Finite Group Databasebeta

← #29689552

Subgroups — #29689552

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic
3#21448212C211yesmaximal, characteristic
4#21446793C211yesmaximal, characteristic, complemented
5#21446792C211yesmaximal, characteristic, complemented
6#21446791C211yesmaximal, characteristic, complemented
7#21446775C214yesmaximal, complemented
8#21445913C214yesmaximal
9C61×#1993C211yesmaximal, characteristic, complemented
10#16840317—24no—
11#16840317C418yescomplemented
12C2×#14519777C2211yescharacteristic
13C2×#14519768C2212yes—
14C2×#14519757—21no—
15C2×#14519753—21no—
16C2×Q8×Dic61—21no—
17C2×#14519742—21no—
18C2×#14519735C2212yes—
19C2×C42×D61C2211yescharacteristic
20#16839817—24no—
21C2×#14519706C2212yes—
22C2×#14519689C2212yes—
23C2×#14519688C2211yescharacteristic
24#16839629—21no—
25#16839628C2211yescharacteristic
26#16839624—21no—
27#16839619C2212yes—
28C4⋊C4×Dic61C2212yes—
29#16839609—21no—
30#16839607—21no—
31#16839525C2214yes—
32#16839524C2214yes—
33#16839524C2214yes—
34C2×C4⋊Q8×C61C2211yescharacteristic
35C2×#97×C61C2211yescharacteristic
36C61×#382C2211yescharacteristic
37C61×#340C2214yes—
38C22×Dic122—21no—
39C22×C4×D61D411yescharacteristic
40C22×C4×D61D411yescharacteristic
41C22×C4×D61C2311yescharacteristic
42C22×#92418C2312yes—
43#14519777—22no—
44#14519777C2×C414yes—
45#14519768—24no—
46#14519768C2×C418yes—
47#14519757—28no—
48#14519753—28no—
49Q8×Dic61—28no—
50#14519742—28no—
51#14519735—24no—
52#14519735C2×C418yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2312yes—
57C2×#11758196—22no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61—22no—
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61D411yescharacteristic
64C2×C4×Dic61C2311yescharacteristic
65C42×D61C2×C414yes—
66C42×D61D414yes—
67#14519706—24no—
68#14519689—24no—
69#14519688—22no—
70#14519684—21no—
71#14519684—21no—
72#14519684—22no—
73#14519684—22no—
74C22×Q8×C61—21no—
75C22×C8×C61C2312yes—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61C2×C414yes—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61C2312yes—
80C2×C4⋊C4×C61C2312yes—
81C2×C42×C61C2311yescharacteristic
82#97×C61—22no—
83#95×C61—21no—
84#95×C61—21no—
85C23×D61C2×D411yescharacteristic
86C2×Q8×C61—24no—
87C2×Q8×C61—44no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2411yescharacteristic
93C22×Dic61—22no—
94C22×Dic61—22no—
95C22×Dic61C2×D411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97C22×Dic61C2×D411yescharacteristic
98C2×C4×D61—22no—
99C2×C4×D61—22no—
100C2×C4×D61—42no—
101C2×C4×D61—42no—
102C2×C4×D61C22⋊C414yes—
103C2×C4×D61C22⋊C414yes—
104C2×C4×D61C22×C412yes—
105C2×C4×D61C2×D412yes—
106C2×Dic122—24no—
107C2×Dic122—44no—
108C2×C8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—24no—
114C4⋊C4×C61C22×C414yes—
115C42×C61C22×C412yes—
116C42×C61C2×D412yes—
117#11758198—28no—
118#11758198—44no—
119C4⋊Dic61—24no—
120C4⋊Dic61—22no—
121C4⋊Dic61—28no—
122C4⋊Dic61—24no—
123C4⋊Dic61C22×C414yes—
124#11758196—28no—
125C4×Dic61—28no—
126C4×Dic61—44no—
127C4×Dic61C22⋊C414yes—
128C4×Dic61C22⋊C414yes—
129C4×Dic61C22×C412yes—
130C4×Dic61C2×D412yes—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—22no—
134C23×C61C22×D411yescharacteristic
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—22no—
138C22×D61—22no—
139C22×D61C2×C22⋊C412yes—
140Q8×C61—48no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—44no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61C2×C22⋊C412yes—
149C2×C4×C61C2×C22⋊C412yes—
150C2×C4×C61C23×C411yescharacteristic
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C612−1+412yes—
153C2×C4×C612−1+412yes—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—24no—
159C2×Dic61—44no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61C2×C22⋊C412yes—
164C2×Dic61C2×C22⋊C412yes—
165C2×Dic61C2×C22⋊C412yes—
166C4×D61—24no—
167C4×D61—24no—
168C4×D61—44no—
169C4×D61—44no—
170Dic122—48no—
171C8×C61—42no—
172#92418—42no—
173#21842—611nomaximal, Sylow
174C22×C61C22×C22⋊C411yescharacteristic
175C22×C61#53611yescharacteristic
176C22×C61#53611yescharacteristic
177C22×C61#57512yes—
178C22×C61#57512yes—
179C2×D61—22no—
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—42no—
183C2×D61—42no—
184C4×C61—22no—
185C4×C61—22no—
186C4×C61—44no—
187C4×C61—82no—
188C4×C61#51912yes—
189C4×C61#53612yes—
190Dic61—22no—
191Dic61—22no—
192Dic61—22no—
193Dic61—44no—
194Dic61—82no—
195C2×#533—611no—
196C2×#432—611no—
197#2283—611no—
198#1993—611no—
199#1993—611no—
200#1993—611no—
201#1993D6111yescharacteristic, complemented
202#1989—614no—
203#1825—614no—
204C2×C61#231011yescharacteristic
205C2×C61#289611yescharacteristic
206C2×C61#289611yescharacteristic
207C2×C61#324012yes—
208C2×C61#324012yes—
209D61—42no—
210#533—618no—
211#533—1224no—
212C2×C4⋊Q8—611no—
213C2×C4⋊Q8C2×D6111yescharacteristic
214C2×#125—612no—
215C2×C22⋊Q8—1221no—
216C2×C22⋊Q8—1221no—
217C2×C4×Q8—1221no—
218C2×C4×Q8—1221no—
219C2×#117—612no—
220C22×C42—611no—
221#432—1224no—
222C2×C4⋊C8—612no—
223C2×C22⋊C8—612no—
224C2×#97—611no—
225C2×#97C2×D6111yescharacteristic
226#383—1221no—
227#382—611no—
228#382C2×D6111yescharacteristic
229#381—1221no—
230#380—612no—
231C4×C4⋊C4—612no—
232C4×C22⋊C4—1221no—
233#376—1221no—
234#340—614no—
235#340—614no—
236#340—614no—
237#340C2×D6114yes—
238C61#2184211yescharacteristic, complemented, Sylow
239C22×Q8—21no—
240C22×Q8—1221no—
241C23×C4—611no—
242C23×C4—611no—
243C23×C4—611no—
244C22×C8—612no—
245C22×C8C22×D6112yes—
246C4⋊Q8—22no—
247C4⋊Q8—614no—
248C4⋊Q8—1222no—
249C4⋊Q8C4×D6114yes—
250#125—618no—
251#125—1224no—
252C22⋊Q8—1228no—
253C22⋊Q8—1228no—
254C4×Q8—1228no—
255C4×Q8—1228no—
256#117—618no—
257#117—1224no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—612no—
260C2×C4⋊C4—612no—
261C2×C4⋊C4—1222no—
262C2×C4⋊C4—1222no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4C22×D6112yes—
265C2×C4⋊C4C22×D6112yes—
266C2×C22⋊C4—1222no—
267C2×C22⋊C4—1222no—
268C2×C42—611no—
269C2×C42—611no—
270C2×C42—611no—
271C2×C42—614no—
272C2×C42—614no—
273C2×C42—1222no—
274C2×C42—1222no—
275C2×C42C22×D6111yescharacteristic
276C4⋊C8—1224no—
277C22⋊C8—1224no—
278#97—22no—
279#97—1222no—
280#95—21no—
281#95—21no—
282#95—1222no—
283#95—1222no—
284#95—1221no—
285#95—1221no—
286C24—611no—
287C2×Q8—24no—
288C2×Q8—44no—
289C2×Q8—1224no—
290C2×Q8—2444no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—612no—
297C22×C4—612no—
298C22×C4—614no—
299C22×C4—614no—
300C22×C4—1222no—
301C22×C4—1222no—
302C22×C4—1222no—
303C22×C4—1222no—
304C22×C4—2442no—
305C22×C4—2442no—
306C22×C4D4×D6111yescharacteristic
307C22×C4D4×D6111yescharacteristic
308C22×C4C23×D6111yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—1222no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—614no—
319C4⋊C4—1228no—
320C4⋊C4—1228no—
321C4⋊C4—1222no—
322C4⋊C4—1224no—
323C4⋊C4—1224no—
324C4⋊C4C2×C4×D6114yes—
325C22⋊C4—1228no—
326C22⋊C4—2444no—
327C42—612no—
328C42—612no—
329C42—614no—
330C42—614no—
331C42—1228no—
332C42—2444no—
333C42C2×C4×D6112yes—
334C42D4×D6112yes—
335C23—612no—
336C23—1222no—
337C23—1222no—
338C23—1221no—
339C23—1221no—
340C23C2×D4×D6111yescharacteristic
341Q8—48no—
342Q8—2448no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—612no—
351C2×C4—612no—
352C2×C4—612no—
353C2×C4—1222no—
354C2×C4—1222no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—1224no—
358C2×C4—1224no—
359C2×C4—1224no—
360C2×C4—2444no—
361C2×C4—2444no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2444no—
366C2×C4C22⋊C4×D6112yes—
367C2×C4C22⋊C4×D6112yes—
368C2×C4C22×C4×D6111yescharacteristic
369C2×C4C2×D4×D6111yescharacteristic
370C2×C4#1451982412yes—
371C2×C4#1451982512yes—
372C8—42no—
373C8—2442no—
374C22—1222no—
375C22—1222no—
376C22—1222no—
377C22—2442no—
378C22—2442no—
379C22C2×C22⋊C4×D6111yescharacteristic
380C22#1684033211yescharacteristic
381C22#1684033711yescharacteristic
382C22#1684055812yes—
383C22#1684056112yes—
384C4—22no—
385C4—22no—
386C4—44no—
387C4—82no—
388C4—1222no—
389C4—1222no—
390C4—1222no—
391C4—4882no—
392C4—2444no—
393C4#1684023812yes—
394C4#1684033512yes—
395C2—2442no—
396C2#2144840211yescharacteristic
397C2#2145433311yescharacteristic
398C2#2145433711yescharacteristic
399C2#2145705612yes—
400C2#2145706212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1993C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1993D61
hypocenterC61#21842
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
4. #21446793C2semidirect
5. #21446792C2semidirect
6. #21446791C2semidirect
7. #21446775C2semidirect
9. C61×#1993C2semidirect
11. #16840317C4semidirect
201. #1993D61semidirect
238. C61#21842semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168398172
#214482122
#214467932
#214467922
#214467912
#214467752
#214459132
C61×#19932
#2184261