Finite Group Databasebeta

← C2×#21461320 #30611712

Subgroups — C2×#21461320 #30611712

Order 15616.

1050 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840780C211yesmaximal, characteristic
2C2×D61⋊(C4⋊C8)C211yesmaximal, characteristic
3#21461320C218yesmaximal, direct product factor
4C2×#16840622C212yesmaximal, complemented
5C2×#16840621C212yesmaximal, complemented
6C2×#16840316C211yesmaximal, characteristic
7C22×D61⋊C8—21no—
8#16840780—22no—
9#16840780C2214yes—
10C2×#14519844C2212yes—
11C2×#14519844C2212yes—
12D61⋊(C4⋊C8)—22no—
13D61⋊(C4⋊C8)C2214yes—
14C2×C4⋊#92420—21no—
15C2×#14519834—21no—
16C2×C4×#92420—21no—
17#16840622C2218yescomplemented
18#16840621C2218yescomplemented
19#16840316—22no—
20#16840316—22no—
21#16840316C414yes—
22#16840316C2214yes—
23C2×#14519776C2211yescharacteristic
24C2×#14519775C2211yescharacteristic
25C2×#14519767—21no—
26C2×#14519757C411yescharacteristic
27C2×#14519755—21no—
28C2×#14519753C411yescharacteristic
29C2×C4×Dic122—21no—
30C2×C4×Dic122C411yescharacteristic
31C2×#14519734—21no—
32C2×C42×D61C2211yescharacteristic
33C2×C4×Q8×C61C411yescharacteristic
34C2×D61⋊C8—24no—
35C2×D61⋊C8—44no—
36C22×#92420—22no—
37C22×#92420—22no—
38#14519844—22no—
39#14519844—22no—
40#14519844C2314yes—
41#14519844C2314yes—
42C4⋊#92420—24no—
43#14519834—24no—
44C4×#92420—24no—
45C22×Dic122—21no—
46C22×Dic122C2×C411yescharacteristic
47C22×C4×D61C2×C411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50#14519776—21no—
51#14519776—21no—
52#14519776C2×C412yes—
53#14519776C2312yes—
54#14519775—21no—
55#14519775—21no—
56#14519775C2×C412yes—
57#14519775C2312yes—
58#14519767—24no—
59#14519767—24no—
60#14519757—22no—
61#14519757C2×C414yes—
62#14519755—24no—
63#14519755—24no—
64#14519753—22no—
65#14519753C2×C414yes—
66C4×Dic122—22no—
67C4×Dic122—24no—
68C4×Dic122—24no—
69C4×Dic122C2×C414yes—
70#14519734—24no—
71#14519734—24no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C412yes—
86C42×D61D412yes—
87C42×D61D412yes—
88C42×D61C2312yes—
89C22×Q8×C61C2×C411yescharacteristic
90C4×Q8×C61—22no—
91C4×Q8×C61C2×C414yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61C2×C411yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61C2×C411yescharacteristic
96C23×D61C22⋊C411yescharacteristic
97C2×Q8×C61—21no—
98C2×Q8×C61—42no—
99C2×Q8×C61C22×C412yes—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61C22⋊C411yescharacteristic
103C22×C4×C61C22⋊C411yescharacteristic
104C22×C4×C61C22×C411yescharacteristic
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61C22⋊C411yescharacteristic
108C22×Dic61C2×D411yescharacteristic
109C22×Dic61C2×D411yescharacteristic
110C2×C4×D61—24no—
111C2×C4×D61—42no—
112C2×C4×D61C22⋊C412yes—
113C2×C4×D61C22⋊C412yes—
114C2×C4×D61C22⋊C412yes—
115C2×C4×D61SD1614yes—
116C2×C4×D61Q1614yes—
117C2×C4×D61C22×C412yes—
118C2×C4×D61C2×D412yes—
119C2×C4×D61C2×D412yes—
120C2×Dic122—21no—
121C2×Dic122—22no—
122C2×Dic122—22no—
123C2×Dic122—44no—
124C2×Dic122—42no—
125C2×Dic122C22×C412yes—
126C2×#92420—24no—
127C2×#92420—24no—
128C2×#92420—44no—
129C2×#92420—44no—
130D61⋊C8—48no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61C22×C412yes—
134C42×C61—24no—
135C42×C61C22⋊C412yes—
136C42×C61C22×C412yes—
137#11758198—24no—
138#11758198—24no—
139C4⋊Dic61—21no—
140C4⋊Dic61—24no—
141C4⋊Dic61C22×C412yes—
142#11758196—24no—
143#11758196—22no—
144#11758196—22no—
145#11758196—24no—
146#11758196—22no—
147#11758196—22no—
148#11758196—22no—
149#11758196—22no—
150C4×Dic61—24no—
151C4×Dic61C22⋊C412yes—
152C4×Dic61C22⋊C412yes—
153C4×Dic61C22×C411yescharacteristic
154C4×Dic61C2×D412yes—
155C4×Dic61C2×D412yes—
156C4×Dic61C2×D411yescharacteristic
157C4×Dic61C2×D411yescharacteristic
158C4×Dic61C2411yescharacteristic
159C23×C61C2×C22⋊C411yescharacteristic
160C22×D61—22no—
161C22×D61—22no—
162C22×D61#10314yes—
163C22×D61C2×C22⋊C412yes—
164Q8×C61—42no—
165Q8×C61—42no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61#10314yes—
173C2×C4×C61C2×C22⋊C412yes—
174C2×C4×C61C2×C22⋊C411yescharacteristic
175C2×C4×C61C2×C22⋊C412yes—
176C2×C4×C61C23×C411yescharacteristic
177C2×Dic61—22no—
178C2×Dic61—22no—
179C2×Dic61—22no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61C2×C22⋊C412yes—
183C2×Dic61C2×C22⋊C411yescharacteristic
184C2×Dic61C2×C22⋊C411yescharacteristic
185C2×Dic61C2×SD1612yes—
186C2×Dic61C2×Q1612yes—
187C2×Dic61#13514yes—
188C2×Dic61C22×D411yescharacteristic
189C2×Dic61C22×D411yescharacteristic
190C4×D61—24no—
191C4×D61—24no—
192C4×D61—42no—
193C4×D61—42no—
194C4×D61#10318yes—
195C4×D61C2×SD1614yes—
196C4×D61C2×Q1614yes—
197Dic122—42no—
198Dic122—44no—
199Dic122—44no—
200Dic122—42no—
201#92420—44no—
202#92420—44no—
203C2×#1542—611nomaximal, Sylow
204C22×C61#41312yes—
205C22×C61C2×#10312yes—
206C22×C61#41612yes—
207C22×C61C22×C22⋊C411yescharacteristic
208C2×D61—28no—
209C2×D61—22no—
210C2×D61—22no—
211C2×D61C2×#10314yes—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—22no—
215C4×C61—42no—
216C4×C61—42no—
217C4×C61C2×#10314yes—
218Dic61—22no—
219Dic61—42no—
220Dic61—42no—
221Dic61C2×#10312yes—
222Dic61#41612yes—
223Dic61C22×SD1611yescharacteristic
224Dic61C22×Q1611yescharacteristic
225Dic61C2×#13512yes—
226C2×#533—611no—
227C2×#431—611no—
228C22×C4⋊C8—611no—
229#1542—618no—
230C2×#333—612no—
231C2×#332—612no—
232C2×C61#154214yescomplemented
233C2×C61C2×#41311yescharacteristic
234C2×C61C22×#10311yescharacteristic
235C2×C61C2×#41611yescharacteristic
236D61—24no—
237C23×C8—1221no—
238#533—614no—
239#533—614no—
240#533—1222no—
241#533—1222no—
242C2×C4⋊Q8—611no—
243C2×C4⋊Q8—611no—
244C2×#125—1221no—
245C2×C22⋊Q8—611no—
246C2×C22⋊Q8—611no—
247C2×C22⋊Q8—1221no—
248C2×C4×Q8—611no—
249C2×C4×Q8—1221no—
250C2×C4×Q8#692311yescharacteristic
251C2×#117—1221no—
252C22×C42—611no—
253#431—614no—
254#431—1222no—
255C2×C4⋊C8—612no—
256C2×C4⋊C8—612no—
257C2×C4⋊C8—614no—
258C2×C4⋊C8—1221no—
259C2×C4⋊C8—1222no—
260C2×C22⋊C8—1221no—
261C2×C4×C8—1221no—
262#333—618no—
263#332—618no—
264C61C2×#154211yescharacteristic, complemented, Sylow
265C22×Q8—611no—
266C22×Q8—1221no—
267C22×Q8C2×#692311yescharacteristic
268C23×C4—611no—
269C23×C4—611no—
270C23×C4—611no—
271C22×C8—1222no—
272C22×C8—1222no—
273C22×C8—1224no—
274C22×C8—2444no—
275C4⋊Q8—612no—
276C4⋊Q8—612no—
277C4⋊Q8—612no—
278C4⋊Q8—612no—
279C4⋊Q8—1221no—
280C4⋊Q8—1221no—
281C4⋊Q8—1221no—
282C4⋊Q8—1221no—
283#125—1224no—
284#125—1224no—
285C22⋊Q8—614no—
286C22⋊Q8—614no—
287C22⋊Q8—1222no—
288C22⋊Q8—1224no—
289C22⋊Q8—1224no—
290C22⋊Q8—1222no—
291C4×Q8—22no—
292C4×Q8—614no—
293C4×Q8—1224no—
294C4×Q8—1224no—
295C4×Q8—1222no—
296C4×Q8C2×#692314yes—
297#117—1224no—
298#117—1224no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1221no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—1221no—
307C2×C4⋊C4C2×#692311yescharacteristic
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C42—21no—
311C2×C42—611no—
312C2×C42—611no—
313C2×C42—611no—
314C2×C42—612no—
315C2×C42—612no—
316C2×C42—612no—
317C2×C42—612no—
318C2×C42—1221no—
319C2×C42C2×#692311yescharacteristic
320C4⋊C8—614no—
321C4⋊C8—614no—
322C4⋊C8—1224no—
323C4⋊C8—1222no—
324C4⋊C8—1222no—
325C22⋊C8—1224no—
326C4×C8—1224no—
327C24—611no—
328C2×Q8—21no—
329C2×Q8—42no—
330C2×Q8—612no—
331C2×Q8—1222no—
332C2×Q8—1222no—
333C2×Q8—1221no—
334C2×Q8—2442no—
335C2×Q8—2444no—
336C2×Q8C22×#692312yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—614no—
342C22×C4—614no—
343C22×C4—612no—
344C22×C4—612no—
345C22×C4—611no—
346C22×C4—612no—
347C22×C4—612no—
348C22×C4—612no—
349C22×C4—612no—
350C22×C4—1224no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—2442no—
354C22×C4C22⋊#692311yescharacteristic
355C22×C4C22⋊#692311yescharacteristic
356C22×C4C22×#692311yescharacteristic
357C2×C8—1224no—
358C2×C8—1224no—
359C2×C8—2448no—
360C2×C8—2444no—
361C2×C8—2444no—
362C4⋊C4—24no—
363C4⋊C4—21no—
364C4⋊C4—612no—
365C4⋊C4—1222no—
366C4⋊C4—1222no—
367C4⋊C4—1222no—
368C4⋊C4—1222no—
369C4⋊C4—1224no—
370C4⋊C4—1222no—
371C4⋊C4—1222no—
372C4⋊C4—1224no—
373C4⋊C4—1221no—
374C4⋊C4—1224no—
375C4⋊C4C22×#692312yes—
376C22⋊C4—1224no—
377C22⋊C4—1224no—
378C42—24no—
379C42—612no—
380C42—612no—
381C42—611no—
382C42—611no—
383C42—612no—
384C42—612no—
385C42—611no—
386C42—611no—
387C42—1224no—
388C42C22⋊#692312yes—
389C42C22×#692312yes—
390C23—614no—
391C23—612no—
392C23—1222no—
393C23—1222no—
394C23C2×C22⋊#692311yescharacteristic
395Q8—42no—
396Q8—42no—
397Q8—2442no—
398Q8—2442no—
399Q8—2444no—
400Q8—2444no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—612no—
408C2×C4—612no—
409C2×C4—614no—
410C2×C4—611no—
411C2×C4—611no—
412C2×C4—611no—
413C2×C4—611no—
414C2×C4—618no—
415C2×C4—614no—
416C2×C4—614no—
417C2×C4—612no—
418C2×C4—1224no—
419C2×C4—1224no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4#1451984014yes—
428C2×C4C2×C22⋊#692311yescharacteristic
429C2×C4C2×C22⋊#692312yes—
430C2×C4C2×C22⋊#692312yes—
431C2×C4C23×#692311yescharacteristic
432C8—2444no—
433C8—2444no—
434C22—614no—
435C22—1228no—
436C22—1222no—
437C22—1222no—
438C22#1684074812yes—
439C22C2×#1451984012yes—
440C22#1684075212yes—
441C22C22×C22⋊#692311yescharacteristic
442C4—21no—
443C4—21no—
444C4—22no—
445C4—42no—
446C4—42no—
447C4—611no—
448C4—611no—
449C4—612no—
450C4—612no—
451C4—612no—
452C4—1222no—
453C4—2442no—
454C4—2442no—
455C4C2×#1451984014yes—
456C2—1224no—
457C2#2146132014yesdirect product factor
458C2C2×#1684074811yescharacteristic
459C2C22×#1451984011yescharacteristic
460C2C2×#1684075211yescharacteristic
subgroupG/H
centerC22×C4C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4×Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC4×Dic61C24
hypercenterC2×C4×Q8#6923
hypocenterC61C2×#1542
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #21461320C2direct (N × H)
4. C2×#16840622C2semidirect
5. C2×#16840621C2semidirect
17. #16840622C22semidirect
18. #16840621C22semidirect
232. C2×C61#1542semidirect
264. C61C2×#1542semidirect
457. C2#21461320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407802
C2×D61⋊(C4⋊C8)2
#214613202
C2×#168406222
C2×#168406212
C2×#168403162
C2×#154261