Finite Group Databasebeta

← #29652799

Subgroups — #29652799

Order 15616.

1106 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455914C212yesmaximal, complemented
2#21449098C211yesmaximal, characteristic
3#21444844C214yesmaximal, complemented
4#21444742C214yesmaximal, complemented
5#21444331C212yesmaximal, complemented
6#21444160C211yesmaximal, characteristic, complemented
7C61×#3501C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×C4×C22⋊D61—22no—
10#16840095C2214yes—
11#16840039—22no—
12#16839964—22no—
13#16839912C2214yes—
14#16839860—24no—
15#16839819C2212yes—
16#16839698C2212yes—
17#16839697—21no—
18#16839671C2212yes—
19#16839670—21no—
20#16839608C2211yescharacteristic
21#16839499C2218yes—
22#16839486C2214yes—
23C61×#554—22no—
24C61×#554—24no—
25D4×Q8×C61—22no—
26C61×#544—21no—
27C61×#543C2214yes—
28C61×#533C2212yes—
29C2×C22⋊Q8×C61—22no—
30C61×#518—21no—
31C2×C4×D4×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C22⋊D61—22no—
34C22×C4×D61—21no—
35C2×#11758201—24no—
36C4×D244—22no—
37C4×D244—22no—
38C4×C22⋊D61—48no—
39C2×C4⋊Dic61C2311yescharacteristic
40C2×#11758196—22no—
41C2×C22⋊Dic61—22no—
42C2×#11758198—22no—
43C2×#11758198—21no—
44C2×C4×Dic61—21no—
45#14519710—24no—
46C4⋊#92418—22no—
47#14519701—24no—
48#14519697—24no—
49C22⋊#92418C2314yes—
50#14519688—22no—
51#14519684—21no—
52#14519684C2312yes—
53C22×Q8×C61—22no—
54C22×D4×C61D411yescharacteristic
55C23×C4×C61D412yes—
56C4⋊Q8×C61—24no—
57C4⋊Q8×C61D412yescomplemented
58C4⋊Q8×C61D412yescomplemented
59#125×C61—24no—
60#125×C61—22no—
61C22⋊Q8×C61—24no—
62C22⋊Q8×C61—24no—
63C22⋊Q8×C61—28no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—28no—
67C22⋊Q8×C61C2314yes—
68C4×Q8×C61—22no—
69C4×Q8×C61D414yescomplemented
70C4×D4×C61—24no—
71C4×D4×C61—24no—
72C4×D4×C61—22no—
73C4×D4×C61—22no—
74#117×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61—24no—
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C22⋊C4×C61—24no—
79C2×C22⋊C4×C61C2312yes—
80C2×C42×C61C2311yescharacteristic
81C24×C61C2×D412yes—
82C23×D61—21no—
83C2×Q8×C61—24no—
84C2×Q8×C61—22no—
85C2×Q8×C61—84no—
86C2×Q8×C61C2×D414yes—
87C2×D4×C61—24no—
88C2×D4×C61—22no—
89C22×C4×C61—24no—
90C22×C4×C61—24no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—44no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2411yescharacteristic
98C2×C22⋊D61—44no—
99C2×C22⋊D61—44no—
100C2×C22⋊D61—44no—
101C22×Dic61—22no—
102C22×Dic61—21no—
103C2×D244—22no—
104C2×D244—42no—
105C2×C4×D61—44no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C4⋊C4×C61—24no—
109C4⋊C4×C61—28no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61C2×D414yes—
117C22⋊C4×C61—28no—
118C22⋊C4×C61—24no—
119C22⋊C4×C61—24no—
120C42×C61—24no—
121C42×C61C2×D412yes—
122C42×C61C2×D412yes—
123C22⋊Dic61—44no—
124#11758201—48no—
125#11758198—44no—
126#11758198—42no—
127C4⋊Dic61—22no—
128#11758196—44no—
129C4×Dic61—24no—
130C2×#92418—24no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61#13714yes—
136C23×C61C22×D411yescharacteristic
137C22×D61—22no—
138C22×D61—42no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—42no—
142Q8×C61—44no—
143Q8×C61—44no—
144Q8×C61—84no—
145D4×C61—28no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—24no—
149C2×C4×C61—24no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—44no—
153C2×C4×C61—42no—
154C2×C4×C61—82no—
155C2×C4×C61—84no—
156C2×C4×C61—84no—
157C2×C4×C61#12012yes—
158C2×C4×C61#12011yescharacteristic
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61C22×D412yes—
161C22⋊D61—416no—
162C2×Dic61—24no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—41no—
167D244—24no—
168D244—42no—
169C4×D61—82no—
170C4×D61—82no—
171#92418—44no—
172#18180—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—24no—
176C22×C61—44no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—82no—
180C22×C61C2×#12011yescharacteristic
181C22×C61C2×#13712yes—
182C2×D61—44no—
183C2×D61—42no—
184C2×D61—82no—
185C2×D61—81no—
186C2×D61—81no—
187C4×C61—22no—
188C4×C61—21no—
189C4×C61—22no—
190C4×C61—42no—
191C4×C61—44no—
192C4×C61—44no—
193C4×C61#45312yes—
194Dic61—44no—
195Dic61—81no—
196#3501D6111yescharacteristic, complemented
197#3110—612no—
198#2397—611no—
199#1614—614no—
200#1596—614no—
201#1506—612no—
202#1466—611no—
203C2×C61—22no—
204C2×C61—44no—
205C2×C61#300411yescharacteristic
206C2×C61C2×#45311yescharacteristic
207C2×C61#302511yescharacteristic
208D61—82no—
209#554—22no—
210#554—24no—
211D4×Q8—22no—
212#544—21no—
213#543C2×D6114yes—
214#533C2×D6112yes—
215C2×C22⋊Q8—22no—
216#518—21no—
217C2×C4×D4—611no—
218C2×C4×D4—1222no—
219C2×C4×D4C2×D6111yescharacteristic
220#484—614no—
221#474—1222no—
222#461—1222no—
223#450—614no—
224#440—1224no—
225#432—612no—
226#399—612no—
227#399—1221no—
228#392—612no—
229#392—1221no—
230#376—611no—
231#328—618no—
232#323—614no—
233C61#1818011yescharacteristic, complemented, Sylow
234C22×Q8—22no—
235C22×D4—611no—
236C22×D4—1222no—
237C22×D4C22⋊D6111yescharacteristic
238C23×C4—1221no—
239C23×C4C22⋊D6112yes—
240C2×SD16—1224no—
241C4⋊Q8—24no—
242C4⋊Q8C22⋊D6112yescomplemented
243C4⋊Q8C22⋊D6112yescomplemented
244#125—24no—
245#125—22no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—28no—
249C22⋊Q8—22no—
250C22⋊Q8—24no—
251C22⋊Q8—28no—
252C22⋊Q8C22×D6114yes—
253C4×Q8—22no—
254C4×Q8C22⋊D6114yescomplemented
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—1222no—
260C4×D4—1222no—
261C4×D4—2448no—
262#117—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—1222no—
267C2×C4⋊C4C22⋊D6111yescharacteristic
268C2×C22⋊C4—24no—
269C2×C22⋊C4—1222no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1222no—
272C2×C22⋊C4C22×D6112yes—
273C2×C42—1221no—
274C2×C42C22×D6111yescharacteristic
275#107—1224no—
276C4⋊C8—1222no—
277#103—1224no—
278#102—1224no—
279C22⋊C8—614no—
280#97—1222no—
281#95—612no—
282#95—1221no—
283C24—1221no—
284C24C2×C22⋊D6112yes—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—84no—
288C2×Q8C2×C22⋊D6114yes—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—1222no—
292C2×D4—2442no—
293C2×D4—2444no—
294C2×D4—2444no—
295C2×D4—2444no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—44no—
301C22×C4—1222no—
302C22×C4—1221no—
303C22×C4—2444no—
304C22×C4—2441no—
305C22×C4—2441no—
306C22×C4C2×C22⋊D6111yescharacteristic
307C22×C4C2×C22⋊D6111yescharacteristic
308C22×C4C2×C22⋊D6112yes—
309C22×C4C23×D6111yescharacteristic
310SD16—2448no—
311C2×C8—1224no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—1222no—
321C4⋊C4—2444no—
322C4⋊C4C2×C22⋊D6114yes—
323C22⋊C4—28no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—2444no—
327C22⋊C4—2444no—
328C22⋊C4—2442no—
329C42—24no—
330C42—1224no—
331C42C2×C22⋊D6112yes—
332C42C2×C22⋊D6112yes—
333C23—22no—
334C23—22no—
335C23—42no—
336C23—42no—
337C23—1222no—
338C23—2442no—
339C23—2441no—
340C23—2441no—
341C23—2442no—
342C23#1451980814yes—
343C23C22×C22⋊D6111yescharacteristic
344Q8—44no—
345Q8—44no—
346Q8—84no—
347D4—28no—
348D4—1224no—
349D4—2442no—
350D4—24416no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—1224no—
363C2×C4—4882no—
364C2×C4—4882no—
365C2×C4—2442no—
366C2×C4—2441no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4#1451974612yes—
370C2×C4#1451974611yescharacteristic
371C2×C4#1451974611yescharacteristic
372C2×C4C22×C22⋊D6112yes—
373C8—2444no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—4882no—
382C22—4881no—
383C22—4881no—
384C22—2444no—
385C22—2442no—
386C22C2×#1451974611yescharacteristic
387C22C2×#1451980812yes—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—42no—
392C4—44no—
393C4—44no—
394C4—4881no—
395C4—2444no—
396C4#1683993012yes—
397C2—22no—
398C2—44no—
399C2—4882no—
400C2#2145505311yescharacteristic
401C2C2×#1683993011yescharacteristic
402C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radical#21449098C2
hypercenter#3501D61
hypocenterC61#18180
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21455914C2semidirect
3. #21444844C2semidirect
4. #21444742C2semidirect
5. #21444331C2semidirect
6. #21444160C2semidirect
7. C61×#3501C2semidirect
57. C4⋊Q8×C61D4semidirect
58. C4⋊Q8×C61D4semidirect
69. C4×Q8×C61D4semidirect
196. #3501D61semidirect
233. C61#18180semidirect
242. C4⋊Q8C22⋊D61semidirect
243. C4⋊Q8C22⋊D61semidirect
254. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214559142
#214490982
#214448442
#214447422
#214443312
#214441602
C61×#35012
#1818061