Finite Group Databasebeta

← C2×#21458207 #30558470

Subgroups — C2×#21458207 #30558470

Order 15616.

1942 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic
2#21458207C2116yesmaximal, direct product factor
3C2×#16840165C211yesmaximal, characteristic, complemented
4C2×#16840097C212yesmaximal, complemented
5C2×#16840095C212yesmaximal, complemented
6C2×#16840057C211yesmaximal, characteristic, complemented
7C2×#16840054C211yesmaximal, characteristic, complemented
8C2×#16840001C212yesmaximal, complemented
9C2×C4×#11758202C211yesmaximal, characteristic, complemented
10C2×D4×#92418C211yesmaximal, characteristic
11C22×#14519710C212yesmaximal, complemented
12C2×C61×#554C211yesmaximal, characteristic
13C22×#11758202C2211yescharacteristic
14C23×#92418C2212yes—
15#16840469—24no—
16#16840469C2218yes—
17C2×#14519777C2211yescharacteristic
18C2×#14519768—21no—
19C2×#14519756C2212yes—
20C2×#14519754—22no—
21C2×C4×Dic122C2211yescharacteristic
22C2×D4×Dic61—21no—
23C22×C4⋊Dic61C2212yes—
24#16840165C2218yescomplemented
25#16840097C22116yescomplemented
26#16840095C22116yescomplemented
27#16840057C2218yescomplemented
28#16840054C2218yescomplemented
29#16840001C22116yescomplemented
30C4×#11758202C2218yescomplemented
31D4×#92418C2218yes—
32C2×#14519710C2212yes—
33C2×#14519710C2211yescharacteristic
34C2×#14519710C2212yes—
35C2×#14519710C2212yes—
36C2×#14519710C2212yes—
37C2×#14519710C22116yescomplemented
38C2×C4⋊#92418C2211yescharacteristic
39C2×#14519701C2212yes—
40C2×#14519701C2211yescharacteristic
41C2×#14519700C2212yes—
42C2×#14519700C2211yescharacteristic
43C2×C22⋊#92418C2212yes—
44C2×C4×#92418C2211yescharacteristic
45C61×#554—24no—
46C61×#554C2218yes—
47C2×C4⋊Q8×C61C2211yescharacteristic
48C2×#125×C61—21no—
49C2×C22⋊Q8×C61—22no—
50C2×C22⋊Q8×C61C2212yes—
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×D4×C61—21no—
53C2×C4×D4×C61C2211yescharacteristic
54C22×C4⋊C4×C61C2212yes—
55C22×Dic122—21no—
56C22×Dic122C2311yescharacteristic
57C23×Dic61—22no—
58C2×#11758202—24no—
59C2×#11758202C2314yes—
60C22×#92418—28no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2312yes—
63C22×#92418C2318yes—
64C22×#92418C2311yescharacteristic
65#14519777—22no—
66#14519777C2314yes—
67#14519768—28no—
68#14519756—24no—
69#14519756C2318yes—
70#14519754—216no—
71C4×Dic122—22no—
72C4×Dic122C2314yes—
73D4×Dic61—28no—
74C2×C4⋊Dic61—24no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2318yes—
79C2×C4⋊Dic61C2312yes—
80C2×#11758196—21no—
81C2×#11758196—22no—
82C2×C22⋊Dic61—22no—
83C2×C4×Dic61—21no—
84#14519710C2318yes—
85#14519710C2314yes—
86#14519710C2318yes—
87#14519710C2318yes—
88#14519710C2318yes—
89C4⋊#92418C2314yes—
90#14519701C2318yes—
91#14519701C2314yes—
92#14519700C2318yes—
93#14519700C2314yes—
94C22⋊#92418C2318yes—
95C4×#92418C2314yes—
96C22×Q8×C61—21no—
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61D411yescharacteristic
99C23×C4×C61—22no—
100C23×C4×C61C2312yes—
101C4⋊Q8×C61—22no—
102C4⋊Q8×C61C2314yes—
103#125×C61—28no—
104C22⋊Q8×C61—24no—
105C22⋊Q8×C61—216no—
106C22⋊Q8×C61C2318yes—
107C4×Q8×C61—22no—
108C4×Q8×C61C2314yes—
109C4×D4×C61—28no—
110C4×D4×C61D414yes—
111C4×D4×C61C2314yes—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—24no—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61C2312yes—
119C2×C4⋊C4×C61C2318yes—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—22no—
122C2×C22⋊C4×C61D412yes—
123C2×C42×C61—21no—
124C2×C42×C61C2311yescharacteristic
125C24×C61C2×D412yes—
126C2×Q8×C61—21no—
127C2×Q8×C61—24no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—82no—
131C2×Q8×C61C2412yes—
132C2×D4×C61Q1618yescomplemented
133C2×D4×C61C2×D414yes—
134C22×C4×C61—24no—
135C22×C4×C61—24no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—28no—
140C22×C4×C61—84no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D414yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2414yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61—28no—
151C22×Dic61—84no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—21no—
155C2×Dic122—24no—
156C2×Dic122—82no—
157C2×Dic122C2412yes—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—28no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—28no—
164C4⋊C4×C61C2×D414yes—
165C4⋊C4×C61C2412yes—
166C4⋊C4×C61C2414yes—
167C22⋊C4×C61—28no—
168C22⋊C4×C61C2×D418yes—
169C42×C61—24no—
170C42×C61C2×D412yes—
171C42×C61C2412yes—
172C22⋊Dic61—28no—
173#11758202—28no—
174C4⋊Dic61—21no—
175C4⋊Dic61—28no—
176C4⋊Dic61—22no—
177C4⋊Dic61—24no—
178C4⋊Dic61C2412yes—
179C4⋊Dic61C2414yes—
180#11758196—24no—
181#11758196—28no—
182C4×Dic61—24no—
183C2×#92418—24no—
184C2×#92418—22no—
185C2×#92418—216no—
186C2×#92418#4114yes—
187C2×#92418C2412yes—
188C2×#92418C2414yes—
189C2×#92418C2412yes—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61C2×Q1618yes—
193C23×C61C22×D414yes—
194C23×C61C22×D411yescharacteristic
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61—42no—
198Q8×C61—84no—
199D4×C61C2×Q16116yescomplemented
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—28no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—44no—
211C2×C4×C61—88no—
212C2×C4×C61C2×Q1614yes—
213C2×C4×C61C22×D414yes—
214C2×C4×C61C22×D412yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D412yes—
217C2×C4×C61C2×#4112yes—
218C2×C4×C612−1+412yes—
219C2×C4×C61C2511yescharacteristic
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—24no—
223C2×Dic61—42no—
224C2×Dic61—44no—
225C2×Dic61—42no—
226C2×Dic61—88no—
227Dic122—22no—
228Dic122—22no—
229Dic122—42no—
230Dic122—84no—
231#92418—24no—
232#92418—22no—
233#92418C2×#4114yes—
234C2×#3338—611nomaximal, Sylow
235C22×C61—216no—
236C22×C61—24no—
237C22×C61—24no—
238C22×C61#54712yes—
239C22×C61C22×Q1618yes—
240C22×C61C22×Q1612yes—
241C22×C61#57612yes—
242C22×C61C23×D411yescharacteristic
243C4×C61—24no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—44no—
248C4×C61#54712yes—
249C4×C61C22×Q1614yes—
250C4×C61C22×#4111yescharacteristic
251C4×C61C2×2−1+411yescharacteristic
252Dic61—42no—
253Dic61—44no—
254Dic61—42no—
255C2×#554—611no—
256C2×#554D6111yescharacteristic
257#3338—6116no—
258C2×#500—611no—
259C2×#484—612no—
260C2×#484—612no—
261C2×#477—611no—
262C2×#477—611no—
263C2×#467—612no—
264C2×C4×Q16—611no—
265C2×C8×D4—611no—
266C22×#107—612no—
267C2×C61—28no—
268C2×C61#333814yescomplemented
269C2×C61C2×#54711yescharacteristic
270C2×C61C23×Q1611yescharacteristic
271C2×C61C2×#57611yescharacteristic
272C22×Q16—611no—
273C23×C8—612no—
274#554—24no—
275#554—618no—
276#554—1224no—
277#554C2×D6118yes—
278C2×C4⋊Q8—611no—
279C2×C4⋊Q8C2×D6111yescharacteristic
280C2×#125—21no—
281C2×#125—1221no—
282C2×C22⋊Q8—22no—
283C2×C22⋊Q8—612no—
284C2×C22⋊Q8—1222no—
285C2×C22⋊Q8C2×D6112yes—
286C2×C4×Q8—611no—
287C2×C4×Q8C2×D6111yescharacteristic
288C2×C4×D4—21no—
289C2×C4×D4—1221no—
290C2×C4×D4C2×D6111yescharacteristic
291C22×C4⋊C4—612no—
292C22×C4⋊C4C2×D6112yes—
293#500—618no—
294#484—6116no—
295#484—6116no—
296#477—618no—
297#477—618no—
298#467—6116no—
299C4×Q16—618no—
300C8×D4—618no—
301C2×#107—612no—
302C2×#107—612no—
303C2×#107—6116no—
304C2×#107—612no—
305C2×#107—612no—
306C2×#107—611no—
307C2×C4⋊C8—611no—
308C2×#103—612no—
309C2×#103—611no—
310C2×#103—612no—
311C2×#103—611no—
312C2×C22⋊C8—612no—
313C2×C4×C8—611no—
314C61C2×#333811yescharacteristic, complemented, Sylow
315C22×Q8—21no—
316C22×Q8—611no—
317C22×Q8—1221no—
318C22×Q8C22×D6111yescharacteristic
319C22×D4C22⋊D6111yescharacteristic
320C23×C4—22no—
321C23×C4—1222no—
322C23×C4C22×D6112yes—
323C2×Q16—614no—
324C2×Q16—1224no—
325C22×C8—611no—
326C22×C8—612no—
327C22×C8—618no—
328C22×C8—611no—
329C22×C8—1228no—
330C4⋊Q8—22no—
331C4⋊Q8—614no—
332C4⋊Q8—1222no—
333C4⋊Q8C22×D6114yes—
334#125—28no—
335#125—1228no—
336C22⋊Q8—24no—
337C22⋊Q8—216no—
338C22⋊Q8—618no—
339C22⋊Q8—1224no—
340C22⋊Q8—12216no—
341C22⋊Q8C22×D6118yes—
342C4×Q8—22no—
343C4×Q8—614no—
344C4×Q8—1222no—
345C4×Q8C22×D6114yes—
346C4×D4—28no—
347C4×D4—1228no—
348C4×D4C22⋊D6114yes—
349C4×D4C22×D6114yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—618no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1224no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4C22⋊D6111yescharacteristic
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C4⋊C4C22×D6112yes—
366C2×C4⋊C4C22×D6118yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4C22⋊D6112yes—
370C2×C42—21no—
371C2×C42—1221no—
372C2×C42C22×D6111yescharacteristic
373#107—614no—
374#107—618no—
375#107—618no—
376#107—618no—
377#107—618no—
378C4⋊C8—614no—
379#103—614no—
380#103—618no—
381#103—614no—
382#103—618no—
383C22⋊C8—618no—
384C4×C8—614no—
385C24C2×C22⋊D6112yes—
386C2×Q8—21no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—82no—
391C2×Q8—612no—
392C2×Q8—1224no—
393C2×Q8—1221no—
394C2×Q8—1222no—
395C2×Q8—1222no—
396C2×Q8—4882no—
397C2×Q8C23×D6112yes—
398C2×D4#1175820218yescomplemented
399C2×D4C2×C22⋊D6114yes—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—84no—
407C22×C4—1228no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4—4884no—
412C22×C4C2×C22⋊D6114yes—
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6112yes—
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C23×D6114yes—
417C22×C4C23×D6111yescharacteristic
418Q16—1228no—
419C2×C8—614no—
420C2×C8—612no—
421C2×C8—612no—
422C2×C8—614no—
423C2×C8—12216no—
424C2×C8—1224no—
425C2×C8—1222no—
426C4⋊C4—21no—
427C4⋊C4—28no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—612no—
433C4⋊C4—614no—
434C4⋊C4—1221no—
435C4⋊C4—1228no—
436C4⋊C4—1222no—
437C4⋊C4—1224no—
438C4⋊C4—1228no—
439C4⋊C4—1224no—
440C4⋊C4C2×C22⋊D6114yes—
441C4⋊C4C23×D6112yes—
442C4⋊C4C23×D6114yes—
443C22⋊C4—28no—
444C22⋊C4—1228no—
445C22⋊C4C2×C22⋊D6118yes—
446C42—24no—
447C42—1224no—
448C42C2×C22⋊D6112yes—
449C42C23×D6112yes—
450C23—24no—
451C23—24no—
452C23C2×#1175820218yes—
453C23C22×C22⋊D6114yes—
454C23C22×C22⋊D6111yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—42no—
458Q8—84no—
459Q8—1222no—
460Q8—1222no—
461Q8—4884no—
462Q8—2442no—
463D4C2×#11758202116yescomplemented
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—88no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1224no—
479C2×C4—4888no—
480C2×C4—2442no—
481C2×C4—2444no—
482C2×C4—2442no—
483C2×C4C2×#1175820214yes—
484C2×C4C22×C22⋊D6112yes—
485C2×C4C22×C22⋊D6114yes—
486C2×C4C22×C22⋊D6112yes—
487C2×C4C22×C22⋊D6111yescharacteristic
488C2×C4#41×D6112yes—
489C2×C4#1451982412yes—
490C2×C4C24×D6111yescharacteristic
491C8—614no—
492C8—1224no—
493C8—1222no—
494C22—216no—
495C22—24no—
496C22—24no—
497C22#1684041112yes—
498C22C22×#1175820218yes—
499C22C22×#1175820212yes—
500C22#1684056512yes—
501C22C23×C22⋊D6111yescharacteristic
502C4—24no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—44no—
507C4—2442no—
508C4—2442no—
509C4—2444no—
510C4#1684041112yes—
511C4C22×#1175820214yes—
512C4C2×#41×D6111yescharacteristic
513C4C2×#1451982411yescharacteristic
514C2—28no—
515C2#2145820714yesdirect product factor
516C2C2×#1684041111yescharacteristic
517C2C23×#1175820211yescharacteristic
518C2C2×#1684056511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×#554D61
hypocenterC61C2×#3338
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. #21458207C2direct (N × H)
3. C2×#16840165C2semidirect
4. C2×#16840097C2semidirect
5. C2×#16840095C2semidirect
6. C2×#16840057C2semidirect
7. C2×#16840054C2semidirect
8. C2×#16840001C2semidirect
9. C2×C4×#11758202C2semidirect
11. C22×#14519710C2semidirect
24. #16840165C22semidirect
25. #16840097C22semidirect
26. #16840095C22semidirect
27. #16840057C22semidirect
28. #16840054C22semidirect
29. #16840001C22semidirect
30. C4×#11758202C22semidirect
37. C2×#14519710C22semidirect
132. C2×D4×C61Q16semidirect
199. D4×C61C2×Q16semidirect
268. C2×C61#3338semidirect
314. C61C2×#3338semidirect
398. C2×D4#11758202semidirect
463. D4C2×#11758202semidirect
515. C2#21458207direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404692
#214582072
C2×#168401652
C2×#168400972
C2×#168400952
C2×#168400572
C2×#168400542
C2×#168400012
C2×C4×#117582022
C2×D4×#924182
C22×#145197102
C2×C61×#5542
C2×#333861