Finite Group Databasebeta

← C2×D61⋊#356 #30624000

Subgroups — C2×D61⋊#356 #30624000

Order 15616.

1254 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D61⋊#101C212yesmaximal
2C2×#16840673C212yesmaximal, complemented
3D61⋊#356C218yesmaximal, direct product factor
4C2×#16840672C212yesmaximal, complemented
5C2×C4⋊Q8×D61C211yesmaximal, characteristic
6C2×D61⋊M16—22no—
7D61⋊#101—24no—
8D61⋊#101C2218yes—
9C2×#14519838—22no—
10C2×#14519838C2214yes—
11#16840673C2218yescomplemented
12#16840672C2218yescomplemented
13C22×Q8×D61C411yescharacteristic
14C22×Q8×D61C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519776—22no—
17C2×#14519775C411yescharacteristic
18C2×#14519775C2211yescharacteristic
19C4⋊Q8×D61—24no—
20C4⋊Q8×D61C414yes—
21C4⋊Q8×D61C2214yes—
22C2×C4⋊C4×D61—22no—
23C2×C42×D61C411yescharacteristic
24C2×C4⋊Q8×C61C411yescharacteristic
25C2×#11758213—42no—
26D61⋊M16—28no—
27D61⋊M16—44no—
28C2×#11758217—24no—
29C2×D61⋊C8—42no—
30#14519838—28no—
31#14519838—24no—
32#14519838C2318yes—
33C22×Dic122—21no—
34C22×Dic122C2×C411yescharacteristic
35C22×Dic122D412yes—
36C22×Dic122C2311yescharacteristic
37C2×Q8×D61—22no—
38C2×Q8×D61—41no—
39C2×Q8×D61—42no—
40C2×Q8×D61—41no—
41C2×Q8×D61—42no—
42C2×Q8×D61—41no—
43C2×Q8×D61—41no—
44C2×Q8×D61C2×C414yes—
45C2×Q8×D61C2×C412yes—
46C2×Q8×D61D414yes—
47C2×Q8×D61C2312yes—
48C22×C4×D61—21no—
49C22×C4×D61—21no—
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61D412yes—
52#14519777—22no—
53#14519777C2×C412yes—
54#14519777C2312yes—
55#14519776—28no—
56#14519776—28no—
57#14519775—22no—
58#14519775—22no—
59#14519775C2×C412yes—
60#14519775C2×C414yes—
61#14519775C2312yes—
62C2×C4⋊Dic61—22no—
63C2×#11758196—22no—
64C2×#11758196—22no—
65C4⋊C4×D61—28no—
66C4⋊C4×D61—28no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61C2×C411yescharacteristic
69C42×D61—22no—
70C42×D61C2×C414yes—
71C22×Q8×C61C2×C411yescharacteristic
72C22×Q8×C61C2×C411yescharacteristic
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61C2×C414yes—
75C2×C4⋊C4×C61—22no—
76C2×C42×C61C2×C411yescharacteristic
77C23×D61C22⋊C411yescharacteristic
78C2×Q8×C61—21no—
79C2×Q8×C61—41no—
80C2×Q8×C61—41no—
81C2×Q8×C61—41no—
82C2×Q8×C61—41no—
83C2×Q8×C61C22⋊C412yes—
84C2×Q8×C61C22×C412yes—
85C2×Q8×C61C22×C412yes—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61C22⋊C412yes—
89C22×C4×C61C22×C411yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C2×D412yes—
94Q8×D61—42no—
95Q8×D61—42no—
96Q8×D61—44no—
97Q8×D61—44no—
98Q8×D61—42no—
99Q8×D61—42no—
100Q8×D61—44no—
101Q8×D61—44no—
102Q8×D61—44no—
103Q8×D61—44no—
104C2×C4×D61—28no—
105C2×C4×D61—21no—
106C2×C4×D61—22no—
107C2×C4×D61—21no—
108C2×C4×D61—22no—
109C2×C4×D61—24no—
110C2×C4×D61—22no—
111C2×C4×D61—24no—
112C2×C4×D61—24no—
113C2×C4×D61—41no—
114C2×C4×D61—42no—
115C2×C4×D61—41no—
116C2×C4×D61C22⋊C412yes—
117C2×C4×D61C22⋊C414yes—
118C2×C4×D61C22×C412yes—
119C2×C4×D61C2×D414yes—
120C2×Dic122—21no—
121C2×Dic122—24no—
122C2×Dic122—42no—
123C2×Dic122—42no—
124C2×Dic122—41no—
125C2×Dic122—41no—
126C2×Dic122—42no—
127C2×Dic122—42no—
128C2×Dic122—41no—
129C2×Dic122—41no—
130C2×Dic122C22⋊C414yes—
131C2×Dic122C22×C412yes—
132C2×Dic122C22×C411yescharacteristic
133C2×Dic122C2×D412yes—
134C2×Dic122C2×D414yes—
135C2×Dic122C2411yescharacteristic
136#11758217—28no—
137#11758217—44no—
138C2×#92420—44no—
139#11758213—48no—
140D61⋊C8—48no—
141C4⋊C4×C61—28no—
142C42×C61—21no—
143C42×C61C22×C412yes—
144C4⋊Dic61—24no—
145C4⋊Dic61—24no—
146#11758196—28no—
147#11758196—24no—
148#11758196—24no—
149C4×Dic61—21no—
150C4×Dic61—24no—
151C4×Dic61C22×C412yes—
152C23×C61C2×C22⋊C411yescharacteristic
153C22×D61—24no—
154C22×D61#9911yescharacteristic
155C22×D61#9912yes—
156C22×D61#9911yescharacteristic
157C22×D61C2×C22⋊C412yes—
158Q8×C61—42no—
159Q8×C61—42no—
160Q8×C61—42no—
161Q8×C61—42no—
162C2×C4×C61—24no—
163C2×C4×C61—22no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61C2×C22⋊C411yescharacteristic
172C2×C4×C61C2×C22⋊C414yes—
173C2×C4×C61C23×C411yescharacteristic
174C2×Dic61—24no—
175C2×Dic61—22no—
176C2×Dic61—22no—
177C2×Dic61—22no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—22no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61C2×C22⋊C412yes—
184C2×Dic61C2×C22⋊C412yes—
185C2×Dic61C22×D412yes—
186C4×D61—216no—
187C4×D61—24no—
188C4×D61—24no—
189C4×D61—24no—
190C4×D61—24no—
191C4×D61—22no—
192C4×D61—22no—
193C4×D61—22no—
194C4×D61—22no—
195C4×D61—44no—
196C4×D61—44no—
197Dic122—44no—
198Dic122—41no—
199Dic122—44no—
200Dic122—42no—
201Dic122—41no—
202Dic122—48no—
203Dic122—42no—
204Dic122—42no—
205#92420—48no—
206C22×#356—611nomaximal, Sylow
207C22×C61—22no—
208C22×C61C2×#9911yescharacteristic
209C22×C61C2×#9911yescharacteristic
210C22×C61C22×C22⋊C411yescharacteristic
211C2×D61—22no—
212C2×D61—24no—
213C2×D61—24no—
214C2×D61#35614yescomplemented
215C2×D61C2×#9912yes—
216C2×D61C2×#9912yes—
217C4×C61—24no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—24no—
221C4×C61—42no—
222Dic61—24no—
223Dic61—22no—
224Dic61—22no—
225Dic61—22no—
226Dic61—42no—
227C22×C4⋊Q8—611no—
228C22×#101—612no—
229C2×#356—612no—
230C2×#356—618no—
231C2×#356—612no—
232C2×C61—21no—
233C2×C61—21no—
234C2×C61C2×#35612yescomplemented
235C2×C61C22×#9911yescharacteristic
236D61—22no—
237D61C2×#35614yescomplemented
238C23×Q8—611no—
239C23×Q8—611no—
240C22×M16—1222no—
241C2×C4⋊Q8—611no—
242C2×C4⋊Q8—611no—
243C2×C4⋊Q8—614no—
244C2×C4⋊Q8—614no—
245C2×C4⋊Q8—611no—
246C2×C4⋊Q8—1222no—
247C2×C4⋊Q8—1224no—
248C2×C4⋊Q8#692311yescharacteristic
249C22×C4⋊C4—1222no—
250C22×C42—611no—
251C2×#101—614no—
252C2×#101—618no—
253C2×#101—1222no—
254C2×#101—1224no—
255#356—618no—
256#356—618no—
257C61C22×#35611yescharacteristic, complemented, Sylow
258C22×Q8—614no—
259C22×Q8—612no—
260C22×Q8—612no—
261C22×Q8—612no—
262C22×Q8—611no—
263C22×Q8—614no—
264C22×Q8—611no—
265C22×Q8—1221no—
266C22×Q8—1222no—
267C22×Q8—2441no—
268C22×Q8—2442no—
269C22×Q8—2441no—
270C22×Q8—2442no—
271C22×Q8—2441no—
272C22×Q8—2441no—
273C22×Q8C2×#692311yescharacteristic
274C22×Q8C2×#692311yescharacteristic
275C23×C4—612no—
276C23×C4—611no—
277C23×C4—1221no—
278C23×C4—1221no—
279C2×M16—1224no—
280C2×M16—1228no—
281C2×M16—2444no—
282C2×M16—2442no—
283C22×C8—2442no—
284C4⋊Q8—22no—
285C4⋊Q8—612no—
286C4⋊Q8—612no—
287C4⋊Q8—614no—
288C4⋊Q8—612no—
289C4⋊Q8—612no—
290C4⋊Q8—1222no—
291C4⋊Q8—1228no—
292C4⋊Q8—1228no—
293C4⋊Q8—1222no—
294C4⋊Q8—1222no—
295C4⋊Q8C2×#692314yes—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—1228no—
298C2×C4⋊C4—1228no—
299C2×C4⋊C4—1222no—
300C2×C4⋊C4—1222no—
301C2×C4⋊C4—1222no—
302C2×C42—611no—
303C2×C42—614no—
304C2×C42—1222no—
305C2×C42—1221no—
306C2×C42C2×#692311yescharacteristic
307#101—618no—
308#101—1228no—
309#101—1224no—
310C24—611no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—614no—
317C2×Q8—612no—
318C2×Q8—614no—
319C2×Q8—611no—
320C2×Q8—611no—
321C2×Q8—612no—
322C2×Q8—1224no—
323C2×Q8—1221no—
324C2×Q8—2442no—
325C2×Q8—2441no—
326C2×Q8—2442no—
327C2×Q8—2441no—
328C2×Q8—2441no—
329C2×Q8—2441no—
330C2×Q8—2442no—
331C2×Q8—2442no—
332C2×Q8—2442no—
333C2×Q8—2444no—
334C2×Q8—2442no—
335C2×Q8—2444no—
336C2×Q8—2442no—
337C2×Q8—2442no—
338C2×Q8—2444no—
339C2×Q8—2444no—
340C2×Q8—2444no—
341C2×Q8—2444no—
342C2×Q8C22⋊#692312yes—
343C2×Q8C22×#692312yes—
344C2×Q8C22×#692312yes—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—611no—
348C22×C4—612no—
349C22×C4—612no—
350C22×C4—614no—
351C22×C4—614no—
352C22×C4—612no—
353C22×C4—1224no—
354C22×C4—1222no—
355C22×C4—1224no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1228no—
361C22×C4—1224no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—2442no—
365C22×C4—2441no—
366C22×C4—2441no—
367C22×C4C22⋊#692312yes—
368C22×C4C22×#692311yescharacteristic
369M16—1228no—
370M16—2448no—
371M16—2444no—
372C2×C8—2448no—
373C2×C8—2444no—
374C4⋊C4—28no—
375C4⋊C4—1224no—
376C4⋊C4—1224no—
377C4⋊C4—1224no—
378C4⋊C4—1224no—
379C4⋊C4—1228no—
380C42—21no—
381C42—612no—
382C42—1224no—
383C42—1221no—
384C42C22×#692312yes—
385C23—611no—
386C23—612no—
387C23—611no—
388C23—612no—
389C23—1224no—
390C23C2×C22⋊#692311yescharacteristic
391Q8—42no—
392Q8—42no—
393Q8—42no—
394Q8—42no—
395Q8—2442no—
396Q8—2444no—
397Q8—2444no—
398Q8—2441no—
399Q8—2441no—
400Q8—2442no—
401Q8—2442no—
402Q8—2448no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—612no—
413C2×C4—612no—
414C2×C4—612no—
415C2×C4—1224no—
416C2×C4—1224no—
417C2×C4—1224no—
418C2×C4—1224no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—12216no—
424C2×C4—1224no—
425C2×C4—1222no—
426C2×C4—1221no—
427C2×C4—1222no—
428C2×C4—1221no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—2444no—
432C2×C4—2444no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4C2×C22⋊#692311yescharacteristic
436C2×C4C2×C22⋊#692314yes—
437C2×C4C23×#692311yescharacteristic
438C8—2448no—
439C22—22no—
440C22—614no—
441C22—612no—
442C22—612no—
443C22—1224no—
444C22—1224no—
445C22—1222no—
446C22D61⋊#9911yescharacteristic
447C22D61⋊#9911yescharacteristic
448C22C22×C22⋊#692311yescharacteristic
449C4—24no—
450C4—21no—
451C4—24no—
452C4—21no—
453C4—42no—
454C4—1224no—
455C4—1222no—
456C4—1222no—
457C4—1222no—
458C4—2442no—
459C2—21no—
460C2—21no—
461C2—614no—
462C2—1222no—
463C2D61⋊#35612yesdirect product factor
464C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC2×Dic122C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61C22×#356
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16840673C2semidirect
3. D61⋊#356C2direct (N × H)
4. C2×#16840672C2semidirect
11. #16840673C22semidirect
12. #16840672C22semidirect
214. C2×D61#356semidirect
234. C2×C61C2×#356semidirect
237. D61C2×#356semidirect
257. C61C22×#356semidirect
463. C2D61⋊#356direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D61⋊#1012
C2×#168406732
D61⋊#3562
C2×#168406722
C2×C4⋊Q8×D612
C22×#35661