Finite Group Databasebeta

← #29686953

Subgroups — #29686953

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454722C212yesmaximal, complemented
2C2×#16839879C211yesmaximal, characteristic, complemented
3#131×Dic61C211yesmaximal, characteristic, complemented
4#21446718C211yesmaximal, characteristic, complemented
5#21446610C211yesmaximal, characteristic, complemented
6#21446490C212yesmaximal, complemented
7#21446322C212yesmaximal, complemented
8#21445873C211yesmaximal, characteristic, complemented
9#21445701C211yesmaximal, characteristic, complemented
10#21445622C212yesmaximal, complemented
11C61×#3067C211yesmaximal, characteristic
12C2×#14519784C2212yes—
13C2×#14519784C2211yescharacteristic
14#41×Dic61C2211yescharacteristic
15#16839884—24no—
16#16839879—24no—
17#16839879—21no—
18#16839879—21no—
19#16839879—21no—
20#16839879—21no—
21#16839811C2212yes—
22C2×#14519713C2211yescharacteristic
23#16839800C2211yescharacteristic
24#16839779C2212yes—
25#16839779C2212yes—
26C2×#14519704C2212yes—
27#16839736C2212yes—
28#16839720—21no—
29#16839720C2212yes—
30#16839720C2211yescharacteristic
31#16839717—21no—
32#16839717C2211yescharacteristic
33M16×Dic61—21no—
34M16×Dic61C2211yescharacteristic
35C2×C8×Dic61—21no—
36C2×C8×Dic61C2211yescharacteristic
37#16839534C2212yes—
38#16839534C2212yes—
39#16839523C2212yes—
40C61×#575C2212yes—
41C2×#135×C61C2211yescharacteristic
42C2×#131×C61C2211yescharacteristic
43C61×#473C412yescomplemented
44C61×#472C412yescomplemented
45C61×#471C414yescomplemented
46C2×#108×C61C2211yescharacteristic
47C61×#413C2212yes—
48#14519784—21no—
49#14519784—21no—
50#14519784—22no—
51#14519784—22no—
52#14519784—22no—
53#14519784—22no—
54#14519784D412yes—
55#14519784D412yes—
56C2×#11758194—21no—
57C2×#11758194—22no—
58C2×#11758194C2312yes—
59C2×#11758194C2311yescharacteristic
60C22×#92418—21no—
61C22×#92418—22no—
62C22×#92418C2311yescharacteristic
63Q8×Dic61—21no—
64D4×Dic61—21no—
65D4×Dic61—22no—
66#14519736—21no—
67#14519736C2312yes—
68#14519736C2311yescharacteristic
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2311yescharacteristic
71#14519713—21no—
72#14519713—21no—
73#14519713—24no—
74#14519711—21no—
75#14519709—21no—
76#14519704—24no—
77#14519704—22no—
78#14519704—22no—
79#14519704—24no—
80#14519692—24no—
81#14519687—21no—
82#14519687—21no—
83#14519687—21no—
84#14519687—22no—
85C8×Dic61—21no—
86C8×Dic61—21no—
87C8×Dic61—21no—
88C8×Dic61—22no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C8×Dic61D411yescharacteristic
93C2×#41×C61C2312yes—
94C2×#41×C61C2311yescharacteristic
95#137×C61—22no—
96#137×C61C2×C414yes—
97AGL(1,ℤ/8)×C61—22no—
98AGL(1,ℤ/8)×C61C2×C414yes—
99#135×C61—24no—
100#135×C61—21no—
101#135×C61—21no—
102#135×C61—21no—
103#135×C61—21no—
104C2×Q16×C61C2×C411yescharacteristic, complemented
105C2×SD16×C61C2×C412yescomplemented
106C2×D8×C61C2×C411yescharacteristic, complemented
107#131×C61—21no—
108#131×C61—21no—
109#131×C61C2×C412yes—
110#131×C61C2×C412yes—
111C2×M16×C61—21no—
112C2×M16×C61C2311yescharacteristic
113C2×M16×C61C2312yes—
114C22×C8×C61—21no—
115C22×C8×C61C2311yescharacteristic
116#108×C61C2×C412yescomplemented
117#108×C61C2×C412yescomplemented
118#101×C61C2×C414yescomplemented
119#100×C61C2×C414yescomplemented
120#41×C61—21no—
121#41×C61—21no—
122#41×C61—22no—
123#41×C61—22no—
124#41×C61—22no—
125#41×C61—22no—
126#41×C61C22×C412yes—
127#41×C61C2×D412yes—
128C2×Q8×C61C22×C412yes—
129C2×Q8×C61#4111yescharacteristic
130C2×D4×C61—21no—
131C2×D4×C61—22no—
132C2×D4×C61C22×C412yes—
133C2×D4×C61#4111yescharacteristic
134C22×C4×C61—21no—
135C22×C4×C61—22no—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—22no—
138C22×Dic61—21no—
139Q16×C61—24no—
140Q16×C61—21no—
141Q16×C61—21no—
142SD16×C61—24no—
143SD16×C61—24no—
144SD16×C61—22no—
145SD16×C61—22no—
146D8×C61—24no—
147D8×C61—21no—
148D8×C61—21no—
149M16×C61—21no—
150M16×C61—22no—
151M16×C61—22no—
152M16×C61—22no—
153M16×C61C22×C412yes—
154M16×C61C22×C414yes—
155C2×C8×C61—21no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61C22×C411yescharacteristic
160C2×C8×C61C22×C411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61#4111yescharacteristic
164C2×C8×C61#4111yescharacteristic
165C22⋊Dic61—22no—
166C22⋊Dic61—22no—
167C22⋊Dic61#4112yes—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4×Dic61—22no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—22no—
175C4×Dic61C2×D411yescharacteristic
176C4×Dic61C2×D411yescharacteristic
177#11758194—21no—
178#11758194—22no—
179#11758194—22no—
180#11758194—22no—
181#11758194—22no—
182#11758194—42no—
183#11758194—42no—
184#11758194C2×D412yes—
185C2×#92418—21no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—42no—
190C2×#92418—42no—
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195C2×#92418#4111yescharacteristic
196C2×#92418#4111yescharacteristic
197C23×C61—21no—
198C23×C61—22no—
199C23×C61C2×#4111yescharacteristic
200Q8×C61—21no—
201Q8×C61—22no—
202Q8×C61—22no—
203Q8×C61C4×D412yes—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—22no—
207D4×C61—22no—
208D4×C61—21no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61C4×D412yes—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—22no—
226C2×Dic61—21no—
227C2×Dic61—42no—
228C2×Dic61—44no—
229C2×Dic61—42no—
230C8×C61—22no—
231C8×C61—24no—
232C8×C61C4×D411yescharacteristic
233C8×C61C4×D411yescharacteristic
234C8×C61C4×D411yescharacteristic
235C8×C61C4×D411yescharacteristic
236#92418—22no—
237#92418—42no—
238#92418#12111yescharacteristic
239#92418#12111yescharacteristic
240#92418#12111yescharacteristic
241#92418#12111yescharacteristic
242#21558—611nomaximal, Sylow
243C22×C61—22no—
244C22×C61—41no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—82no—
248C22×C61C4×#4111yescharacteristic
249C22×C61#52511yescharacteristic
250C22×C61#53211yescharacteristic
251C4×C61—22no—
252C4×C61—42no—
253C4×C61C2×C4×D411yescharacteristic
254C4×C61C2×C4×D411yescharacteristic
255C4×C61C2×#12111yescharacteristic
256C4×C61#52911yescharacteristic
257Dic61—22no—
258Dic61—42no—
259Dic61—41no—
260Dic61—82no—
261#3067D6111yescharacteristic
262#2955—612no—
263C2×#443—611no—
264C4×#131—611no—
265#1976—611no—
266#1953—611no—
267#1932—612no—
268#1906—612no—
269#1814—611no—
270#1772—611no—
271#1757—612no—
272C2×C61—21no—
273C2×C61—21no—
274C2×C61—21no—
275C2×C61—42no—
276C2×C61—82no—
277C2×C61C4×#12111yescharacteristic
278#575C2×D6112yes—
279C2×#135C2×D6111yescharacteristic
280C2×#131—611no—
281C2×#131—612no—
282C2×#131C2×D6111yescharacteristic
283C4×#41—611no—
284#473Dic6112yescomplemented
285#472Dic6112yescomplemented
286#471Dic6114yescomplemented
287#444—1224no—
288#443—1221no—
289#443—1221no—
290#443—1221no—
291#443—1221no—
292#443—1224no—
293#430—612no—
294C2×#108—611no—
295C2×#108C2×D6111yescharacteristic
296#427—611no—
297#421—612no—
298#421—612no—
299C2×#104—612no—
300#413C2×D6112yes—
301#410—612no—
302#405—611no—
303#405—612no—
304#405—611no—
305#405—1221no—
306#405—1221no—
307C4×M16—611no—
308C4×M16—1221no—
309C2×C4×C8—611no—
310C2×C4×C8—1221no—
311#344—612no—
312#344—612no—
313#339—612no—
314C61#2155811yescharacteristic, complemented, Sylow
315C2×#41C22×D6111yescharacteristic
316C2×#41C22×D6112yes—
317#137—22no—
318#137C2×Dic6114yes—
319AGL(1,ℤ/8)—22no—
320AGL(1,ℤ/8)C2×Dic6114yes—
321#135—24no—
322#135—21no—
323#135—21no—
324#135—21no—
325#135—21no—
326C2×Q16C2×Dic6111yescharacteristic, complemented
327C2×SD16C2×Dic6112yescomplemented
328C2×D8C2×Dic6111yescharacteristic, complemented
329#131—21no—
330#131—21no—
331#131—612no—
332#131—612no—
333#131—1221no—
334#131—1221no—
335#131—1222no—
336#131—1222no—
337#131—1222no—
338#131—1222no—
339#131C2×Dic6112yes—
340#131C2×Dic6112yes—
341C2×M16—21no—
342C2×M16—611no—
343C2×M16—612no—
344C2×M16—1221no—
345C2×M16—1222no—
346C2×M16C22×D6111yescharacteristic
347C2×M16C22×D6112yes—
348C22×C8—21no—
349C22×C8—611no—
350C22×C8—1221no—
351C22×C8—1222no—
352C22×C8C22×D6111yescharacteristic
353C4×Q8—1221no—
354C4×D4—1222no—
355C4×D4—1221no—
356#117—611no—
357#117—612no—
358#117—1221no—
359C2×C42—611no—
360C2×C42—1221no—
361#108—1224no—
362#108—1221no—
363#108—1221no—
364#108C2×Dic6112yescomplemented
365#108C2×Dic6112yescomplemented
366#107—1221no—
367#106—1221no—
368#104—1222no—
369#104—1222no—
370#104—1224no—
371#104—1224no—
372#101C2×Dic6114yescomplemented
373#100C2×Dic6114yescomplemented
374#99—1224no—
375#97—1221no—
376#97—1221no—
377#97—1222no—
378#97—1221no—
379C4×C8—611no—
380C4×C8—611no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8—1221no—
384C4×C8—1221no—
385C4×C8—1222no—
386C4×C8—1221no—
387#41—21no—
388#41—21no—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—22no—
393#41D4×D6112yes—
394#41C22×Dic6112yes—
395C2×Q8#1175822411yescharacteristic
396C2×Q8C22×Dic6112yes—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4#1175822411yescharacteristic
400C2×D4C22×Dic6112yes—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—1222no—
404C22×C4—1221no—
405C22×C4C23×D6111yescharacteristic
406Q16—24no—
407Q16—21no—
408Q16—21no—
409SD16—24no—
410SD16—24no—
411SD16—22no—
412SD16—22no—
413D8—24no—
414D8—21no—
415D8—21no—
416M16—21no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—612no—
421M16—1222no—
422M16—1222no—
423M16—1221no—
424M16—1222no—
425M16—1222no—
426M16—2442no—
427M16—2442no—
428M16C22×Dic6112yes—
429M16C22×Dic6114yes—
430C2×C8—21no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—611no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—611no—
438C2×C8—611no—
439C2×C8—611no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1221no—
443C2×C8—1222no—
444C2×C8—2442no—
445C2×C8—2442no—
446C2×C8D4×D6111yescharacteristic
447C2×C8D4×D6111yescharacteristic
448C2×C8#1175822411yescharacteristic
449C2×C8#1175822411yescharacteristic
450C2×C8C22×Dic6111yescharacteristic
451C2×C8C22×Dic6111yescharacteristic
452C4⋊C4—1222no—
453C4⋊C4—1221no—
454C4⋊C4—1222no—
455C22⋊C4—612no—
456C22⋊C4—1222no—
457C22⋊C4—1222no—
458C42—611no—
459C42—611no—
460C42—1222no—
461C42—1221no—
462C42—1221no—
463C42—1222no—
464C23—21no—
465C23—22no—
466C23#41×D6111yescharacteristic
467Q8—21no—
468Q8—22no—
469Q8—22no—
470Q8D4×Dic6112yes—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—22no—
475D4—21no—
476D4—42no—
477D4—42no—
478D4D4×Dic6112yes—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—1222no—
488C2×C4—2442no—
489C2×C4—2444no—
490C2×C4—2442no—
491C2×C4C23×Dic6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4C2×#1175822411yescharacteristic
496C2×C4C2×#1175822411yescharacteristic
497C8—22no—
498C8—24no—
499C8—611no—
500C8—611no—
501C8—611no—
502C8—611no—
503C8—1222no—
504C8—2442no—
505C8D4×Dic6111yescharacteristic
506C8D4×Dic6111yescharacteristic
507C8D4×Dic6111yescharacteristic
508C8D4×Dic6111yescharacteristic
509C22—22no—
510C22—41no—
511C22—42no—
512C22—42no—
513C22—82no—
514C22#41×Dic6111yescharacteristic
515C22#1684027211yescharacteristic
516C22#1684030711yescharacteristic
517C4—22no—
518C4—42no—
519C4—1222no—
520C4—4882no—
521C4—2441no—
522C4—2442no—
523C4C2×D4×Dic6111yescharacteristic
524C4C2×D4×Dic6111yescharacteristic
525C4#121×D6111yescharacteristic
526C4#1684029311yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—82no—
532C2#121×Dic6111yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3067C2
socleC2×C61C4×#121
cosocleC22×C4×C61C24
hypercenter#3067D61
hypocenterC61#21558
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #21454722C2semidirect
2. C2×#16839879C2semidirect
3. #131×Dic61C2semidirect
4. #21446718C2semidirect
5. #21446610C2semidirect
6. #21446490C2semidirect
7. #21446322C2semidirect
8. #21445873C2semidirect
9. #21445701C2semidirect
10. #21445622C2semidirect
43. C61×#473C4semidirect
44. C61×#472C4semidirect
45. C61×#471C4semidirect
104. C2×Q16×C61C2×C4semidirect
105. C2×SD16×C61C2×C4semidirect
106. C2×D8×C61C2×C4semidirect
116. #108×C61C2×C4semidirect
117. #108×C61C2×C4semidirect
118. #101×C61C2×C4semidirect
119. #100×C61C2×C4semidirect
284. #473Dic61semidirect
285. #472Dic61semidirect
286. #471Dic61semidirect
314. C61#21558semidirect
326. C2×Q16C2×Dic61semidirect
327. C2×SD16C2×Dic61semidirect
328. C2×D8C2×Dic61semidirect
364. #108C2×Dic61semidirect
365. #108C2×Dic61semidirect
372. #101C2×Dic61semidirect
373. #100C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214547222
C2×#168398792
#131×Dic612
#214467182
#214466102
#214464902
#214463222
#214458732
#214457012
#214456222
C61×#30672
#2155861