Finite Group Databasebeta

← C2×#21460814 #30604808

Subgroups — C2×#21460814 #30604808

Order 15616.

776 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 104 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#21460814C214yesmaximal, direct product factor
4C2×#16839673C211yesmaximal, characteristic
5C22×C22⋊#6923—21no—
6C2×C22⋊#92420—21no—
7#16840640—21no—
8#16840640C2212yes—
9#16840606—21no—
10#16840606C2212yes—
11C22×#11758198—21no—
12C22×#11758198C2211yescharacteristic
13#16839673—21no—
14#16839673—21no—
15#16839673C412yescomplemented
16#16839673C2212yes—
17C2×#14519684—21no—
18C2×#14519684C411yescharacteristic, complemented
19C23×Q8×C61C411yescharacteristic, complemented
20C23×#6923—41no—
21C22×#92420—41no—
22C2×C22⋊#6923—24no—
23C2×C22⋊#6923—22no—
24C2×C22⋊#6923—41no—
25C2×C22⋊#6923—42no—
26C22⋊#92420—22no—
27C22⋊#92420—41no—
28C24×D61D411yescharacteristic
29C23×Dic61—21no—
30C23×Dic61D411yescharacteristic
31C2×#11758198—22no—
32C2×#11758198—22no—
33C2×#11758198—41no—
34C2×#11758198—41no—
35C2×#11758198—44no—
36C2×#11758198C2×C411yescharacteristic
37C2×#11758198D411yescharacteristic
38C2×#11758198D411yescharacteristic
39C2×#11758198C2311yescharacteristic
40#14519684—21no—
41#14519684—22no—
42#14519684—22no—
43#14519684C2×C412yescomplemented
44C22×Q8×C61—21no—
45C22×Q8×C61—44no—
46C22×Q8×C61—41no—
47C22×Q8×C61—41no—
48C22×Q8×C61C2×C412yescomplemented
49C23×C4×C61—21no—
50C23×C4×C61C2×C411yescharacteristic
51C24×C61C22⋊C411yescharacteristic
52C23×D61—21no—
53C23×D61—21no—
54C23×D61—22no—
55C23×D61—42no—
56C23×D61—42no—
57C23×D61C22⋊C411yescharacteristic
58C23×D61SD1612yes—
59C23×D61Q1612yes—
60C23×D61C2×D411yescharacteristic
61C22×#6923—44no—
62C22×#6923—42no—
63C22×#6923—82no—
64C22×#6923—82no—
65C2×Q8×C61—44no—
66C2×Q8×C61—42no—
67C2×Q8×C61—44no—
68C2×Q8×C61—42no—
69C2×Q8×C61—48no—
70C2×Q8×C61—44no—
71C2×Q8×C61—44no—
72C22×C4×C61—24no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—22no—
76C22×C4×C61—42no—
77C22×C4×C61—41no—
78C22×C4×C61—41no—
79C22×C4×C61—42no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—22no—
85C22×Dic61—42no—
86C22×Dic61—42no—
87C22×Dic61—42no—
88C22×Dic61—42no—
89C22×Dic61C22⋊C411yescharacteristic
90C22×Dic61C2×D411yescharacteristic
91C22⋊#6923—24no—
92C22⋊#6923—44no—
93C22⋊#6923—42no—
94C2×#92420—42no—
95C2×#92420—82no—
96#11758198—41no—
97#11758198—41no—
98#11758198—41no—
99#11758198—41no—
100#11758198—44no—
101#11758198—44no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61#9911yescharacteristic
105C23×C61#9911yescharacteristic
106C23×C61#10312yes—
107C23×C61#10412yes—
108C23×C61C2×C22⋊C411yescharacteristic
109C22×D61—22no—
110C22×D61—22no—
111C22×D61—21no—
112C22×D61—21no—
113C22×D61—22no—
114C22×D61—22no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—42no—
118C22×D61—42no—
119C22×D61—41no—
120C22×D61—41no—
121C22×D61—41no—
122C22×D61—44no—
123C22×D61—44no—
124C22×D61—41no—
125C22×D61—81no—
126C22×D61—84no—
127C22×D61—81no—
128C22×D61#10312yes—
129C22×D61C2×SD1611yescharacteristic
130C22×D61C2×Q1611yescharacteristic
131C2×#6923—44no—
132C2×#6923—82no—
133C2×#6923—88no—
134C2×#6923—82no—
135Q8×C61—48no—
136Q8×C61—44no—
137Q8×C61—44no—
138C2×C4×C61—24no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—42no—
142C2×C4×C61—44no—
143C2×C4×C61—42no—
144C2×C4×C61—44no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—42no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—81no—
158C2×Dic61—84no—
159#92420—82no—
160C2×#1351—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—24no—
164C22×C61—24no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61#32812yes—
168C22×C61#32812yes—
169C22×C61AΣL(1,𝔽16)12yes—
170C22×C61#35412yes—
171C22×C61C2×#9911yescharacteristic
172C22×C61C2×#10311yescharacteristic
173C22×C61C2×#10411yescharacteristic
174C2×D61—21no—
175C2×D61—22no—
176C2×D61—21no—
177C2×D61—41no—
178C2×D61—41no—
179C2×D61—44no—
180C2×D61—44no—
181C2×D61—41no—
182C2×D61—41no—
183C2×D61—42no—
184C2×D61—41no—
185C2×D61—41no—
186C2×D61—81no—
187C2×D61—82no—
188C2×D61—81no—
189C2×D61—82no—
190#6923—84no—
191C4×C61—44no—
192C4×C61—41no—
193C4×C61—41no—
194Dic61—41no—
195Dic61—41no—
196Dic61—82no—
197C2×#393—611no—
198C2×#342—611no—
199C2×#323—611no—
200#1351—614no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—22no—
204C2×C61#135114yescomplemented
205C2×C61C2×#32811yescharacteristic
206C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
207C2×C61C2×#35411yescharacteristic
208D61—41no—
209D61—42no—
210D61—41no—
211C23×Q8#692311yescharacteristic, complemented
212C22×C22⋊C4—611no—
213C22×C22⋊C4—1221no—
214C22×C22⋊C4—1221no—
215C2×C22⋊C8—1221no—
216#393—612no—
217#393—612no—
218#393—1221no—
219#393—1221no—
220C2×#95—611no—
221C2×#95—1221no—
222#342—612no—
223#342—1221no—
224#323—612no—
225#323—1221no—
226C61C2×#135111yescharacteristic, complemented, Sylow
227C25—611no—
228C22×Q8—21no—
229C22×Q8—44no—
230C22×Q8—41no—
231C22×Q8—41no—
232C22×Q8C2×#692312yescomplemented
233C23×C4—21no—
234C23×C4—611no—
235C23×C4—1221no—
236C23×C4—2441no—
237C23×C4C2×#692311yescharacteristic
238C22×C8—2441no—
239C2×C22⋊C4—611no—
240C2×C22⋊C4—611no—
241C2×C22⋊C4—611no—
242C2×C22⋊C4—611no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1224no—
245C2×C22⋊C4—1222no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—2441no—
248C2×C22⋊C4—2441no—
249C2×C22⋊C4—2444no—
250C2×C22⋊C4—2442no—
251C2×C22⋊C4—2441no—
252C22⋊C8—1222no—
253C22⋊C8—2441no—
254#95—612no—
255#95—1222no—
256#95—1222no—
257#95—1221no—
258C24—612no—
259C24—612no—
260C24—611no—
261C24—611no—
262C24—1221no—
263C24—1222no—
264C24—1221no—
265C24—2442no—
266C24—2442no—
267C24C22⋊#692311yescharacteristic
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—48no—
273C2×Q8—44no—
274C2×Q8—44no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—4882no—
289C22×C4—4882no—
290C22×C4—2442no—
291C22×C4—2442no—
292C22×C4—2442no—
293C22×C4—2444no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4C22⋊#692311yescharacteristic
297C22×C4C22×#692311yescharacteristic
298C2×C8—4882no—
299C2×C8—2442no—
300C22⋊C4—1224no—
301C22⋊C4—2441no—
302C22⋊C4—2441no—
303C22⋊C4—2441no—
304C22⋊C4—2441no—
305C22⋊C4—2444no—
306C22⋊C4—2444no—
307C22⋊C4—2442no—
308C22⋊C4—2444no—
309C23—22no—
310C23—22no—
311C23—612no—
312C23—611no—
313C23—611no—
314C23—1221no—
315C23—1221no—
316C23—1222no—
317C23—1222no—
318C23—1221no—
319C23—1221no—
320C23—1222no—
321C23—1222no—
322C23—4881no—
323C23—4881no—
324C23—4884no—
325C23—2441no—
326C23—2441no—
327C23—2441no—
328C23—2441no—
329C23—2444no—
330C23—2444no—
331C23—2442no—
332C23—2442no—
333C23#1451983511yescharacteristic
334C23#1451983511yescharacteristic
335C23#1451984012yes—
336C23#1451984212yes—
337C23C2×C22⋊#692311yescharacteristic
338Q8—44no—
339Q8—44no—
340Q8—48no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—4884no—
355C2×C4—4882no—
356C2×C4—4882no—
357C2×C4—4888no—
358C2×C4—4881no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4—2444no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2441no—
365C2×C4—2441no—
366C8—4882no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—24no—
372C22—24no—
373C22—1221no—
374C22—1222no—
375C22—1221no—
376C22—4881no—
377C22—4881no—
378C22—4882no—
379C22—4882no—
380C22—2441no—
381C22—2442no—
382C22—2441no—
383C22—2441no—
384C22—2441no—
385C22—2441no—
386C22—2441no—
387C22—2444no—
388C22—2444no—
389C22#1684061612yes—
390C22#1684061612yes—
391C22#1684066212yes—
392C22#1684066812yes—
393C22C2×#1451983511yescharacteristic
394C22C2×#1451984011yescharacteristic
395C22C2×#1451984211yescharacteristic
396C4—44no—
397C4—41no—
398C4—41no—
399C4—4882no—
400C4—4884no—
401C4—2441no—
402C4—2441no—
403C2—21no—
404C2—22no—
405C2—21no—
406C2—2441no—
407C2—2441no—
408C2—2442no—
409C2#2146081414yesdirect product factor
410C2C2×#1684061611yescharacteristic
411C2C2×#1684066211yescharacteristic
412C2C2×#1684066811yescharacteristic
subgroupG/H
centerC23#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC23×C61#99
prime radicalC24×D61D4
cosocleC2×#11758198C23
hypercenterC23×Q8#6923
hypocenterC61C2×#1351
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #21460814C2direct (N × H)
15. #16839673C4semidirect
18. C2×#14519684C4semidirect
19. C23×Q8×C61C4semidirect
43. #14519684C2×C4semidirect
48. C22×Q8×C61C2×C4semidirect
204. C2×C61#1351semidirect
211. C23×Q8#6923semidirect
226. C61C2×#1351semidirect
232. C22×Q8C2×#6923semidirect
409. C2#21460814direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168406402
C2×#168406062
#214608142
C2×#168396732
C2×#135161