristic
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61D412yes—
75C2×C42×C61D412yes—
76C2×C42×C61C2312yes—
77#95×C61—24no—
78C24×C61C2×D411yescharacteristic
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61—42no—
82C22×C4×C61—82no—
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×Q814yes—
89C22×C4×C61C2×Q812yes—
90C22×C4×C61#4112yes—
91C22×C4×C61C2411yescharacteristic
92C22×Dic61—22no—
93C22×Dic61—22no—
94C2×Dic122—24no—
95C2×Dic122—24no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—84no—
103C22⋊C4×C61—24no—
104C22⋊C4×C61—24no—
105C42×C61—22no—
106C42×C61—22no—
107C42×C61—22no—
108C42×C61C2×Q814yes—
109C22⋊Dic61—24no—
110C22⋊Dic61—24no—
111C4⋊Dic61—24no—
112C4⋊Dic61—24no—
113C4⋊Dic61—22no—
114C4⋊Dic61—22no—
115C4⋊Dic61C2×D414yes—
116C4⋊Dic61C2×D414yes—
117#11758196—28no—
118#11758196—28no—
119C4×Dic61—24no—
120C4×Dic61—24no—
121#11758194—28no—
122#11758194—24no—
123#11758194C2×Q818yes—
124C2×#92418—28no—
125C2×#92418—24no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61#12012yes—
129C23×C61C22⋊Q812yes—
130C23×C61C4⋊Q812yes—
131C23×C61C22×D411yescharacteristic
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—24no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—44no—
140C2×C4×C61—42no—
141C2×C4×C61—84no—
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12011yescharacteristic
144C2×C4×C61C22⋊Q814yes—
145C2×C4×C61C22⋊Q814yes—
146C2×C4×C61C22⋊Q812yes—
147C2×C4×C61C4⋊Q814yes—
148C2×C4×C61C4⋊Q814yes—
149C2×C4×C61C4⋊Q812yes—
150C2×C4×C61C22×D412yes—
151C2×C4×C61C22×Q812yes—
152C2×C4×C61C22×Q811yescharacteristic
153C2×C4×C61C2×#4111yescharacteristic
154C2×Dic61—24no—
155C2×Dic61—24no—
156C2×Dic61—82no—
157C2×Dic61—82no—
158Dic122—84no—
159Dic122—84no—
160#92418—28no—
161#23562—611nomaximal, Sylow
162C22×C61—22no—
163C22×C61—21no—
164C22×C61—22no—
165C22×C61—21no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61#39514yes—
169C22×C61C2×#12011yescharacteristic
170C22×C61C2×C22⋊Q811yescharacteristic
171C22×C61C2×C4⋊Q811yescharacteristic
172C4×C61—44no—
173C4×C61—42no—
174C4×C61—42no—
175C4×C61—82no—
176C4×C61#39512yes—
177C4×C61#39512yes—
178C4×C61C2×C22⋊Q812yes—
179C4×C61C2×C4⋊Q812yes—
180Dic61—44no—
181Dic61—44no—
182#3458—611no—
183C2×#428—612no—
184#2317D6111yescharacteristic
185#2061—614no—
186#2060—614no—
187#1742—611no—
188#1733—612no—
189C2×C61—21no—
190C2×C61—21no—
191C2×C61—24no—
192C2×C61C2×#39511yescharacteristic
193C2×C61#302111yescharacteristic
194C2×C61#302211yescharacteristic
195C22×M16—612no—
196#544—612no—
197#544—612no—
198#544—1222no—
199#544—1222no—
200#543—1222no—
201#533—1224no—
202#533—1222no—
203C2×#117—612no—
204C2×#117C2×D6111yescharacteristic
205C2×#117C2×D6111yescharacteristic
206C22×C4⋊C4C2×D6111yescharacteristic
207#428—618no—
208#428—1224no—
209C2×#107—1222no—
210C2×#106—1222no—
211#389—22no—
212#385—22no—
213#384C2×D6112yes—
214#382C2×D6112yes—
215#343—612no—
216#343—612no—
217#339—618no—
218C61#2356211yescharacteristic, complemented, Sylow
219C23×C4—21no—
220C23×C4C22×D6111yescharacteristic
221C2×M16—618no—
222C2×M16—614no—
223C2×M16—1224no—
224C2×M16—1222no—
225C22×C8—1222no—
226C4⋊Q8—1222no—
227C4⋊Q8—1222no—
228C4⋊Q8—1224no—
229C4⋊Q8—1224no—
230#125—1224no—
231#125—1224no—
232#125—1224no—
233C22⋊Q8—1228no—
234C22⋊Q8—1228no—
235C22⋊Q8—1224no—
236C22⋊Q8—1224no—
237C4×Q8—1224no—
238C4×Q8—1224no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—614no—
243#117—614no—
244#117—1222no—
245#117—1222no—
246#117Dic12214yescomplemented
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—612no—
253C2×C4⋊C4—612no—
254C2×C4⋊C4C22×D6112yes—
255C2×C4⋊C4C22×D6111yescharacteristic
256C2×C4⋊C4C22×D6111yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1222no—
259C2×C22⋊C4C22⋊D6112yes—
260C2×C42—1222no—
261C2×C42C22⋊D6112yes—
262C2×C42C22×D6112yes—
263#107—1228no—
264#106—1228no—
265#95—24no—
266C24C2×C22⋊D6111yescharacteristic
267C2×Q8—1224no—
268C2×Q8—1224no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—82no—
273C22×C4—1222no—
274C22×C4—1222no—
275C22×C4C2×Dic12212yes—
276C22×C4#1175822212yes—
277C22×C4D4×D6112yes—
278C22×C4D4×D6112yes—
279C22×C4Q8×D6114yes—
280C22×C4C2×C22⋊D6112yes—
281C22×C4C2×C22⋊D6112yes—
282C22×C4C2×C22⋊D6111yescharacteristic
283C22×C4C23×D6111yescharacteristic
284M16—618no—
285M16—1224no—
286M16—1228no—
287C2×C8—1224no—
288C2×C8—1228no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—84no—
296C4⋊C4—614no—
297C4⋊C4—614no—
298C4⋊C4—1228no—
299C4⋊C4—1228no—
300C4⋊C4—1224no—
301C4⋊C4—1224no—
302C4⋊C4—1222no—
303C4⋊C4—1222no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—1224no—
307C22⋊C4—1224no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—1224no—
312C42—1224no—
313C42C2×Dic12214yes—
314C23—41no—
315C23—41no—
316C23#1451974512yes—
317C23#1451975612yes—
318C23#1451977512yes—
319C23C22×C22⋊D6111yescharacteristic
320Q8—4884no—
321Q8—4884no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—84no—
332C2×C4—1224no—
333C2×C4—1224no—
334C2×C4—4882no—
335C2×C4—4882no—
336C2×C4#1451974511yescharacteristic
337C2×C4#1451974511yescharacteristic
338C2×C4#1451975514yes—
339C2×C4#1451975514yes—
340C2×C4#1451975612yes—
341C2×C4#1451977512yes—
342C2×C4#1451977614yes—
343C2×C4#1451977614yes—
344C2×C4C2×D4×D6112yes—
345C2×C4C2×Q8×D6112yes—
346C2×C4C22×Dic12211yescharacteristic
347C2×C4C2×#1175822211yescharacteristic
348C8—1228no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—22no—
353C22—42no—
354C22—42no—
355C22#1683968214yes—
356C22C2×#1451974511yescharacteristic
357C22C2×#1451975611yescharacteristic
358C22C2×#1451977511yescharacteristic
359C4—44no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—2444no—
364C4—2444no—
365C4#1683968212yes—
366C4#1683968212yes—
367C4C2×#1451975512yes—
368C4C2×#1451977612yes—
369C2—21no—
370C2—24no—
371C2—21no—
372C2C2×#1683968211yescharacteristic
373C2#2145519011yescharacteristic
374C2#2145519711yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2317C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#2317D61
hypocenterC61#23562
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #21447169C2semidirect
4. #21447164C2semidirect
33. C2×#11758194Q8semidirect
64. #117×C61Q8semidirect
218. C61#23562semidirect
246. #117Dic122semidirect

Maximal subgroups

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

subgroupindex
#214592242
C2×#168398022
#214471692
#214471642
#214455582
#214455292
C61×#23172
#2356261