Finite Group Databasebeta

← #30600491

Subgroups — #30600491

Order 15616.

446 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 84 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#21462339C211yesmaximal, characteristic
2#21461253C211yesmaximal, characteristic
3#21461202C211yesmaximal, characteristic
4#21461122C211yesmaximal, characteristic
5#21460715C211yesmaximal, characteristic
6#21460674C211yesmaximal, characteristic
7#21455625C211yesmaximal, characteristic
8#16840781C2211yescharacteristic
9#16840780C2211yescharacteristic
10(C2×#6923)⋊Q8—21no—
11#16840711—22no—
12#16840707—21no—
13#16840706C2211yescharacteristic
14C4⋊C4×#6923—21no—
15C4×C4⋊#6923C2211yescharacteristic
16#16840698—21no—
17#16840616—22no—
18#16840615—21no—
19#16840615—21no—
20#16840612—21no—
21Q8⋊#92420—21no—
22#16840317C2211yescharacteristic
23C4×Q8×D61C2211yescharacteristic
24#16840068—21no—
25#16840067—21no—
26#16840054C411yescharacteristic, complemented
27#16840048C411yescharacteristic, complemented
28#16840047C411yescharacteristic, complemented
29#16839803—21no—
30#16839784C2211yescharacteristic
31#16839769—21no—
32C61×#477C411yescharacteristic, complemented
33C2×C4⋊#6923—21no—
34C2×C4⋊#6923—21no—
35C2×C4×#6923—22no—
36C2×C4×#6923—21no—
37#14519844—21no—
38C4⋊#92420—21no—
39#14519834—21no—
40#14519834—21no—
41C4×#92420—21no—
42C4×#92420—21no—
43(C2×#6923)⋊C4—22no—
44(C2×#6923)⋊C4—22no—
45(C2×#6923)⋊C4—22no—
46(C2×#6923)⋊C4—21no—
47(C2×#6923)⋊C4D412yes—
48(C2×#6923)⋊C4D411yescharacteristic
49(C2×#6923)⋊C4D411yescharacteristic
50C2×Q8×D61D411yescharacteristic
51C2×#11758189—21no—
52#14519777C2×C411yescharacteristic
53#14519768—21no—
54#14519757—21no—
55#14519753—21no—
56Q8×Dic61—21no—
57Q8×Dic61D411yescharacteristic
58C4×Dic122—21no—
59C4×Dic122C2×C411yescharacteristic
60#14519742—21no—
61#14519742—21no—
62#14519742D411yescharacteristic
63#14519735—21no—
64C4⋊C4×D61—21no—
65C4⋊C4×D61D411yescharacteristic
66C42×D61—21no—
67C42×D61C2311yescharacteristic
68#14519710—21no—
69#14519709—21no—
70#14519708—21no—
71C4⋊#92418C2×C411yescharacteristic
72#14519706—21no—
73#14519702—21no—
74#14519701—21no—
75#14519700—21no—
76C4⋊Q8×C61C2×C411yescharacteristic
77C4×Q8×C61C2×C411yescharacteristic
78#107×C61—21no—
79C4⋊C8×C61C2×C411yescharacteristic
80#103×C61—21no—
81C22×#6923—21no—
82C22×#6923—22no—
83C22×#6923—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61C22⋊C411yescharacteristic
86Q8×D61—21no—
87Q8×D61—21no—
88Q8×D61—21no—
89Q8×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61C2×D411yescharacteristic
93C2×C4×D61C2×D411yescharacteristic
94C2×C4×D61C2×D411yescharacteristic
95C2×Dic122—21no—
96C2×Dic122—21no—
97C2×Dic122C22⋊C411yescharacteristic
98C2×#92420—41no—
99C2×#92420—41no—
100C4⋊#6923—42no—
101C4⋊#6923—41no—
102C4⋊#6923—41no—
103C4×#6923—44no—
104C4×#6923—41no—
105C4×#6923—41no—
106C2×C8×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61C22⋊C411yescharacteristic
111C42×C61—21no—
112C42×C61C22×C411yescharacteristic
113#11758198—41no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—41no—
118C4⋊Dic61C22⋊C411yescharacteristic
119#11758196—21no—
120#11758196—21no—
121#11758196—21no—
122#11758196—41no—
123C4×Dic61—21no—
124C4×Dic61—21no—
125C4×Dic61—21no—
126C4×Dic61—41no—
127C4×Dic61C2×D411yescharacteristic
128C4×Dic61#4111yescharacteristic
129C4×Dic61#4111yescharacteristic
130C2×#92418—21no—
131#11758189—41no—
132#11758189—41no—
133C22×D61#12011yescharacteristic
134C2×#6923—22no—
135C2×#6923—41no—
136C2×#6923—41no—
137C2×#6923—42no—
138C2×#6923—41no—
139C2×#6923—41no—
140C2×#6923—42no—
141C2×#6923—41no—
142C2×#6923—42no—
143Q8×C61—21no—
144Q8×C61—21no—
145Q8×C61—81no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—41no—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C4×D411yescharacteristic
151C2×C4×C61C4×D411yescharacteristic
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—41no—
155C2×Dic61#12111yescharacteristic
156C2×Dic61#12111yescharacteristic
157C2×Dic61#12311yescharacteristic
158C4×D61—41no—
159C4×D61—41no—
160C4×D61—42no—
161C4×D61—41no—
162C4×D61—41no—
163C4×D61#13511yescharacteristic
164C4×D61#13511yescharacteristic
165C4×D61#13711yescharacteristic
166C4×D61#13711yescharacteristic
167Dic122—21no—
168Dic122—21no—
169Dic122—41no—
170Dic122—41no—
171Dic122—81no—
172#92420—81no—
173#92420—81no—
174C8×C61—41no—
175#92418—41no—
176#9178—611nomaximal, Sylow
177C22×C61#38611yescharacteristic
178C2×D61—21no—
179C2×D61—21no—
180C2×D61#45211yescharacteristic
181C2×D61#45211yescharacteristic
182#6923—42no—
183#6923—81no—
184#6923—82no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—41no—
188C4×C61—41no—
189C4×C61—81no—
190C4×C61#41611yescharacteristic
191C4×C61#41911yescharacteristic
192Dic61—21no—
193Dic61—21no—
194Dic61—41no—
195Dic61—41no—
196Dic61—81no—
197Dic61#48511yescharacteristic
198Dic61#48511yescharacteristic
199#3078—611no—
200#2356—611no—
201#1518—611no—
202#1503—611no—
203#1479—611no—
204#1303—611no—
205#1286—611no—
206C2×C61#187311yescharacteristic
207C2×C61#190511yescharacteristic
208C2×C61#190611yescharacteristic
209D61—21no—
210D61—21no—
211#533—611no—
212C2×C4×Q8—611no—
213#479—1221no—
214#479—1221no—
215#477—611no—
216#477#692311yescharacteristic, complemented
217#475—611no—
218#475—611no—
219#431—611no—
220#431—611no—
221#428—1221no—
222#423—611no—
223#419—1221no—
224#391—1221no—
225#387—1222no—
226#383—1221no—
227#382—611no—
228C4×C4⋊C4—611no—
229C4×C4⋊C4—1221no—
230#376—1221no—
231#328—1221no—
232#328—1221no—
233#328—1222no—
234Q8⋊C8—1221no—
235Q8⋊C8—1221no—
236C61#917811yescharacteristic, complemented, Sylow
237C22×Q8—611no—
238C2×M16—1221no—
239C4⋊Q8—611no—
240C4⋊Q8C2×#692311yescharacteristic
241#125—1221no—
242C22⋊Q8—1221no—
243C22⋊Q8—1221no—
244C4×Q8—611no—
245C4×Q8—611no—
246C4×Q8—611no—
247C4×Q8—1221no—
248C4×Q8—1221no—
249C4×Q8—1221no—
250C4×Q8—1221no—
251C4×Q8C2×#692311yescharacteristic
252#117—1221no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C42—611no—
258C2×C42—1221no—
259C2×C42—1221no—
260C2×C42—1222no—
261#107—21no—
262#107—1221no—
263#106—1221no—
264#106—1221no—
265C4⋊C8—611no—
266C4⋊C8—1221no—
267C4⋊C8—1221no—
268C4⋊C8—1221no—
269C4⋊C8C2×#692311yescharacteristic
270#103—21no—
271#103—1221no—
272#103—1221no—
273#103—1221no—
274C22⋊C8—1221no—
275C22⋊C8—1221no—
276C4×C8—1221no—
277C4×C8—1221no—
278#95—611no—
279#95—611no—
280#95—612no—
281#95—1221no—
282#95—1222no—
283#95—1222no—
284#95—1222no—
285C2×Q8—21no—
286C2×Q8—611no—
287C2×Q8—1221no—
288C2×Q8—1221no—
289C2×Q8—1221no—
290C2×Q8—1221no—
291C2×Q8—1221no—
292C2×Q8—1221no—
293C2×Q8C22⋊#692311yescharacteristic
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—611no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—1222no—
302M16—2441no—
303M16—2441no—
304C2×C8—21no—
305C2×C8—1221no—
306C2×C8—2441no—
307C2×C8—2441no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—611no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C4⋊C4—1221no—
316C4⋊C4—1221no—
317C4⋊C4—1221no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2442no—
323C4⋊C4C22⋊#692311yescharacteristic
324C22⋊C4—2441no—
325C42—21no—
326C42—611no—
327C42—611no—
328C42—611no—
329C42—1221no—
330C42—1221no—
331C42—1221no—
332C42—2441no—
333C42—2441no—
334C42—2441no—
335C42—2444no—
336C42C22×#692311yescharacteristic
337C23—611no—
338Q8—21no—
339Q8—21no—
340Q8—81no—
341Q8—1221no—
342Q8—1221no—
343Q8—4881no—
344Q8—2441no—
345Q8—2441no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—611no—
353C2×C4—611no—
354C2×C4—611no—
355C2×C4—611no—
356C2×C4—1222no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2442no—
364C2×C4—2441no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2441no—
368C2×C4—2442no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4C2×C22⋊#692311yescharacteristic
374C2×C4D4×#692311yescharacteristic
375C2×C4D4×#692311yescharacteristic
376C8—41no—
377C8—4881no—
378C8—4881no—
379C8—2441no—
380C22—611no—
381C22—611no—
382C22—1221no—
383C22—1221no—
384C22#1684071011yescharacteristic
385C4—21no—
386C4—21no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—611no—
391C4—611no—
392C4—1221no—
393C4—1221no—
394C4—4882no—
395C4—4881no—
396C4—4881no—
397C4—2442no—
398C4—2441no—
399C4—2441no—
400C4#1684075211yescharacteristic
401C4#1684075511yescharacteristic
402C2—1221no—
403C2—1221no—
404C2#2146180511yescharacteristic
405C2#2146185411yescharacteristic
406C2#2146185611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#477C4
socleC22×C61#386
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#477#6923
hypocenterC61#9178
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
26. #16840054C4semidirect
27. #16840048C4semidirect
28. #16840047C4semidirect
32. C61×#477C4semidirect
216. #477#6923semidirect
236. C61#9178semidirect

Maximal subgroups

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

subgroupindex
#214623392
#214612532
#214612022
#214611222
#214607152
#214606742
#214556252
#917861