Finite Group Databasebeta

← #29725097

Subgroups — #29725097

Order 15616.

646 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
1#135×Dic61C211yesmaximal, characteristic, complemented
2#21447272C212yesmaximal, complemented
3#21447122C212yesmaximal, complemented
4#21446718C211yesmaximal, characteristic, complemented
5#21446565C211yesmaximal, characteristic, complemented
6#21446322C212yesmaximal, complemented
7#21445944C211yesmaximal, characteristic, complemented
8#21445924C212yesmaximal, complemented
9#21445865C211yesmaximal, characteristic, complemented
10#21445654C211yesmaximal, characteristic, complemented
11C61×#3067C211yesmaximal, characteristic
12#41×Dic61C2212yes—
13Q16×Dic61—21no—
14SD16×Dic61—22no—
15D8×Dic61—21no—
16#16839800C2211yescharacteristic
17#16839800C2211yescharacteristic
18#16839790C2212yes—
19#16839779C2212yes—
20C2×#14519704C2211yescharacteristic
21#16839758C2212yes—
22#16839758C2211yescharacteristic
23#16839736C2212yes—
24#16839736C2212yes—
25#16839717C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839534C2212yes—
28#16839526C2211yescharacteristic
29#16839523C2211yescharacteristic
30#16839523C2212yes—
31#16839519C2212yes—
32#16839519C2212yes—
33C61×#575C2212yes—
34C2×#135×C61C2211yescharacteristic
35C2×#131×C61C2211yescharacteristic
36C61×#473—21no—
37C61×#472—21no—
38C61×#471—22no—
39C2×#108×C61C2211yescharacteristic
40C61×#413C2212yes—
41C2×#11758194C2311yescharacteristic
42C22×#92418C2311yescharacteristic
43Q8×Dic61—22no—
44D4×Dic61—22no—
45D4×Dic61—42no—
46#14519736—22no—
47#14519736C2311yescharacteristic
48#14519736C2312yes—
49#14519736C2312yes—
50C2×C4×Dic61—22no—
51C2×C4×Dic61C2311yescharacteristic
52#14519711—21no—
53#14519711—21no—
54#14519709—21no—
55#14519709—21no—
56#14519706—42no—
57#14519704—21no—
58#14519704—21no—
59#14519704—42no—
60#14519701—22no—
61#14519701—21no—
62#14519698—22no—
63#14519698—21no—
64#14519692—42no—
65#14519692—42no—
66#14519687—21no—
67C8×Dic61—21no—
68C8×Dic61—21no—
69C8×Dic61—21no—
70C2×#41×C61C2312yes—
71C2×#41×C61C2311yescharacteristic
72#137×C61—22no—
73#137×C61—22no—
74AGL(1,ℤ/8)×C61—22no—
75AGL(1,ℤ/8)×C61—22no—
76#135×C61—21no—
77#135×C61—21no—
78#135×C61—21no—
79#135×C61—21no—
80#135×C61—42no—
81C2×Q16×C61D411yescharacteristic
82C2×SD16×C61D412yes—
83C2×D8×C61D411yescharacteristic
84#131×C61—21no—
85#131×C61—21no—
86#131×C61—21no—
87#131×C61—21no—
88C2×M16×C61—21no—
89C2×M16×C61C2311yescharacteristic
90C2×M16×C61C2312yes—
91C22×C8×C61—21no—
92C22×C8×C61C2311yescharacteristic
93#108×C61—21no—
94#108×C61—21no—
95#101×C61—22no—
96#100×C61—22no—
97#41×C61—21no—
98#41×C61—21no—
99#41×C61—22no—
100#41×C61—22no—
101#41×C61—22no—
102#41×C61—21no—
103#41×C61—21no—
104#41×C61—22no—
105C2×Q8×C61C2×D412yes—
106C2×Q8×C61#4111yescharacteristic
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61C2×D412yes—
110C2×D4×C61#4111yescharacteristic
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—42no—
116Q16×C61—21no—
117Q16×C61—21no—
118Q16×C61—42no—
119SD16×C61—22no—
120SD16×C61—22no—
121SD16×C61—42no—
122SD16×C61—42no—
123D8×C61—21no—
124D8×C61—21no—
125D8×C61—42no—
126M16×C61—21no—
127M16×C61—22no—
128M16×C61—21no—
129M16×C61—22no—
130M16×C61—41no—
131M16×C61—41no—
132C2×C8×C61—21no—
133C2×C8×C61—22no—
134C2×C8×C61—41no—
135C2×C8×C61—41no—
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61#4111yescharacteristic
139C2×C8×C61#4111yescharacteristic
140C2×C8×C61#4111yescharacteristic
141C2×C8×C61#4111yescharacteristic
142C22⋊Dic61—22no—
143C22⋊Dic61—21no—
144C22⋊Dic61—22no—
145C22⋊Dic61—42no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—42no—
150C4×Dic61—21no—
151C4×Dic61—21no—
152C4×Dic61—22no—
153C4×Dic61—22no—
154C4×Dic61—42no—
155C4×Dic61#4111yescharacteristic
156C4×Dic61#4111yescharacteristic
157#11758194—21no—
158#11758194—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—21no—
163C23×C61—22no—
164C23×C61—21no—
165C23×C61H(5,𝔽2)11yescharacteristic
166Q8×C61—22no—
167Q8×C61—21no—
168Q8×C61—21no—
169Q8×C61—22no—
170D4×C61—22no—
171D4×C61—21no—
172D4×C61—21no—
173D4×C61—22no—
174D4×C61—41no—
175D4×C61—42no—
176D4×C61—41no—
177D4×C61—42no—
178C2×C4×C61—22no—
179C2×C4×C61—21no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61H(5,𝔽2)11yescharacteristic
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—41no—
196C2×Dic61—84no—
197C8×C61—41no—
198C8×C61—42no—
199C8×C61#12311yescharacteristic
200C8×C61#12311yescharacteristic
201C8×C61#12311yescharacteristic
202C8×C61#12311yescharacteristic
203#92418—41no—
204#92418—41no—
205#24985—611nomaximal, Sylow
206C22×C61—41no—
207C22×C61—41no—
208C22×C61—42no—
209C22×C61—82no—
210C22×C61—81no—
211C22×C61#54011yescharacteristic
212C22×C61#54611yescharacteristic
213C22×C61#55311yescharacteristic
214C4×C61—42no—
215C4×C61—41no—
216C4×C61C2×#12311yescharacteristic
217C4×C61#52911yescharacteristic
218C4×C61#54611yescharacteristic
219C4×C61#55511yescharacteristic
220Dic61—22no—
221Dic61—41no—
222Dic61—82no—
223Dic61—81no—
224Dic61—82no—
225#3067D6111yescharacteristic
226C4×#135—611no—
227#2080—612no—
228#2053—612no—
229#1976—611no—
230#1945—611no—
231#1906—612no—
232#1833—611no—
233#1827—612no—
234#1812—611no—
235#1764—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—82no—
240C2×C61—81no—
241C2×C61#242011yescharacteristic
242#575C2×D6112yes—
243C2×#135C2×D6111yescharacteristic
244C2×#131C2×D6111yescharacteristic
245C4×#41—612no—
246#473—21no—
247#472—21no—
248#471—22no—
249C4×Q16—1221no—
250C4×SD16—1222no—
251C4×D8—1221no—
252C2×#108C2×D6111yescharacteristic
253#427—611no—
254#427—611no—
255#424—612no—
256#421—612no—
257C2×#104—611no—
258#416—611no—
259#416—612no—
260#413C2×D6112yes—
261#410—612no—
262#410—612no—
263#405—611no—
264C2×C4×C8—611no—
265#344—612no—
266#341—611no—
267#339—611no—
268#339—612no—
269H(3,ℤ/4)—612no—
270H(3,ℤ/4)—612no—
271C61#2498511yescharacteristic, complemented, Sylow
272C2×#41C22×D6111yescharacteristic
273C2×#41C22×D6112yes—
274#137—22no—
275#137—22no—
276AGL(1,ℤ/8)—22no—
277AGL(1,ℤ/8)—22no—
278#135—21no—
279#135—21no—
280#135—21no—
281#135—21no—
282#135—42no—
283C2×Q16C22⋊D6111yescharacteristic
284C2×SD16C22⋊D6112yes—
285C2×D8C22⋊D6111yescharacteristic
286#131—21no—
287#131—21no—
288#131—21no—
289#131—21no—
290C2×M16—21no—
291C2×M16—611no—
292C2×M16C22×D6111yescharacteristic
293C2×M16C22×D6112yes—
294C22×C8—21no—
295C22×C8—611no—
296C22×C8C22×D6111yescharacteristic
297C4×Q8—1222no—
298C4×D4—1222no—
299C4×D4—2442no—
300#117—612no—
301#117—611no—
302#117—612no—
303#117—1222no—
304C2×C42—611no—
305C2×C42—1222no—
306#108—21no—
307#108—21no—
308#107—1221no—
309#107—1221no—
310#106—1221no—
311#106—1221no—
312C4⋊C8—2442no—
313#104—1221no—
314#104—1221no—
315#104—2442no—
316#103—1221no—
317#103—1222no—
318#102—1221no—
319#102—1222no—
320#101—22no—
321#100—22no—
322#99—2442no—
323#99—2442no—
324#97—1221no—
325C4×C8—1221no—
326C4×C8—1221no—
327C4×C8—1221no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—21no—
336C2×Q8#1175822411yescharacteristic
337C2×Q8C2×C22⋊D6112yes—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4#1175822411yescharacteristic
341C2×D4C2×C22⋊D6112yes—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—1221no—
345C22×C4—2442no—
346C22×C4C23×D6111yescharacteristic
347Q16—21no—
348Q16—21no—
349Q16—42no—
350SD16—22no—
351SD16—22no—
352SD16—42no—
353SD16—42no—
354D8—21no—
355D8—21no—
356D8—42no—
357M16—21no—
358M16—22no—
359M16—21no—
360M16—22no—
361M16—41no—
362M16—41no—
363M16—1221no—
364M16—1221no—
365C2×C8—21no—
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—1221no—
372C2×C8—1221no—
373C2×C8#1175822411yescharacteristic
374C2×C8#1175822411yescharacteristic
375C2×C8#1175822411yescharacteristic
376C2×C8#1175822411yescharacteristic
377C2×C8C2×C22⋊D6111yescharacteristic
378C2×C8C2×C22⋊D6111yescharacteristic
379C4⋊C4—1222no—
380C4⋊C4—1221no—
381C4⋊C4—1222no—
382C4⋊C4—2442no—
383C22⋊C4—1221no—
384C22⋊C4—1222no—
385C22⋊C4—1222no—
386C22⋊C4—2442no—
387C42—611no—
388C42—611no—
389C42—1221no—
390C42—1222no—
391C42—1221no—
392C42—1222no—
393C42—2442no—
394C23—21no—
395C23—22no—
396C23#1451982311yescharacteristic
397Q8—22no—
398Q8—21no—
399Q8—21no—
400Q8—22no—
401D4—22no—
402D4—21no—
403D4—21no—
404D4—22no—
405D4—42no—
406D4—41no—
407D4—42no—
408D4—41no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—1221no—
416C2×C4—1221no—
417C2×C4—4884no—
418C2×C4—2442no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4C22×C22⋊D6111yescharacteristic
423C2×C4#41×D6111yescharacteristic
424C2×C4C2×#1175822411yescharacteristic
425C2×C4C2×#1175822411yescharacteristic
426C2×C4C2×#1175822411yescharacteristic
427C2×C4#1451982211yescharacteristic
428C8—41no—
429C8—42no—
430C8—2441no—
431C8—2441no—
432C8#1451975811yescharacteristic
433C8#1451975811yescharacteristic
434C8#1451975811yescharacteristic
435C8#1451975811yescharacteristic
436C22—41no—
437C22—41no—
438C22—42no—
439C22—82no—
440C22—81no—
441C22#1684036511yescharacteristic
442C22#1684040011yescharacteristic
443C22#1684046411yescharacteristic
444C4—41no—
445C4—42no—
446C4—1222no—
447C4—4882no—
448C4—4882no—
449C4—4881no—
450C4—2441no—
451C4C2×#1451975811yescharacteristic
452C4#1684029311yescharacteristic
453C4#1684040611yescharacteristic
454C4#1684047611yescharacteristic
455C2—21no—
456C2—21no—
457C2—21no—
458C2—82no—
459C2—81no—
460C2#2144930711yescharacteristic
subgroupG/H
centerC4#16840476
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3067C2
socleC2×C61#2420
hypercenter#3067D61
hypocenterC61#24985
LCSG ▷ C22×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. #135×Dic61C2semidirect
2. #21447272C2semidirect
3. #21447122C2semidirect
4. #21446718C2semidirect
5. #21446565C2semidirect
6. #21446322C2semidirect
7. #21445944C2semidirect
8. #21445924C2semidirect
9. #21445865C2semidirect
10. #21445654C2semidirect
271. C61#24985semidirect

Maximal subgroups

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

subgroupindex
#135×Dic612
#214472722
#214471222
#214467182
#214465652
#214463222
#214459442
#214459242
#214458652
#214456542
C61×#30672
#2498561