Finite Group Databasebeta

← C2×#21461345 #30611761

Subgroups — C2×#21461345 #30611761

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840782C211yesmaximal, characteristic
2C2×#16840779C211yesmaximal, characteristic
3#21461345C218yesmaximal, direct product factor
4C2×#16840622C212yesmaximal, complemented
5C2×#16840621C212yesmaximal, complemented
6C2×#16840314C211yesmaximal, characteristic
7#16840782—22no—
8#16840782C2214yes—
9#16840779—22no—
10#16840779C2214yes—
11C2×#14519844C2212yes—
12C2×#14519844C2212yes—
13C2×C22⋊#92420—21no—
14C2×C22⋊#92420—21no—
15C2×#14519832—21no—
16C2×C4×#92420—21no—
17#16840622C2218yescomplemented
18#16840621C2218yescomplemented
19#16840314—22no—
20#16840314—22no—
21#16840314C414yes—
22#16840314C2214yes—
23C2×#14519776C2211yescharacteristic
24C2×#14519775C2211yescharacteristic
25C2×#14519767—21no—
26C2×#14519756C411yescharacteristic
27C2×#14519754—21no—
28C2×Q8×Dic61C411yescharacteristic
29C2×#14519742—21no—
30C2×#14519742C411yescharacteristic
31C2×#14519733—21no—
32C22×C4×Dic61C2211yescharacteristic
33C2×C22⋊Q8×C61C411yescharacteristic
34C22×#92420—22no—
35C22×#92420—22no—
36#14519844—22no—
37#14519844—22no—
38#14519844C2314yes—
39#14519844C2314yes—
40C22⋊#92420—24no—
41C22⋊#92420—24no—
42#14519832—24no—
43C4×#92420—24no—
44C22×Dic122—21no—
45C22×Dic122C2×C411yescharacteristic
46C23×Dic61D411yescharacteristic
47C23×Dic61D411yescharacteristic
48#14519776—21no—
49#14519776—21no—
50#14519776C2×C412yes—
51#14519776C2312yes—
52#14519775—21no—
53#14519775—21no—
54#14519775C2×C412yes—
55#14519775C2312yes—
56#14519767—24no—
57#14519767—24no—
58#14519756—22no—
59#14519756C2×C414yes—
60#14519754—24no—
61#14519754—24no—
62Q8×Dic61—22no—
63Q8×Dic61C2×C414yes—
64#14519742—24no—
65#14519742—22no—
66#14519742—24no—
67#14519742C2×C414yes—
68#14519733—24no—
69#14519733—24no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2×C411yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61C2×C411yescharacteristic
81C2×C4×Dic61C2×C412yes—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D412yes—
85C2×C4×Dic61D412yes—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2312yes—
88C22×Q8×C61C2×C411yescharacteristic
89C23×C4×C61C2×C411yescharacteristic
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61C2×C414yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61C2×C411yescharacteristic
94C2×C22⋊C4×C61—21no—
95C24×C61C22⋊C411yescharacteristic
96C2×Q8×C61—21no—
97C2×Q8×C61—42no—
98C2×Q8×C61C22×C412yes—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—42no—
102C22×C4×C61C22⋊C411yescharacteristic
103C22×C4×C61C22⋊C412yes—
104C22×C4×C61C22×C411yescharacteristic
105C22×C4×C61C22×C412yes—
106C22×Dic61—24no—
107C22×Dic61—24no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61C22⋊C411yescharacteristic
111C22×Dic61C22⋊C411yescharacteristic
112C22×Dic61C22⋊C412yes—
113C22×Dic61C22⋊C412yes—
114C22×Dic61C2×D412yes—
115C22×Dic61C2×D412yes—
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C2×Dic122—21no—
119C2×Dic122—22no—
120C2×Dic122—22no—
121C2×Dic122—44no—
122C2×Dic122—42no—
123C2×Dic122C22×C412yes—
124C2×#92420—24no—
125C2×#92420—24no—
126C2×#92420—44no—
127C2×#92420—44no—
128C4⋊C4×C61—24no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61C22×C412yes—
131C22⋊C4×C61—24no—
132C22⋊Dic61—24no—
133C4⋊Dic61—21no—
134C4⋊Dic61—24no—
135C4⋊Dic61C22×C412yes—
136#11758196—24no—
137#11758196—22no—
138#11758196—22no—
139#11758196—22no—
140#11758196—24no—
141#11758196—22no—
142#11758196—22no—
143#11758196—22no—
144C4×Dic61—24no—
145C4×Dic61—24no—
146C4×Dic61C22⋊C412yes—
147C4×Dic61C22⋊C412yes—
148C4×Dic61C22⋊C412yes—
149C4×Dic61C22×C412yes—
150C4×Dic61C22×C411yescharacteristic
151C4×Dic61C2×D412yes—
152C4×Dic61C2×D412yes—
153C4×Dic61C2×D411yescharacteristic
154C4×Dic61C2×D411yescharacteristic
155C4×Dic61C2411yescharacteristic
156C23×C61—22no—
157C23×C61—22no—
158C23×C61—22no—
159C23×C61C2×C22⋊C412yes—
160C23×C61C2×C22⋊C411yescharacteristic
161Q8×C61—42no—
162Q8×C61—42no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61C2×C22⋊C412yes—
172C2×C4×C61C2×C22⋊C411yescharacteristic
173C2×C4×C61C23×C411yescharacteristic
174C2×Dic61—22no—
175C2×Dic61—22no—
176C2×Dic61—24no—
177C2×Dic61—24no—
178C2×Dic61—24no—
179C2×Dic61—24no—
180C2×Dic61—22no—
181C2×Dic61—22no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61C2×C22⋊C412yes—
185C2×Dic61C2×C22⋊C412yes—
186C2×Dic61C2×C22⋊C411yescharacteristic
187C2×Dic61C2×C22⋊C411yescharacteristic
188C2×Dic61#13514yes—
189C2×Dic61#13714yes—
190C2×Dic61C22×D411yescharacteristic
191C2×Dic61C22×D411yescharacteristic
192Dic122—42no—
193Dic122—44no—
194Dic122—44no—
195Dic122—42no—
196#92420—44no—
197#92420—44no—
198C2×#1554—611nomaximal, Sylow
199C22×C61—22no—
200C22×C61—22no—
201C22×C61—22no—
202C22×C61—44no—
203C22×C61#41312yes—
204C22×C61#41612yes—
205C22×C61#41912yes—
206C22×C61C22×C22⋊C411yescharacteristic
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—22no—
210C4×C61—42no—
211C4×C61—42no—
212Dic61—22no—
213Dic61—22no—
214Dic61—42no—
215Dic61—42no—
216Dic61#41612yes—
217Dic61#41912yes—
218Dic61C2×#13512yes—
219Dic61C2×#13712yes—
220C2×#533—611no—
221C2×#432—611no—
222C2×#431—611no—
223#1554—618no—
224C2×#333—612no—
225C2×#332—612no—
226C2×C61—42no—
227C2×C61#155414yescomplemented
228C2×C61C2×#41311yescharacteristic
229C2×C61C2×#41611yescharacteristic
230C2×C61C2×#41911yescharacteristic
231#533—614no—
232#533—614no—
233#533—1222no—
234#533—1222no—
235C2×C4⋊Q8—611no—
236C2×C4⋊Q8—611no—
237C2×#125—1221no—
238C2×C22⋊Q8—611no—
239C2×C22⋊Q8—1221no—
240C2×C22⋊Q8#692311yescharacteristic
241C2×C4×Q8—611no—
242C2×C4×Q8—611no—
243C2×C4×Q8—1221no—
244C2×#117—1221no—
245C22×C42—611no—
246#432—614no—
247#432—1222no—
248#431—614no—
249#431—1222no—
250C2×C4⋊C8—612no—
251C2×C4⋊C8—612no—
252C2×C22⋊C8—1221no—
253C2×C22⋊C8—1221no—
254C2×#97—1221no—
255C2×C4×C8—1221no—
256#333—618no—
257#332—618no—
258C61C2×#155411yescharacteristic, complemented, Sylow
259C22×Q8—611no—
260C22×Q8—1221no—
261C22×Q8C2×#692311yescharacteristic
262C23×C4—611no—
263C23×C4—611no—
264C23×C4C2×#692311yescharacteristic
265C22×C8—1222no—
266C22×C8—1222no—
267C4⋊Q8—612no—
268C4⋊Q8—612no—
269C4⋊Q8—612no—
270C4⋊Q8—612no—
271C4⋊Q8—1221no—
272C4⋊Q8—1221no—
273C4⋊Q8—1221no—
274C4⋊Q8—1221no—
275#125—1224no—
276#125—1224no—
277C22⋊Q8—22no—
278C22⋊Q8—614no—
279C22⋊Q8—1224no—
280C22⋊Q8—1224no—
281C22⋊Q8—1222no—
282C22⋊Q8C2×#692314yes—
283C4×Q8—614no—
284C4×Q8—614no—
285C4×Q8—1222no—
286C4×Q8—1224no—
287C4×Q8—1224no—
288C4×Q8—1222no—
289#117—1224no—
290#117—1224no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4—1221no—
299C2×C4⋊C4C2×#692311yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—1221no—
302C2×C42—611no—
303C2×C42—611no—
304C2×C42—612no—
305C2×C42—612no—
306C2×C42—611no—
307C2×C42—611no—
308C2×C42—612no—
309C2×C42—612no—
310C2×C42—1221no—
311C2×C42—1221no—
312C4⋊C8—614no—
313C4⋊C8—614no—
314C4⋊C8—1222no—
315C4⋊C8—1222no—
316C22⋊C8—1224no—
317C22⋊C8—1224no—
318#97—1224no—
319C4×C8—1224no—
320C24C22⋊#692311yescharacteristic
321C2×Q8—21no—
322C2×Q8—42no—
323C2×Q8—612no—
324C2×Q8—1222no—
325C2×Q8—1222no—
326C2×Q8—1221no—
327C2×Q8—2442no—
328C2×Q8—2444no—
329C2×Q8C22×#692312yes—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—42no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—612no—
336C22×C4—612no—
337C22×C4—612no—
338C22×C4—612no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—1224no—
342C22×C4—1224no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4C22⋊#692311yescharacteristic
346C22×C4C22⋊#692312yes—
347C22×C4C22×#692311yescharacteristic
348C22×C4C22×#692312yes—
349C2×C8—1224no—
350C2×C8—1224no—
351C2×C8—2444no—
352C2×C8—2444no—
353C4⋊C4—24no—
354C4⋊C4—21no—
355C4⋊C4—612no—
356C4⋊C4—1222no—
357C4⋊C4—1222no—
358C4⋊C4—1222no—
359C4⋊C4—1222no—
360C4⋊C4—1224no—
361C4⋊C4—1222no—
362C4⋊C4—1222no—
363C4⋊C4—1224no—
364C4⋊C4—1221no—
365C4⋊C4—1224no—
366C4⋊C4C22×#692312yes—
367C22⋊C4—24no—
368C22⋊C4—1224no—
369C42—612no—
370C42—612no—
371C42—611no—
372C42—611no—
373C42—612no—
374C42—612no—
375C42—612no—
376C42—611no—
377C42—612no—
378C42—611no—
379C42—1224no—
380C42—1224no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23C2×C22⋊#692312yes—
385C23C2×C22⋊#692311yescharacteristic
386Q8—42no—
387Q8—42no—
388Q8—2442no—
389Q8—2442no—
390Q8—2444no—
391Q8—2444no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—614no—
401C2×C4—614no—
402C2×C4—611no—
403C2×C4—611no—
404C2×C4—611no—
405C2×C4—611no—
406C2×C4—612no—
407C2×C4—612no—
408C2×C4—1224no—
409C2×C4—1224no—
410C2×C4—1224no—
411C2×C4—1224no—
412C2×C4—1222no—
413C2×C4—1222no—
414C2×C4—1222no—
415C2×C4—1222no—
416C2×C4—2442no—
417C2×C4—2442no—
418C2×C4C2×C22⋊#692312yes—
419C2×C4C2×C22⋊#692311yescharacteristic
420C2×C4C23×#692311yescharacteristic
421C8—2444no—
422C8—2444no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—44no—
427C22#1684074912yes—
428C22#1684075212yes—
429C22#1684075512yes—
430C22C22×C22⋊#692311yescharacteristic
431C4—22no—
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—612no—
437C4—612no—
438C4—612no—
439C4—612no—
440C4—1222no—
441C4—1222no—
442C4—2442no—
443C4—2442no—
444C2—42no—
445C2#2146134514yesdirect product factor
446C2C2×#1684074911yescharacteristic
447C2C2×#1684075211yescharacteristic
448C2C2×#1684075511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC4×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61C2×#1554
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21461345C2direct (N × H)
4. C2×#16840622C2semidirect
5. C2×#16840621C2semidirect
17. #16840622C22semidirect
18. #16840621C22semidirect
227. C2×C61#1554semidirect
258. C61C2×#1554semidirect
445. C2#21461345direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407822
C2×#168407792
#214613452
C2×#168406222
C2×#168406212
C2×#168403142
C2×#155461