Finite Group Databasebeta

← #30600487

Subgroups — #30600487

Order 15616.

490 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462334C211yesmaximal, characteristic
2#21461253C211yesmaximal, characteristic
3#21461202C211yesmaximal, characteristic
4#21461122C211yesmaximal, characteristic
5#21460715C211yesmaximal, characteristic
6#21460674C211yesmaximal, characteristic
7#21455625C211yesmaximal, characteristic
8C2×Q8×#6923—21no—
9#16840781C2211yescharacteristic
10#16840780C2211yescharacteristic
11#16840713—21no—
12#16840713—22no—
13#16840706—21no—
14#16840706C2211yescharacteristic
15C4×C4⋊#6923—21no—
16C4×C4⋊#6923C2211yescharacteristic
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
33Q8×#6923—42no—
34Q8×#6923—42no—
35C2×C4⋊#6923—22no—
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923—21no—
38C2×C4⋊#6923—21no—
39C2×C4×#6923—22no—
40C2×C4×#6923—21no—
41#14519844—21no—
42C4⋊#92420—21no—
43#14519834—21no—
44#14519834—21no—
45C4×#92420—21no—
46C4×#92420—21no—
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—21no—
49(C2×#6923)⋊C4D411yescharacteristic
50(C2×#6923)⋊C4D412yes—
51(C2×#6923)⋊C4D411yescharacteristic
52C2×Q8×D61D411yescharacteristic
53C2×#11758189—21no—
54#14519777C2×C411yescharacteristic
55#14519768—21no—
56#14519757—21no—
57#14519753—21no—
58Q8×Dic61—21no—
59Q8×Dic61D411yescharacteristic
60C4×Dic122—21no—
61C4×Dic122C2×C411yescharacteristic
62#14519742—21no—
63#14519742—21no—
64#14519742D411yescharacteristic
65#14519735—21no—
66C4⋊C4×D61—21no—
67C4⋊C4×D61D411yescharacteristic
68C42×D61—21no—
69C42×D61C2311yescharacteristic
70#14519710—21no—
71#14519709—21no—
72#14519708—21no—
73C4⋊#92418C2×C411yescharacteristic
74#14519706—21no—
75#14519702—21no—
76#14519701—21no—
77#14519700—21no—
78C4⋊Q8×C61C2×C411yescharacteristic
79C4×Q8×C61C2×C411yescharacteristic
80#107×C61—21no—
81C4⋊C8×C61C2×C411yescharacteristic
82#103×C61—21no—
83C22×#6923—21no—
84C22×#6923—21no—
85C22×#6923—22no—
86C2×Q8×C61—21no—
87C2×Q8×C61C22⋊C411yescharacteristic
88Q8×D61—21no—
89Q8×D61—21no—
90Q8×D61SD1612yes—
91Q8×D61Q1612yes—
92C2×C4×D61—21no—
93C2×C4×D61—21no—
94C2×C4×D61C2×D411yescharacteristic
95C2×C4×D61C2×D411yescharacteristic
96C2×C4×D61C2×D411yescharacteristic
97C2×Dic122—21no—
98C2×Dic122—21no—
99C2×Dic122C22⋊C411yescharacteristic
100C2×#92420—41no—
101C2×#92420—41no—
102C4⋊#6923—44no—
103C4⋊#6923—42no—
104C4⋊#6923—44no—
105C4⋊#6923—41no—
106C4⋊#6923—41no—
107C4×#6923—44no—
108C4×#6923—41no—
109C4×#6923—41no—
110C2×C8×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61C22⋊C411yescharacteristic
115C42×C61—21no—
116C42×C61C22×C411yescharacteristic
117#11758198—41no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—41no—
122C4⋊Dic61C22⋊C411yescharacteristic
123#11758196—21no—
124#11758196—21no—
125#11758196—21no—
126#11758196—41no—
127C4×Dic61—21no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130C4×Dic61—41no—
131C4×Dic61C2×D411yescharacteristic
132C4×Dic61#4111yescharacteristic
133C4×Dic61#4111yescharacteristic
134C2×#92418—21no—
135#11758189—41no—
136#11758189—41no—
137C22×D61#12011yescharacteristic
138C2×#6923—22no—
139C2×#6923—41no—
140C2×#6923—41no—
141C2×#6923—41no—
142C2×#6923—42no—
143C2×#6923—41no—
144C2×#6923—42no—
145C2×#6923—42no—
146C2×#6923—41no—
147Q8×C61—21no—
148Q8×C61—81no—
149Q8×C61#10312yes—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—41no—
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—41no—
159C2×Dic61#12111yescharacteristic
160C2×Dic61#12111yescharacteristic
161C2×Dic61#12311yescharacteristic
162C4×D61—22no—
163C4×D61—22no—
164C4×D61—41no—
165C4×D61—41no—
166C4×D61—42no—
167C4×D61C2×SD1611yescharacteristic
168C4×D61C2×Q1611yescharacteristic
169C4×D61AGL(1,ℤ/8)11yescharacteristic
170C4×D61#13711yescharacteristic
171Dic122—21no—
172Dic122—22no—
173Dic122—22no—
174Dic122—81no—
175Dic122#10312yes—
176#92420—81no—
177#92420—81no—
178C8×C61—41no—
179#92418—41no—
180#9176—611nomaximal, Sylow
181C22×C61#38611yescharacteristic
182C2×D61—21no—
183C2×D61—21no—
184C2×D61#45011yescharacteristic
185C2×D61#45111yescharacteristic
186#6923—42no—
187#6923—82no—
188#6923—81no—
189C4×C61—22no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—41no—
193C4×C61—81no—
194C4×C61C2×#10311yescharacteristic
195C4×C61#41711yescharacteristic
196Dic61—22no—
197Dic61—21no—
198Dic61—21no—
199Dic61—41no—
200Dic61—81no—
201Dic61#48311yescharacteristic
202Dic61#48411yescharacteristic
203#3078—611no—
204#2351—611no—
205#1518—611no—
206#1503—611no—
207#1479—611no—
208#1303—611no—
209#1286—611no—
210C2×C61#187211yescharacteristic
211C2×C61#190511yescharacteristic
212C2×C61#190611yescharacteristic
213D61—21no—
214D61—21no—
215#533—611no—
216C2×C4×Q8—611no—
217C2×C4×Q8—1221no—
218#479—1221no—
219#479—1221no—
220#477—611no—
221#477#692311yescharacteristic, complemented
222#475—611no—
223#475—611no—
224#431—611no—
225#431—611no—
226#428—1221no—
227#423—611no—
228#419—1221no—
229#389—1222no—
230#389—1221no—
231#382—611no—
232#382—1221no—
233C4×C4⋊C4—611no—
234C4×C4⋊C4—1221no—
235#328—1221no—
236#328—1221no—
237#328—1222no—
238Q8⋊C8—1221no—
239Q8⋊C8—1221no—
240C61#917611yescharacteristic, complemented, Sylow
241C22×Q8—611no—
242C2×M16—1221no—
243C4⋊Q8—611no—
244C4⋊Q8C2×#692311yescharacteristic
245#125—1221no—
246C22⋊Q8—1221no—
247C22⋊Q8—1221no—
248C4×Q8—611no—
249C4×Q8—611no—
250C4×Q8—611no—
251C4×Q8—1221no—
252C4×Q8—1221no—
253C4×Q8—1221no—
254C4×Q8—1221no—
255C4×Q8—2442no—
256C4×Q8—2442no—
257C4×Q8C2×#692311yescharacteristic
258#117—1221no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4—1222no—
265C2×C42—611no—
266C2×C42—1221no—
267C2×C42—1221no—
268C2×C42—1222no—
269#107—21no—
270#107—1221no—
271#106—1221no—
272#106—1221no—
273C4⋊C8—611no—
274C4⋊C8—1221no—
275C4⋊C8—1221no—
276C4⋊C8—1221no—
277C4⋊C8C2×#692311yescharacteristic
278#103—21no—
279#103—1221no—
280#103—1221no—
281#103—1221no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284C4×C8—1221no—
285C4×C8—1221no—
286#95—611no—
287#95—611no—
288#95—612no—
289#95—1221no—
290#95—1222no—
291C2×Q8—21no—
292C2×Q8—612no—
293C2×Q8—612no—
294C2×Q8—611no—
295C2×Q8—1221no—
296C2×Q8—1221no—
297C2×Q8—1221no—
298C2×Q8—1221no—
299C2×Q8C22⋊#692311yescharacteristic
300C22×C4—611no—
301C22×C4—611no—
302C22×C4—611no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1222no—
308M16—2441no—
309M16—2441no—
310C2×C8—21no—
311C2×C8—1221no—
312C2×C8—2441no—
313C2×C8—2441no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—611no—
318C4⋊C4—1221no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C4⋊C4—1221no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C4⋊C4—2444no—
329C4⋊C4—2442no—
330C4⋊C4—2444no—
331C4⋊C4C22⋊#692311yescharacteristic
332C22⋊C4—2441no—
333C42—21no—
334C42—611no—
335C42—611no—
336C42—611no—
337C42—1221no—
338C42—1221no—
339C42—1221no—
340C42—2441no—
341C42—2441no—
342C42—2441no—
343C42—2444no—
344C42C22×#692311yescharacteristic
345C23—611no—
346Q8—21no—
347Q8—81no—
348Q8—612no—
349Q8—1222no—
350Q8—1222no—
351Q8—1221no—
352Q8—4881no—
353Q8#1451984012yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—611no—
358C2×C4—611no—
359C2×C4—611no—
360C2×C4—611no—
361C2×C4—611no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—1222no—
365C2×C4—1222no—
366C2×C4—1221no—
367C2×C4—1222no—
368C2×C4—1221no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2442no—
372C2×C4—2441no—
373C2×C4—2442no—
374C2×C4—2441no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4—2442no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4C2×C22⋊#692311yescharacteristic
382C2×C4D4×#692311yescharacteristic
383C2×C4D4×#692311yescharacteristic
384C8—41no—
385C8—4881no—
386C8—4881no—
387C8—2441no—
388C22—611no—
389C22—611no—
390C22—1221no—
391C22—1221no—
392C22#1684071011yescharacteristic
393C4—22no—
394C4—21no—
395C4—21no—
396C4—41no—
397C4—81no—
398C4—611no—
399C4—611no—
400C4—1222no—
401C4—1221no—
402C4—1221no—
403C4—4881no—
404C4—4882no—
405C4—4881no—
406C4—2442no—
407C4—2441no—
408C4C2×#1451984011yescharacteristic
409C4#1684075311yescharacteristic
410C2—1221no—
411C2—1221no—
412C2#2146180411yescharacteristic
413C2#2146185411yescharacteristic
414C2#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#9176
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
221. #477#6923semidirect
240. C61#9176semidirect

Maximal subgroups

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

subgroupindex
#214623342
#214612532
#214612022
#214611222
#214607152
#214606742
#214556252
#917661