Finite Group Databasebeta

← #30600925

Subgroups — #30600925

Order 15616.

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

normal subgroups only

This group has 456 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#21462745C211yesmaximal, characteristic
2#21461962C211yesmaximal, characteristic, complemented
3#21461334C211yesmaximal, characteristic, complemented
4#21461154C211yesmaximal, characteristic
5#21460727C211yesmaximal, characteristic, complemented
6#21460703C211yesmaximal, characteristic, complemented
7#21455630C211yesmaximal, characteristic
8#16840813—21no—
9#16840793—22no—
10D4×#92420—21no—
11#16840782—21no—
12#16840779C2211yescharacteristic
13#16840765C2211yescharacteristic
14#16840765C2211yescharacteristic
15C2×#14519844—21no—
16C2×#14519838—22no—
17C4×#11758217C2211yescharacteristic
18#16840621—21no—
19#16840621—21no—
20Q8⋊#92420—21no—
21D4⋊#92420—21no—
22C2×#14519776C2211yescharacteristic
23#41×Dic61C2211yescharacteristic
24#16840055—21no—
25#16840050—21no—
26#16840045—21no—
27#16840040—21no—
28#16839804C411yescharacteristic
29#16839802C411yescharacteristic
30#16839785C2211yescharacteristic
31#16839763C411yescharacteristic
32C61×#417C411yescharacteristic
33C2×#11758217—21no—
34C2×#11758217—21no—
35C2×#11758217D412yes—
36C22×#92420—22no—
37#14519844—22no—
38#14519844—21no—
39#14519844—21no—
40#14519844D411yescharacteristic
41#14519844D411yescharacteristic
42C4⋊#92420—21no—
43C4⋊#92420—21no—
44#14519838—44no—
45C22⋊#92420—22no—
46C22⋊#92420—21no—
47#14519832—22no—
48#14519832—21no—
49C4×#92420—21no—
50C4×#92420—21no—
51C22×Dic122—21no—
52C2×#11758194C2×C411yescharacteristic
53#14519776—21no—
54#14519776—21no—
55#14519776—21no—
56#14519776—21no—
57Q8×Dic61D411yescharacteristic
58Q8×Dic61D411yescharacteristic
59D4×Dic61—21no—
60D4×Dic61—21no—
61D4×Dic61D411yescharacteristic
62D4×Dic61D411yescharacteristic
63#14519736—21no—
64#14519736C2×C411yescharacteristic
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×#11758196—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61C2311yescharacteristic
69#14519711—21no—
70#14519710—21no—
71#14519709—21no—
72#14519708—21no—
73#14519706—21no—
74#14519706—21no—
75#14519701—21no—
76#14519698—21no—
77C2×#41×C61C2×C411yescharacteristic
78C2×M16×C61C2×C411yescharacteristic
79C2×C4⋊C4×C61C2×C411yescharacteristic
80#103×C61—21no—
81#102×C61—21no—
82#41×C61—22no—
83#41×C61—41no—
84C2×Q8×C61C22⋊C411yescharacteristic
85C2×D4×C61—21no—
86C2×D4×C61C22⋊C411yescharacteristic
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61C22×C411yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61C2×D411yescharacteristic
94C22×Dic61#4111yescharacteristic
95C2×Dic122—42no—
96C2×Dic122—42no—
97C2×Dic122—41no—
98C2×Dic122—41no—
99#11758217—22no—
100#11758217—41no—
101#11758217—42no—
102#11758217—81no—
103C2×#92420—21no—
104C2×#92420—21no—
105C2×#92420—22no—
106C2×#92420—41no—
107C2×#92420—41no—
108C2×#92420—82no—
109M16×C61—41no—
110C2×C8×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C22⋊Dic61—21no—
114C22⋊Dic61—21no—
115C22⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61C22⋊C411yescharacteristic
121C4⋊Dic61C22⋊C411yescharacteristic
122#11758196—41no—
123#11758196—41no—
124C4×Dic61—21no—
125C4×Dic61—21no—
126C4×Dic61—21no—
127C4×Dic61C2×D411yescharacteristic
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C4×Dic61#4111yescharacteristic
131#11758194—41no—
132C2×#92418—21no—
133C23×C61—21no—
134C23×C61C4×D411yescharacteristic
135Q8×C61—21no—
136Q8×C61—21no—
137D4×C61—21no—
138D4×C61—42no—
139D4×C61#10112yescomplemented
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61C2×C22⋊C411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61#12011yescharacteristic
161C2×Dic61#12111yescharacteristic
162C2×Dic61#12111yescharacteristic
163C2×Dic61#12311yescharacteristic
164Dic122—82no—
165Dic122—81no—
166Dic122—81no—
167#92420—42no—
168#92420—41no—
169#92420—42no—
170#92420—41no—
171#92420—81no—
172C8×C61—41no—
173#92418—41no—
174#9357—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—22no—
178C22×C61—21no—
179C22×C61—41no—
180C22×C61—41no—
181C22×C61#38611yescharacteristic
182C4×C61—21no—
183C4×C61—41no—
184C4×C61—81no—
185C4×C61C2×#10111yescharacteristic
186C4×C61#41311yescharacteristic
187Dic61—21no—
188Dic61—21no—
189Dic61—41no—
190Dic61—41no—
191Dic61—81no—
192Dic61#45311yescharacteristic
193Dic61#45411yescharacteristic
194Dic61#47111yescharacteristic
195Dic61#47311yescharacteristic
196#3079—611no—
197#2990—611no—
198#1966—611no—
199#1549—611no—
200#1489—611no—
201#1309—611no—
202#1299—611no—
203C2×C61—41no—
204C2×C61—42no—
205C2×C61#188311yescharacteristic
206C2×C61#190111yescharacteristic
207C2×C61#190811yescharacteristic
208C2×C4⋊Q8—611no—
209C4×#41—611no—
210#477—1221no—
211#476—1221no—
212#475—1221no—
213#474—1221no—
214#446—1221no—
215#435—1222no—
216C8×D4—1221no—
217#432—1221no—
218#431—611no—
219#428—611no—
220#428—611no—
221#423—611no—
222#423—611no—
223#423—611no—
224C2×C4⋊C8—1221no—
225#417—611no—
226#417#692311yescharacteristic
227C2×#101—1222no—
228C4×M16—611no—
229#332—1221no—
230#332—1221no—
231Q8⋊C8—1221no—
232D4⋊C8—1221no—
233C61#935711yescharacteristic, complemented, Sylow
234C2×#41C2×#692311yescharacteristic
235C22×Q8—1221no—
236C2×M16—612no—
237C2×M16—611no—
238C2×M16—1221no—
239C2×M16—1221no—
240C2×M16C2×#692311yescharacteristic
241C22×C8—1222no—
242C4⋊Q8—1221no—
243C4⋊Q8—1221no—
244C4⋊Q8—1221no—
245C4⋊Q8—1221no—
246C4×Q8—611no—
247C4×Q8—611no—
248C4×D4—611no—
249C4×D4—611no—
250C4×D4—1221no—
251C4×D4—1221no—
252#117—611no—
253#117—1221no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4C2×#692311yescharacteristic
257C2×C42—611no—
258C2×C42—1221no—
259#107—1221no—
260#107—1221no—
261#106—1221no—
262#106—1221no—
263C4⋊C8—611no—
264C4⋊C8—611no—
265C4⋊C8—1221no—
266C4⋊C8—1221no—
267C4⋊C8—1221no—
268C4⋊C8—1221no—
269C4⋊C8—1221no—
270C4⋊C8—1221no—
271C4⋊C8—1222no—
272#103—21no—
273#103—1221no—
274#102—21no—
275#102—1221no—
276#101—2444no—
277C22⋊C8—1221no—
278C22⋊C8—1222no—
279#97—1221no—
280#97—1222no—
281C4×C8—1221no—
282C4×C8—1221no—
283#41—22no—
284#41—41no—
285C2×Q8—2441no—
286C2×Q8—2441no—
287C2×Q8—2442no—
288C2×Q8—2442no—
289C2×Q8C22⋊#692311yescharacteristic
290C2×D4—21no—
291C2×D4C22⋊#692311yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4C22×#692311yescharacteristic
300M16—41no—
301M16—1222no—
302M16—4881no—
303M16—2441no—
304M16—2441no—
305M16—2442no—
306C2×C8—21no—
307C2×C8—1221no—
308C2×C8—1221no—
309C2×C8—1222no—
310C2×C8—1221no—
311C2×C8—4882no—
312C2×C8—2441no—
313C2×C8—2441no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—611no—
317C4⋊C4—611no—
318C4⋊C4—1221no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C22⋊C4—1221no—
325C22⋊C4—1221no—
326C22⋊C4—1221no—
327C42—611no—
328C42—611no—
329C42—611no—
330C42—611no—
331C42—1221no—
332C42—1221no—
333C42—1221no—
334C23—21no—
335C23D4×#692311yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—4881no—
339Q8—4881no—
340Q8—4882no—
341D4—21no—
342D4—42no—
343D4#1451983812yescomplemented
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—611no—
353C2×C4—611no—
354C2×C4—1221no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2442no—
365C2×C4—2441no—
366C2×C4C2×C22⋊#692311yescharacteristic
367C2×C4D4×#692311yescharacteristic
368C8—41no—
369C8—4881no—
370C8—2441no—
371C8—2441no—
372C8—2442no—
373C8—2442no—
374C8—2441no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22#1684071011yescharacteristic
382C4—21no—
383C4—41no—
384C4—81no—
385C4—611no—
386C4—611no—
387C4—611no—
388C4—611no—
389C4—1221no—
390C4—1221no—
391C4—4881no—
392C4—2441no—
393C4—2441no—
394C4C2×#1451983811yescharacteristic
395C4#1684074911yescharacteristic
396C2—41no—
397C2—42no—
398C2#2146181911yescharacteristic
399C2#2146184611yescharacteristic
400C2#2146185911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#417C4
socleC22×C61#386
prime radicalC2×#41×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#417#6923
hypocenterC61#9357
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #21461962C2semidirect
3. #21461334C2semidirect
5. #21460727C2semidirect
6. #21460703C2semidirect
139. D4×C61#101semidirect
233. C61#9357semidirect
343. D4#14519838semidirect

Maximal subgroups

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

subgroupindex
#214627452
#214619622
#214613342
#214611542
#214607272
#214607032
#214556302
#935761