Finite Group Databasebeta

← #30604818

Subgroups — #30604818

Order 15616.

592 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#16840640C211yesmaximal, characteristic
2C2×#16840606C211yesmaximal, characteristic
3#21460816C212yesmaximal, complemented
4#21460814C212yesmaximal, complemented
5#21451626C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7C2×C22⋊#92420—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840606—21no—
11#16840606C2212yes—
12C22×#11758198C2211yescharacteristic
13#16839697C411yescharacteristic, complemented
14#16839697C2211yescharacteristic
15#16839684—21no—
16#16839684—21no—
17#16839673C411yescharacteristic, complemented
18#16839673C2211yescharacteristic
19#16839634—21no—
20#16839622C411yescharacteristic, complemented
21#16839617—21no—
22C2×C22⋊Q8×C61C411yescharacteristic, complemented
23C23×#6923—41no—
24C22×#92420—41no—
25C2×C22⋊#6923—24no—
26C2×C22⋊#6923—22no—
27C2×C22⋊#6923—41no—
28C2×C22⋊#6923—42no—
29C22⋊#92420—22no—
30C22⋊#92420—41no—
31C24×D61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C2×C4⋊Dic61—21no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×C22⋊Dic61—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—41no—
42C2×#11758198—41no—
43C2×#11758198C2×C411yescharacteristic
44C2×#11758198D411yescharacteristic
45C2×#11758198D411yescharacteristic
46C2×#11758198C2311yescharacteristic
47#14519684—21no—
48#14519684—21no—
49#14519684C2×C411yescharacteristic, complemented
50#14519684C2×C411yescharacteristic, complemented
51C22×Q8×C61C2×C411yescharacteristic, complemented
52C23×C4×C61C2×C411yescharacteristic
53C22⋊Q8×C61—41no—
54C22⋊Q8×C61—41no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
57C2×C22⋊C4×C61—21no—
58C24×C61C22⋊C411yescharacteristic
59C23×D61—21no—
60C23×D61—21no—
61C23×D61—22no—
62C23×D61—81no—
63C23×D61—81no—
64C23×D61C22⋊C411yescharacteristic
65C23×D61SD1612yes—
66C23×D61Q1612yes—
67C23×D61C2×D411yescharacteristic
68C22×#6923—44no—
69C22×#6923—42no—
70C22×#6923—82no—
71C22×#6923—82no—
72C2×Q8×C61—41no—
73C2×Q8×C61—81no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—81no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22×C411yescharacteristic
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—81no—
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C2×D411yescharacteristic
88C22⋊#6923—24no—
89C22⋊#6923—44no—
90C22⋊#6923—42no—
91C2×#92420—42no—
92C2×#92420—82no—
93C4⋊C4×C61—42no—
94C4⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C22⋊Dic61—42no—
98#11758198—41no—
99#11758198—41no—
100#11758198—41no—
101#11758198—41no—
102#11758198—81no—
103#11758198—81no—
104#11758198—81no—
105#11758198—81no—
106C4⋊Dic61—81no—
107#11758196—81no—
108#11758196—81no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61#9911yescharacteristic
112C23×C61#9911yescharacteristic
113C23×C61#10312yes—
114C23×C61#10412yes—
115C23×C61C2×C22⋊C411yescharacteristic
116C22×D61—22no—
117C22×D61—22no—
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—22no—
121C22×D61—22no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—42no—
125C22×D61—42no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—82no—
131C22×D61—82no—
132C22×D61—81no—
133C22×D61—81no—
134C22×D61—84no—
135C22×D61#10312yes—
136C22×D61C2×SD1611yescharacteristic
137C22×D61C2×Q1611yescharacteristic
138C2×#6923—44no—
139C2×#6923—88no—
140C2×#6923—82no—
141C2×#6923—82no—
142Q8×C61—82no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—81no—
154C2×C4×C61—81no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—81no—
167#92420—82no—
168#11716—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—42no—
176C22×C61—42no—
177C22×C61#32812yes—
178C22×C61#32812yes—
179C22×C61C2×#9911yescharacteristic
180C22×C61C2×#10311yescharacteristic
181C22×C61C2×#10411yescharacteristic
182C2×D61—21no—
183C2×D61—22no—
184C2×D61—21no—
185C2×D61—44no—
186C2×D61—44no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—41no—
190C2×D61—41no—
191C2×D61—42no—
192C2×D61—41no—
193C2×D61—41no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—82no—
197C2×D61—82no—
198#6923—84no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—81no—
202C4×C61—81no—
203Dic61—41no—
204Dic61—41no—
205Dic61—81no—
206Dic61—81no—
207#2626—611no—
208C2×#342—611no—
209C2×#323—611no—
210#1352—612no—
211#1351—612no—
212C2×C61—21no—
213C2×C61—22no—
214C2×C61—21no—
215C2×C61—22no—
216C2×C61C2×#32811yescharacteristic
217C2×C61#212111yescharacteristic
218C2×C61#212711yescharacteristic
219D61—41no—
220D61—42no—
221D61—41no—
222C2×C22⋊Q8#692311yescharacteristic, complemented
223C22×C22⋊C4—611no—
224C22×C22⋊C4—1221no—
225C2×C22⋊C8—1221no—
226#399—611no—
227#399—611no—
228#396—1221no—
229#396—1221no—
230#393—611no—
231#393—611no—
232#385—1221no—
233#380—611no—
234#379—1221no—
235#342—612no—
236#342—1221no—
237#323—612no—
238#323—1221no—
239C61#1171611yescharacteristic, complemented, Sylow
240C25—611no—
241C22×Q8C2×#692311yescharacteristic, complemented
242C23×C4—611no—
243C23×C4—2441no—
244C23×C4C2×#692311yescharacteristic
245C22×C8—2441no—
246C22⋊Q8—41no—
247C22⋊Q8—41no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4C2×#692311yescharacteristic, complemented
253C2×C22⋊C4—21no—
254C2×C22⋊C4—611no—
255C2×C22⋊C4—611no—
256C2×C22⋊C4—611no—
257C2×C22⋊C4—611no—
258C2×C22⋊C4—1222no—
259C2×C22⋊C4—1224no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—2441no—
266C2×C22⋊C4—2441no—
267C2×C22⋊C4—2442no—
268C2×C22⋊C4—2441no—
269C22⋊C8—1222no—
270C22⋊C8—2441no—
271#95—611no—
272#95—611no—
273#95—1221no—
274#95—1221no—
275C24—612no—
276C24—612no—
277C24—611no—
278C24—611no—
279C24—1221no—
280C24—1222no—
281C24—1221no—
282C24—4881no—
283C24—4881no—
284C24C22⋊#692311yescharacteristic
285C2×Q8—41no—
286C2×Q8—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—611no—
293C22×C4—611no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—4882no—
299C22×C4—4882no—
300C22×C4—4881no—
301C22×C4—2442no—
302C22×C4—2444no—
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22×#692311yescharacteristic
305C2×C8—4882no—
306C2×C8—2442no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—4881no—
310C4⋊C4—4881no—
311C4⋊C4—4881no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—1224no—
315C22⋊C4—4881no—
316C22⋊C4—4881no—
317C22⋊C4—4881no—
318C22⋊C4—4881no—
319C22⋊C4—2441no—
320C22⋊C4—2441no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C22⋊C4—2442no—
324C22⋊C4—2444no—
325C22⋊C4—2442no—
326C23—41no—
327C23—41no—
328C23—612no—
329C23—611no—
330C23—611no—
331C23—1221no—
332C23—1221no—
333C23—1222no—
334C23—1222no—
335C23—1221no—
336C23—1221no—
337C23—1222no—
338C23—1222no—
339C23—4881no—
340C23—4881no—
341C23—4884no—
342C23—4882no—
343C23—4882no—
344C23—2441no—
345C23—2441no—
346C23—2441no—
347C23—2441no—
348C23—2442no—
349C23—2442no—
350C23#1451983511yescharacteristic
351C23#1451983511yescharacteristic
352C23#1451984012yes—
353C23#1451984212yes—
354C23C2×C22⋊#692311yescharacteristic
355Q8—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—1221no—
369C2×C4—1221no—
370C2×C4—4888no—
371C2×C4—4882no—
372C2×C4—4882no—
373C2×C4—4881no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2444no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4—2441no—
384C8—4882no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—1221no—
394C22—1222no—
395C22—1221no—
396C22—4882no—
397C22—4882no—
398C22—4881no—
399C22—4881no—
400C22—2441no—
401C22—2442no—
402C22—2441no—
403C22—2444no—
404C22—2444no—
405C22—2441no—
406C22—2441no—
407C22—2441no—
408C22—2441no—
409C22#1684061612yes—
410C22#1684061612yes—
411C22C2×#1451983511yescharacteristic
412C22C2×#1451984011yescharacteristic
413C22C2×#1451984211yescharacteristic
414C4—41no—
415C4—41no—
416C4—81no—
417C4—81no—
418C4—4881no—
419C4—4881no—
420C4—4884no—
421C4—2441no—
422C4—2441no—
423C2—21no—
424C2—22no—
425C2—21no—
426C2—22no—
427C2—2441no—
428C2—2442no—
429C2—2441no—
430C2C2×#1684061611yescharacteristic
431C2#2146207411yescharacteristic
432C2#2146209411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC24×D61D4
cosocleC2×#11758198C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#11716
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21460816C2semidirect
4. #21460814C2semidirect
13. #16839697C4semidirect
17. #16839673C4semidirect
20. #16839622C4semidirect
22. C2×C22⋊Q8×C61C4semidirect
49. #14519684C2×C4semidirect
50. #14519684C2×C4semidirect
51. C22×Q8×C61C2×C4semidirect
56. C2×C4⋊C4×C61C2×C4semidirect
222. C2×C22⋊Q8#6923semidirect
239. C61#11716semidirect
241. C22×Q8C2×#6923semidirect
252. C2×C4⋊C4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406402
C2×#168406062
#214608162
#214608142
#214516262
#1171661