Finite Group Databasebeta

← C2×C61×#3523 #29566220

Subgroups — C2×C61×#3523 #29566220

Order 15616.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3523C2132yesmaximal, direct product factor
2C2×C61×#557C213yesmaximal, complemented
3C2×C61×#557C211yesmaximal, characteristic, complemented
4C2×C61×#556C213yesmaximal, complemented
5C2×C61×#552C216yesmaximal, complemented
6C2×D4×Q8×C61C211yesmaximal, characteristic, complemented
7C2×D4×Q8×C61C212yesmaximal, complemented
8C2×C61×#541C216yesmaximal, complemented
9C2×C61×#533C213yesmaximal, complemented
10C2×C61×#529C213yesmaximal
11C2×C61×#519C212yesmaximal, complemented
12C22×C4×Q8×C61C211yesmaximal, characteristic, complemented
13C23×Q8×C61C2211yescharacteristic
14C23×Q8×C61C2212yes—
15C61×#557C22148yescomplemented
16C61×#557C22116yescomplemented
17C61×#556C22148yescomplemented
18C61×#552C22196yescomplemented
19D4×Q8×C61C22132yescomplemented
20D4×Q8×C61C22116yescomplemented
21C61×#541C22196yescomplemented
22C61×#533C22148yescomplemented
23C2×C4⋊Q8×C61C2213yes—
24C2×C4⋊Q8×C61C2213yes—
25C2×C4⋊Q8×C61C2216yes—
26C61×#529C22148yes—
27C2×#126×C61C2216yes—
28C2×#126×C61C22112yes—
29C2×#125×C61C2213yes—
30C2×#125×C61C2216yes—
31C2×#125×C61C2216yes—
32C2×#125×C61C22112yes—
33C2×#124×C61C2213yes—
34C2×#123×C61C2216yes—
35C2×C22⋊Q8×C61C2213yes—
36C2×C22⋊Q8×C61C2216yes—
37C2×C22⋊Q8×C61C2216yes—
38C2×C22⋊Q8×C61C2216yes—
39C2×#121×C61C2211yescharacteristic
40C61×#519C22132yescomplemented
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2213yes—
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2213yes—
45C2×C4×Q8×C61C2216yes—
46C2×C4×Q8×C61C2212yes—
47C2×C4×Q8×C61C2216yes—
48C2×C4×Q8×C61C2213yes—
49C2×C4×Q8×C61C2216yes—
50C2×C4×Q8×C61C22116yescomplemented
51C2×C4×D4×C61C2216yes—
52C2×C4×D4×C61C2213yes—
53C2×#117×C61C2216yes—
54C2×#117×C61C2216yes—
55C2×#117×C61C2216yes—
56C22×C4⋊C4×C61C2213yes—
57C22×C42×C61C2213yes—
58C22×Q8×C61—28no—
59C22×Q8×C61—216no—
60C22×Q8×C61C2316yes—
61C22×Q8×C61C2313yes—
62C22×Q8×C61C2313yes—
63C22×Q8×C61C2318yes—
64C22×Q8×C61C23116yes—
65C22×Q8×C61C2311yescharacteristic
66C22×D4×C61C2312yes—
67C22×D4×C61C2311yescharacteristic
68C23×C4×C61C2313yes—
69C23×C4×C61C2316yes—
70C23×C4×C61C2311yescharacteristic
71C23×C4×C61C2313yes—
72C4⋊Q8×C61C23124yes—
73C4⋊Q8×C61C23124yes—
74C4⋊Q8×C61C23148yes—
75#126×C61C23148yes—
76#126×C61C23196yes—
77#125×C61C23148yes—
78#125×C61C23196yes—
79#125×C61C23148yes—
80#125×C61C23124yes—
81#124×C61C23124yes—
82#123×C61C23148yes—
83C22⋊Q8×C61C23124yes—
84C22⋊Q8×C61C23148yes—
85C22⋊Q8×C61C23148yes—
86C22⋊Q8×C61C23148yes—
87#121×C61C2318yes—
88C4×Q8×C61C23148yes—
89C4×Q8×C61C23148yes—
90C4×Q8×C61C23124yes—
91C4×Q8×C61C2318yes—
92C4×Q8×C61C23116yes—
93C4×Q8×C61C23148yes—
94C4×Q8×C61C23124yes—
95C4×Q8×C61C23124yes—
96C4×Q8×C61C2318yes—
97C4×D4×C61C23148yes—
98C4×D4×C61C23124yes—
99#117×C61C23148yes—
100#117×C61C23148yes—
101#117×C61C23148yes—
102C2×C4⋊C4×C61C2313yes—
103C2×C4⋊C4×C61C2316yes—
104C2×C4⋊C4×C61C2316yes—
105C2×C4⋊C4×C61C2316yes—
106C2×C4⋊C4×C61C2316yes—
107C2×C4⋊C4×C61C23112yes—
108C2×C4⋊C4×C61C23124yes—
109C2×C4⋊C4×C61C2313yes—
110C2×C4⋊C4×C61C2313yes—
111C2×C4⋊C4×C61C2316yes—
112C2×C4⋊C4×C61C2316yes—
113C2×C4⋊C4×C61C2316yes—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2316yes—
116C2×C22⋊C4×C61C2312yes—
117C2×C22⋊C4×C61C2316yes—
118C2×C22⋊C4×C61C2316yes—
119C2×C22⋊C4×C61C2316yes—
120C2×C22⋊C4×C61C2316yes—
121C2×C42×C61C2313yes—
122C2×C42×C61C2316yes—
123C2×C42×C61C23124yes—
124C2×C42×C61C2313yes—
125C2×C42×C61C2313yes—
126C2×C42×C61C2313yes—
127C2×C42×C61C23112yes—
128C24×C61C2412yes—
129C24×C61C2411yescharacteristic
130C2×Q8×C61—232no—
131C2×Q8×C61—264no—
132C2×Q8×C61—212no—
133C2×Q8×C61—224no—
134C2×Q8×C61—212no—
135C2×Q8×C61#4118yes—
136C2×Q8×C61C24112yes—
137C2×Q8×C61C24124yes—
138C2×Q8×C61C24112yes—
139C2×Q8×C61C2414yes—
140C2×D4×C61—28no—
141C2×D4×C61—24no—
142C2×D4×C61C2418yes—
143C2×D4×C61C2414yes—
144C22×C4×C61—212no—
145C22×C4×C61—212no—
146C22×C4×C61—224no—
147C22×C4×C61—24no—
148C22×C4×C61C2414yes—
149C22×C4×C61C24112yes—
150C22×C4×C61C24112yes—
151C22×C4×C61C24124yes—
152C22×C4×C61C2411yescharacteristic
153C22×C4×C61C2413yes—
154C22×C4×C61C2416yes—
155C22×C4×C61C2412yes—
156C22×C4×C61C2413yes—
157C22×C4×C61C2413yes—
158C22×C4×C61C2416yes—
159C22×C4×C61C2413yes—
160C22×C4×C61C2411yescharacteristic
161C4⋊C4×C61C24112yes—
162C4⋊C4×C61C24112yes—
163C4⋊C4×C61C24124yes—
164C4⋊C4×C61C24124yes—
165C4⋊C4×C61C24124yes—
166C4⋊C4×C61C24124yes—
167C4⋊C4×C61C24124yes—
168C4⋊C4×C61C24124yes—
169C4⋊C4×C61C24112yes—
170C4⋊C4×C61C24124yes—
171C4⋊C4×C61C2414yes—
172C4⋊C4×C61C24148yes—
173C4⋊C4×C61C24124yes—
174C22⋊C4×C61C2418yes—
175C22⋊C4×C61C24124yes—
176C22⋊C4×C61C24124yes—
177C22⋊C4×C61C24124yes—
178C22⋊C4×C61C24124yes—
179C42×C61C24112yes—
180C42×C61C24112yes—
181C42×C61C24112yes—
182C42×C61C24112yes—
183C42×C61C24148yes—
184C42×C61C24124yes—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C612−1+414yes—
191C23×C61C2514yes—
192C23×C61C2512yes—
193C23×C61C2511yescharacteristic
194Q8×C61—224no—
195Q8×C61—248no—
196Q8×C61—224no—
197Q8×C61C2×#41116yes—
198D4×C61—216no—
199D4×C61—28no—
200C2×C4×C61—26no—
201C2×C4×C61—212no—
202C2×C4×C61—224no—
203C2×C4×C61—26no—
204C2×C4×C61—26no—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—212no—
208C2×C4×C61—224no—
209C2×C4×C61—248no—
210C2×C4×C61—28no—
211C2×C4×C61C2×#41112yes—
212C2×C4×C612−1+414yes—
213C2×C4×C61C2512yes—
214C2×C4×C61C2516yes—
215C2×C4×C61C25112yes—
216C2×C4×C61C2514yes—
217C2×C4×C61C2516yes—
218C2×C4×C61C2516yes—
219C2×C4×C61C25112yes—
220C2×C4×C61C2516yes—
221C2×C4×C61C2512yes—
222C2×#3523C6111yesmaximal, characteristic, direct product factor, Sylow
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—28no—
226C22×C61—24no—
227C22×C61—24no—
228C22×C61—24no—
229C22×C61—48no—
230C22×C61C22×#4112yes—
231C22×C61C2×2−1+414yes—
232C22×C61C2×2−1+412yes—
233C22×C61C2×2−1+412yes—
234C22×C61C2611yescharacteristic
235C4×C61—26no—
236C4×C61—212no—
237C4×C61—26no—
238C4×C61—26no—
239C4×C61—22no—
240C4×C61—24no—
241C4×C61—212no—
242C4×C61C22×#41112yes—
243C4×C61C2×2−1+414yes—
244#3523C2×C61132yesdirect product factor
245C2×#557C2×C6111yescharacteristic, complemented
246C2×#557C2×C6113yescomplemented
247C2×#556C2×C6113yescomplemented
248C2×#552C2×C6116yescomplemented
249C2×D4×Q8C2×C6112yescomplemented
250C2×D4×Q8C2×C6111yescharacteristic, complemented
251C2×#541C2×C6116yescomplemented
252C2×#533C2×C6113yescomplemented
253C2×#529C2×C6113yes—
254C2×#519C2×C6112yescomplemented
255C22×C4×Q8C2×C6111yescharacteristic, complemented
256C2×C61—24no—
257C2×C61—44no—
258C2×C61#352314yesdirect product factor
259C2×C61C23×#4111yescharacteristic
260C2×C61C22×2−1+411yescharacteristic
261C2×C61C22×2−1+411yescharacteristic
262C23×Q8C22×C6111yescharacteristic
263C23×Q8C22×C6112yes—
264#557C22×C61116yescomplemented
265#557C22×C61148yescomplemented
266#556C22×C61148yescomplemented
267#552C22×C61196yescomplemented
268D4×Q8C22×C61132yescomplemented
269D4×Q8C22×C61116yescomplemented
270#541C22×C61196yescomplemented
271#533C22×C61148yescomplemented
272C2×C4⋊Q8C22×C6113yes—
273C2×C4⋊Q8C22×C6113yes—
274C2×C4⋊Q8C22×C6116yes—
275#529C22×C61148yes—
276C2×#126C22×C6116yes—
277C2×#126C22×C61112yes—
278C2×#125C22×C6113yes—
279C2×#125C22×C6116yes—
280C2×#125C22×C61112yes—
281C2×#125C22×C6116yes—
282C2×#124C22×C6113yes—
283C2×#123C22×C6116yes—
284C2×C22⋊Q8C22×C6113yes—
285C2×C22⋊Q8C22×C6116yes—
286C2×C22⋊Q8C22×C6116yes—
287C2×C22⋊Q8C22×C6116yes—
288C2×#121C22×C6111yescharacteristic
289#519C22×C61132yescomplemented
290C2×C4×Q8C22×C6111yescharacteristic
291C2×C4×Q8C22×C6113yes—
292C2×C4×Q8C22×C6111yescharacteristic
293C2×C4×Q8C22×C6113yes—
294C2×C4×Q8C22×C6116yes—
295C2×C4×Q8C22×C6112yes—
296C2×C4×Q8C22×C6116yes—
297C2×C4×Q8C22×C6113yes—
298C2×C4×Q8C22×C6116yes—
299C2×C4×Q8C22×C61116yescomplemented
300C2×C4×D4C22×C6116yes—
301C2×C4×D4C22×C6113yes—
302C2×#117C22×C6116yes—
303C2×#117C22×C6116yes—
304C2×#117C22×C6116yes—
305C22×C4⋊C4C22×C6113yes—
306C22×C42C22×C6113yes—
307C61C2×#352311yescharacteristic, direct product factor, Sylow
308C22×Q8—28no—
309C22×Q8—216no—
310C22×Q8C23×C6113yes—
311C22×Q8C23×C6113yes—
312C22×Q8C23×C6116yes—
313C22×Q8C23×C6118yes—
314C22×Q8C23×C61116yes—
315C22×Q8C23×C6111yescharacteristic
316C22×D4C23×C6112yes—
317C22×D4C23×C6111yescharacteristic
318C23×C4C23×C6113yes—
319C23×C4C23×C6113yes—
320C23×C4C23×C6116yes—
321C23×C4C23×C6111yescharacteristic
322C4⋊Q8C23×C61124yes—
323C4⋊Q8C23×C61124yes—
324C4⋊Q8C23×C61148yes—
325#126C23×C61148yes—
326#126C23×C61196yes—
327#125C23×C61124yes—
328#125C23×C61148yes—
329#125C23×C61196yes—
330#125C23×C61148yes—
331#124C23×C61124yes—
332#123C23×C61148yes—
333C22⋊Q8C23×C61124yes—
334C22⋊Q8C23×C61148yes—
335C22⋊Q8C23×C61148yes—
336C22⋊Q8C23×C61148yes—
337#121C23×C6118yes—
338C4×Q8C23×C6118yes—
339C4×Q8C23×C61124yes—
340C4×Q8C23×C6118yes—
341C4×Q8C23×C61124yes—
342C4×Q8C23×C61148yes—
343C4×Q8C23×C61116yes—
344C4×Q8C23×C61148yes—
345C4×Q8C23×C61124yes—
346C4×Q8C23×C61148yes—
347C4×D4C23×C61148yes—
348C4×D4C23×C61124yes—
349#117C23×C61148yes—
350#117C23×C61148yes—
351#117C23×C61148yes—
352C2×C4⋊C4C23×C6113yes—
353C2×C4⋊C4C23×C6113yes—
354C2×C4⋊C4C23×C6116yes—
355C2×C4⋊C4C23×C61124yes—
356C2×C4⋊C4C23×C6116yes—
357C2×C4⋊C4C23×C6116yes—
358C2×C4⋊C4C23×C6116yes—
359C2×C4⋊C4C23×C6116yes—
360C2×C4⋊C4C23×C6116yes—
361C2×C4⋊C4C23×C6113yes—
362C2×C4⋊C4C23×C6116yes—
363C2×C4⋊C4C23×C6111yescharacteristic
364C2×C4⋊C4C23×C61112yes—
365C2×C4⋊C4C23×C6116yes—
366C2×C22⋊C4C23×C6112yes—
367C2×C22⋊C4C23×C6116yes—
368C2×C22⋊C4C23×C6116yes—
369C2×C22⋊C4C23×C6116yes—
370C2×C22⋊C4C23×C6116yes—
371C2×C42C23×C6113yes—
372C2×C42C23×C61112yes—
373C2×C42C23×C6116yes—
374C2×C42C23×C6113yes—
375C2×C42C23×C6113yes—
376C2×C42C23×C6113yes—
377C2×C42C23×C61124yes—
378C24C24×C6112yes—
379C24C24×C6111yescharacteristic
380C2×Q8—232no—
381C2×Q8—264no—
382C2×Q8—212no—
383C2×Q8—224no—
384C2×Q8—212no—
385C2×Q8#41×C6118yes—
386C2×Q8C24×C61112yes—
387C2×Q8C24×C61112yes—
388C2×Q8C24×C61124yes—
389C2×Q8C24×C6114yes—
390C2×D4—28no—
391C2×D4—24no—
392C2×D4C24×C6118yes—
393C2×D4C24×C6114yes—
394C22×C4—212no—
395C22×C4—212no—
396C22×C4—224no—
397C22×C4—24no—
398C22×C4C24×C6111yescharacteristic
399C22×C4C24×C6113yes—
400C22×C4C24×C6114yes—
401C22×C4C24×C6116yes—
402C22×C4C24×C6112yes—
403C22×C4C24×C6113yes—
404C22×C4C24×C6116yes—
405C22×C4C24×C6113yes—
406C22×C4C24×C61112yes—
407C22×C4C24×C61112yes—
408C22×C4C24×C61124yes—
409C22×C4C24×C6113yes—
410C22×C4C24×C6111yescharacteristic
411C4⋊C4C24×C61112yes—
412C4⋊C4C24×C61112yes—
413C4⋊C4C24×C61124yes—
414C4⋊C4C24×C61124yes—
415C4⋊C4C24×C61124yes—
416C4⋊C4C24×C61124yes—
417C4⋊C4C24×C61124yes—
418C4⋊C4C24×C61124yes—
419C4⋊C4C24×C61112yes—
420C4⋊C4C24×C61124yes—
421C4⋊C4C24×C6114yes—
422C4⋊C4C24×C61148yes—
423C4⋊C4C24×C61124yes—
424C22⋊C4C24×C6118yes—
425C22⋊C4C24×C61124yes—
426C22⋊C4C24×C61124yes—
427C22⋊C4C24×C61124yes—
428C22⋊C4C24×C61124yes—
429C42C24×C61112yes—
430C42C24×C61112yes—
431C42C24×C61112yes—
432C42C24×C61112yes—
433C42C24×C61148yes—
434C42C24×C61124yes—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C232−1+4×C6114yes—
441C23C25×C6114yes—
442C23C25×C6112yes—
443C23C25×C6111yescharacteristic
444Q8—224no—
445Q8—248no—
446Q8—224no—
447Q8C2×#41×C61116yes—
448D4—216no—
449D4—28no—
450C2×C4—26no—
451C2×C4—212no—
452C2×C4—224no—
453C2×C4—26no—
454C2×C4—26no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—212no—
458C2×C4—224no—
459C2×C4—248no—
460C2×C4—28no—
461C2×C4C2×#41×C61112yes—
462C2×C42−1+4×C6114yes—
463C2×C4C25×C6112yes—
464C2×C4C25×C6116yes—
465C2×C4C25×C61112yes—
466C2×C4C25×C6114yes—
467C2×C4C25×C6116yes—
468C2×C4C25×C6116yes—
469C2×C4C25×C61112yes—
470C2×C4C25×C6116yes—
471C2×C4C25×C6112yes—
472C22—22no—
473C22—22no—
474C22—28no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—48no—
479C22C22×#41×C6112yes—
480C22C2×2−1+4×C6114yes—
481C22C2×2−1+4×C6112yes—
482C22C2×2−1+4×C6112yes—
483C22C26×C6111yescharacteristic
484C4—26no—
485C4—212no—
486C4—26no—
487C4—26no—
488C4—22no—
489C4—24no—
490C4—212no—
491C4C22×#41×C61112yes—
492C4C2×2−1+4×C6114yes—
493C2—24no—
494C2—44no—
495C2C61×#352314yesdirect product factor
496C2C23×#41×C6111yescharacteristic
497C2C22×2−1+4×C6111yescharacteristic
498C2C22×2−1+4×C6111yescharacteristic
subgroupG/H
center, socleC23×C61C25
commutator, Frattini, cosocleC22C26×C61
prime radicalC2×#121×C61C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3523C2direct (N × H)
2. C2×C61×#557C2semidirect
3. C2×C61×#557C2semidirect
4. C2×C61×#556C2semidirect
5. C2×C61×#552C2semidirect
6. C2×D4×Q8×C61C2semidirect
7. C2×D4×Q8×C61C2semidirect
8. C2×C61×#541C2semidirect
9. C2×C61×#533C2semidirect
11. C2×C61×#519C2semidirect
12. C22×C4×Q8×C61C2semidirect
15. C61×#557C22semidirect
16. C61×#557C22semidirect
17. C61×#556C22semidirect
18. C61×#552C22semidirect
19. D4×Q8×C61C22semidirect
20. D4×Q8×C61C22semidirect
21. C61×#541C22semidirect
22. C61×#533C22semidirect
40. C61×#519C22semidirect
50. C2×C4×Q8×C61C22semidirect
222. C2×#3523C61direct (N × H)
244. #3523C2×C61direct (N × H)
245. C2×#557C2×C61semidirect
246. C2×#557C2×C61semidirect
247. C2×#556C2×C61semidirect
248. C2×#552C2×C61semidirect
249. C2×D4×Q8C2×C61semidirect
250. C2×D4×Q8C2×C61semidirect
251. C2×#541C2×C61semidirect
252. C2×#533C2×C61semidirect
254. C2×#519C2×C61semidirect
255. C22×C4×Q8C2×C61semidirect
258. C2×C61#3523direct (N × H)
264. #557C22×C61semidirect
265. #557C22×C61semidirect
266. #556C22×C61semidirect
267. #552C22×C61semidirect
268. D4×Q8C22×C61semidirect
269. D4×Q8C22×C61semidirect
270. #541C22×C61semidirect
271. #533C22×C61semidirect
289. #519C22×C61semidirect
299. C2×C4×Q8C22×C61semidirect
307. C61C2×#3523direct (N × H)
495. C2C61×#3523direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#35232
C2×C61×#5572
C2×C61×#5562
C2×C61×#5522
C2×D4×Q8×C612
C2×C61×#5412
C2×C61×#5332
C2×C61×#5292
C2×C61×#5192
C22×C4×Q8×C612
C2×#352361