Finite Group Databasebeta

← C2×#21445010 #29655917

Subgroups — C2×#21445010 #29655917

Order 15616.

1290 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840434C211yesmaximal, characteristic
2C2×#16840110C211yesmaximal, characteristic, complemented
3C2×#16840039C211yesmaximal, characteristic, complemented
4#21445010C218yesmaximal, direct product factor
5C2×#16839505C211yesmaximal, characteristic, complemented
6C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
7C2×#16839489C211yesmaximal, characteristic, complemented
8C2×C61×#462C211yesmaximal, characteristic, complemented
9C22×#11758226—21no—
10#16840434—22no—
11#16840434—22no—
12#16840434—22no—
13#16840434C2214yes—
14C2×#14519774—21no—
15C2×#14519774C2211yescharacteristic
16C2×#14519752—21no—
17C2×#14519752—22no—
18C2×C4×D244—21no—
19C2×C4×D244C2211yescharacteristic
20#16840110—22no—
21#16840110C2214yescomplemented
22#16840039—22no—
23#16840039C2214yescomplemented
24C2×#14519710—21no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519697—21no—
27C2×#14519697—22no—
28C2×C4×#92418C2211yescharacteristic
29#16839505C2214yescomplemented
30Q8⋊#92418C2214yescomplemented
31#16839489C2214yescomplemented
32C22×Q16×C61—21no—
33C2×C4⋊Q8×C61C2211yescharacteristic
34C2×C4×Q8×C61C2211yescharacteristic
35C61×#462—22no—
36C61×#462C2214yescomplemented
37C2×C4⋊C8×C61C2211yescharacteristic
38C2×#103×C61—21no—
39C2×#11758226—24no—
40C2×#11758226—24no—
41C2×#11758226—44no—
42C2×#11758226—44no—
43C22×D244—22no—
44C22×D244—21no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244D411yescharacteristic
48C22×D244D412yes—
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C22×#92418—21no—
52C22×#92418—21no—
53#14519774—24no—
54#14519774—24no—
55#14519774—21no—
56#14519774—21no—
57#14519774D412yes—
58#14519774C2312yes—
59#14519752—24no—
60#14519752—28no—
61#14519752—28no—
62#14519752—24no—
63C4×D244—24no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244—24no—
67C4×D244D412yes—
68C4×D244C2312yes—
69C2×C4⋊Dic61D411yescharacteristic
70C2×#11758198—22no—
71C2×#11758198—21no—
72#14519710—24no—
73C4⋊#92418—21no—
74C4⋊#92418C2312yes—
75#14519697—24no—
76#14519697—28no—
77C4×#92418—21no—
78C4×#92418C2312yes—
79C22×Q8×C61—21no—
80C22×Q8×C61D411yescharacteristic
81C2×Q16×C61—24no—
82C2×Q16×C61—44no—
83C22×C8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2312yes—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61D412yes—
91C4×Q8×C61C2312yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C42×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—21no—
99C4⋊C8×C61C2312yes—
100#103×C61—24no—
101C23×D61—21no—
102C23×D61—22no—
103C2×Q8×C61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—41no—
107C2×Q8×C61—42no—
108C2×Q8×C61—42no—
109C2×Q8×C61D814yes—
110C2×Q8×C61C2×D412yes—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×Dic61—21no—
118#11758226—48no—
119#11758226—416no—
120#11758226—48no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—28no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—21no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—28no—
130C2×D244—22no—
131C2×D244—48no—
132C2×D244—42no—
133C2×D244—42no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244—44no—
137C2×D244—84no—
138C2×D244C2×D412yes—
139C2×D244C2×D414yes—
140C2×C4×D61—28no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—48no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146Q16×C61—48no—
147C2×C8×C61—22no—
148C2×C8×C61—42no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—41no—
155C4⋊C4×C61—41no—
156C4⋊C4×C61C2×D412yes—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2411yescharacteristic
163#11758198—28no—
164#11758198—22no—
165#11758198—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61C2×D412yes—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—42no—
171C2×#92418—42no—
172C23×C61#12011yescharacteristic
173C22×D61—22no—
174C22×D61—24no—
175C22×D61—44no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—44no—
179C22×D61—44no—
180C22×D61—42no—
181Q8×C61—22no—
182Q8×C61—24no—
183Q8×C61—42no—
184Q8×C61—42no—
185Q8×C61—82no—
186Q8×C61C2×D814yes—
187C2×C4×C61—22no—
188C2×C4×C61—24no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—82no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61C2×D812yes—
198C2×C4×C61#13512yes—
199C2×C4×C61AGL(1,ℤ/8)12yes—
200C2×C4×C61#13712yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×Dic61—22no—
205C2×Dic61—42no—
206D244—22no—
207D244—22no—
208D244—44no—
209D244—416no—
210D244—42no—
211D244—42no—
212D244—42no—
213D244—42no—
214D244—44no—
215D244—42no—
216D244—44no—
217D244—82no—
218D244—88no—
219C4×D61—416no—
220C4×D61—42no—
221C4×D61—42no—
222C4×D61—44no—
223C8×C61—42no—
224#92418—42no—
225#92418—42no—
226C2×#1641—611nomaximal, Sylow
227C22×C61#44712yes—
228C22×C61#45212yes—
229C22×C61#45512yes—
230C22×C61C2×#12011yescharacteristic
231C2×D61—44no—
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—44no—
235C2×D61—44no—
236C2×D61—42no—
237C2×D61—84no—
238C2×D61—88no—
239C4×C61—24no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—42no—
243C4×C61—82no—
244C4×C61#44712yes—
245C4×C61#45212yes—
246C4×C61C22×D811yescharacteristic
247C4×C61C2×#13511yescharacteristic
248C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
249C4×C61C2×#13711yescharacteristic
250Dic61—42no—
251C2×#550—611no—
252C2×#486—611no—
253C2×#474—611no—
254C2×#462D6111yescharacteristic, complemented
255#1641—618no—
256C2×#331—611no—
257C2×Q8⋊C8—611no—
258C2×D4⋊C8—611no—
259C2×C61#164114yescomplemented
260C2×C61C2×#44711yescharacteristic
261C2×C61C2×#45211yescharacteristic
262C2×C61C2×#45511yescharacteristic
263D61—82no—
264D61—84no—
265C22×#41—1221no—
266C22×Q16—21no—
267#550—614no—
268#550—1222no—
269#550—1222no—
270#550—1222no—
271C2×C4⋊Q8C2×D6111yescharacteristic
272C2×#127—611no—
273C2×#127—1221no—
274C2×#121—1222no—
275C2×#121—1221no—
276C2×C4×Q8C2×D6111yescharacteristic
277C2×C4×D4—611no—
278C2×C4×D4—1221no—
279#486—614no—
280#486—1222no—
281#474—614no—
282#474—1222no—
283#462—22no—
284#462C2×D6114yescomplemented
285C2×#107—1221no—
286C2×C4⋊C8—611no—
287C2×C4⋊C8C2×D6111yescharacteristic
288C2×#103—21no—
289C2×#102—1222no—
290C2×#102—1221no—
291C2×C4×C8—611no—
292#331—614no—
293Q8⋊C8—614no—
294D4⋊C8—614no—
295C61C2×#164111yescharacteristic, complemented, Sylow
296C2×#41—1224no—
297C2×#41—1224no—
298C2×#41—2444no—
299C2×#41—2444no—
300C22×Q8—21no—
301C22×Q8C22⋊D6111yescharacteristic
302C22×D4—611no—
303C22×D4—612no—
304C22×D4—1222no—
305C22×D4—1222no—
306C22×D4—1221no—
307C22×D4—1221no—
308C23×C4—1222no—
309C23×C4—1221no—
310C2×Q16—24no—
311C2×Q16—44no—
312C22×C8—21no—
313C22×C8—1221no—
314C22×C8—1221no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8C22×D6112yes—
319#127—612no—
320#127—612no—
321#127—1224no—
322#127—1224no—
323#127—1221no—
324#127—1221no—
325#121—1228no—
326#121—1224no—
327#121—1228no—
328#121—1224no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8C22⋊D6112yes—
332C4×Q8C22×D6112yes—
333C4×D4—612no—
334C4×D4—612no—
335C4×D4—1224no—
336C4×D4—1224no—
337C4×D4—1221no—
338C4×D4—1221no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1221no—
346C2×C42—21no—
347C2×C42C22×D6111yescharacteristic
348#107—1224no—
349C4⋊C8—21no—
350C4⋊C8—612no—
351C4⋊C8—1221no—
352C4⋊C8C22×D6112yes—
353#103—24no—
354#102—1224no—
355#102—1228no—
356C4×C8—612no—
357C4×C8—1221no—
358C24—1221no—
359C24—1222no—
360#41—2448no—
361#41—24416no—
362#41—2448no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8#1175819914yes—
370C2×Q8C2×C22⋊D6112yes—
371C2×D4—612no—
372C2×D4—614no—
373C2×D4—1228no—
374C2×D4—1228no—
375C2×D4—1222no—
376C2×D4—1222no—
377C2×D4—1222no—
378C2×D4—1222no—
379C2×D4—1222no—
380C2×D4—1221no—
381C2×D4—1222no—
382C2×D4—1222no—
383C2×D4—4884no—
384C2×D4—2444no—
385C2×D4—2448no—
386C2×D4—2442no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4—2442no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—1228no—
394C22×C4—1221no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—2448no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4D4×D6111yescharacteristic
401C22×C4D4×D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403Q16—48no—
404C2×C8—22no—
405C2×C8—42no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—2442no—
409C2×C8—2442no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—612no—
418C4⋊C4—1221no—
419C4⋊C4C2×C22⋊D6112yes—
420C22⋊C4—1228no—
421C22⋊C4—1222no—
422C22⋊C4—1222no—
423C42—22no—
424C42—22no—
425C42D4×D6111yescharacteristic
426C42D4×D6111yescharacteristic
427C42C2×C22⋊D6111yescharacteristic
428C42C23×D6111yescharacteristic
429C23—1222no—
430C23—1224no—
431C23—2442no—
432C23—2444no—
433C23—2444no—
434C23—2442no—
435C23—2442no—
436C23—2444no—
437C23#1451974511yescharacteristic
438Q8—22no—
439Q8—24no—
440Q8—42no—
441Q8—42no—
442Q8—82no—
443Q8C2×#1175819914yes—
444D4—1222no—
445D4—1222no—
446D4—4882no—
447D4—4888no—
448D4—2444no—
449D4—24416no—
450D4—2442no—
451D4—2442no—
452D4—2442no—
453D4—2442no—
454D4—2444no—
455D4—2442no—
456D4—2444no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—82no—
464C2×C4—1222no—
465C2×C4—24416no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2444no—
470C2×C4#1451974511yescharacteristic
471C2×C4#1451974511yescharacteristic
472C2×C4#1451974511yescharacteristic
473C2×C4C2×#1175819912yes—
474C2×C4#1451980012yes—
475C2×C4#1451980212yes—
476C2×C4#1451980912yes—
477C2×C4C2×D4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4C22×C22⋊D6111yescharacteristic
480C8—42no—
481C8—2442no—
482C8—2442no—
483C22—4884no—
484C22—4888no—
485C22—2442no—
486C22—2444no—
487C22—2444no—
488C22—2442no—
489C22—2442no—
490C22—2444no—
491C22#1683989512yes—
492C22#1683992312yes—
493C22#1683993812yes—
494C22C2×#1451974511yescharacteristic
495C4—21no—
496C4—21no—
497C4—24no—
498C4—42no—
499C4—82no—
500C4—2442no—
501C4#1683989512yes—
502C4#1683992312yes—
503C4C22×#1175819911yescharacteristic
504C4C2×#1451980011yescharacteristic
505C4C2×#1451980211yescharacteristic
506C4C2×#1451980911yescharacteristic
507C2—4882no—
508C2—4884no—
509C2#2144501014yesdirect product factor
510C2C2×#1683989511yescharacteristic
511C2C2×#1683992311yescharacteristic
512C2C2×#1683993811yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#462C2
socleC23×C61#120
prime radicalC2×#16840434C2
hypercenterC2×#462D61
hypocenterC61C2×#1641
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
2. C2×#16840110C2semidirect
3. C2×#16840039C2semidirect
4. #21445010C2direct (N × H)
5. C2×#16839505C2semidirect
6. C2×Q8⋊#92418C2semidirect
7. C2×#16839489C2semidirect
8. C2×C61×#462C2semidirect
21. #16840110C22semidirect
23. #16840039C22semidirect
29. #16839505C22semidirect
30. Q8⋊#92418C22semidirect
31. #16839489C22semidirect
36. C61×#462C22semidirect
254. C2×#462D61semidirect
259. C2×C61#1641semidirect
284. #462C2×D61semidirect
295. C61C2×#1641semidirect
509. C2#21445010direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404342
C2×#168401102
C2×#168400392
#214450102
C2×#168395052
C2×Q8⋊#924182
C2×#168394892
C2×C61×#4622
C2×#164161