Finite Group Databasebeta

← #29663965

Subgroups — #29663965

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840108C211yesmaximal, characteristic, complemented
2#21455634C211yesmaximal, characteristic, complemented
3#41×#92418C211yesmaximal, characteristic, complemented
4#21445233C214yesmaximal, complemented
5#21445226C214yesmaximal, complemented
6#21444574C211yesmaximal, characteristic, complemented
7#21444501C211yesmaximal, characteristic, complemented
8#21444487C211yesmaximal, characteristic, complemented
9C61×#3038C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840108—21no—
12#16840108—21no—
13#16840108C2214yes—
14#16840057C2212yescomplemented
15#16840052C2212yescomplemented
16#16840047—21no—
17#16840042—21no—
18Q8×#92418C2212yescomplemented
19D4×#92418—22no—
20D4×#92418C2212yescomplemented
21#16839815—21no—
22#16839815C2211yescharacteristic
23#16839805—21no—
24#16839786C2211yescharacteristic
25#16839763—21no—
26C2×#14519698—22no—
27C2×C4×#92418—21no—
28C2×C4×#92418C2211yescharacteristic
29#16839512C2214yescomplemented
30#16839510C2214yescomplemented
31#16839506C2214yes—
32C2×AGL(1,ℤ/8)×C61—21no—
33C2×#127×C61C2211yescharacteristic
34C4×#41×C61C2211yescharacteristic
35C61×#460—21no—
36C61×#460C2212yes—
37C61×#459—21no—
38C61×#459C2212yes—
39C61×#423C2211yescharacteristic
40C61×#418—21no—
41C22×Dic122—21no—
42C2×#11758194—21no—
43C22×#92418—22no—
44C22×#92418D412yes—
45#14519777—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777C2312yes—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51#14519710—22no—
52#14519708—22no—
53C4⋊#92418—21no—
54C4⋊#92418—22no—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57#14519701—22no—
58#14519698—22no—
59#14519698—24no—
60#14519698—24no—
61C22⋊#92418—21no—
62C22⋊#92418—22no—
63C4×#92418—21no—
64C4×#92418—22no—
65C4×#92418D411yescharacteristic
66C4×#92418D411yescharacteristic
67C4×#92418C2312yes—
68C2×#41×C61D411yescharacteristic
69C22×D4×C61—21no—
70C22×D4×C61—21no—
71C22×D4×C61—21no—
72AGL(1,ℤ/8)×C61—44no—
73C2×SD16×C61—22no—
74C2×D8×C61—22no—
75C2×M16×C61—21no—
76#127×C61—21no—
77#127×C61—21no—
78#127×C61—21no—
79#127×C61C2312yes—
80C4×Q8×C61D411yescharacteristic
81C4×Q8×C61C2311yescharacteristic
82C4×D4×C61—21no—
83C4×D4×C61—21no—
84C4×D4×C61D411yescharacteristic
85C4×D4×C61C2311yescharacteristic
86#117×C61—21no—
87#117×C61D411yescharacteristic
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61—21no—
91C4⋊C8×C61C2312yes—
92#102×C61—22no—
93#102×C61—22no—
94C24×C61—41no—
95#41×C61—22no—
96#41×C61SD1614yes—
97C2×Q8×C61C2×D411yescharacteristic
98C2×D4×C61—21no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—44no—
103C2×D4×C61—41no—
104C2×D4×C61—41no—
105C2×D4×C61—41no—
106C2×D4×C61—42no—
107C2×D4×C61—42no—
108C2×D4×C61—82no—
109C2×D4×C61C2×D411yescharacteristic
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×Dic61—41no—
117C2×Dic122—22no—
118C2×Dic122—41no—
119C2×Dic122—82no—
120SD16×C61—44no—
121D8×C61—44no—
122M16×C61—42no—
123C2×C8×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61C2×D411yescharacteristic
127C4⋊C4×C61C2×D411yescharacteristic
128C22⋊C4×C61—21no—
129C22⋊C4×C61—21no—
130C22⋊C4×C61—21no—
131C42×C61—21no—
132C42×C61—21no—
133C42×C61—21no—
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C42×C61C2411yescharacteristic
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140C4⋊Dic61—41no—
141C4⋊Dic61—41no—
142#11758194—42no—
143C2×#92418—22no—
144C2×#92418—22no—
145C2×#92418—42no—
146C2×#92418—44no—
147C2×#92418C2×D412yes—
148C2×#92418C2×D412yes—
149C23×C61—21no—
150C23×C61—42no—
151C23×C61—82no—
152C23×C61—82no—
153C23×C61—82no—
154C23×C61#12111yescharacteristic
155Q8×C61—21no—
156Q8×C61C2×SD1612yes—
157D4×C61—24no—
158D4×C61—21no—
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—44no—
162D4×C61—82no—
163D4×C61—82no—
164D4×C61—82no—
165D4×C61—82no—
166D4×C61C2×SD1612yes—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—24no—
170C2×C4×C61—22no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61C2×SD1612yes—
179C2×C4×C61#13512yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C2×Dic61—42no—
184C2×Dic61—82no—
185Dic122—82no—
186Dic122—82no—
187C8×C61—42no—
188#92418—22no—
189#92418—22no—
190#92418—42no—
191#92418—42no—
192#19274—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—22no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—82no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61—164no—
202C22×C61#46512yes—
203C22×C61C2×#12111yescharacteristic
204C4×C61—21no—
205C4×C61—22no—
206C4×C61—21no—
207C4×C61—41no—
208C4×C61—41no—
209C4×C61#46512yes—
210C4×C61#47212yes—
211C4×C61C22×SD1611yescharacteristic
212C4×C61C2×#13511yescharacteristic
213C4×C61#57512yes—
214Dic61—82no—
215C2×#486—611no—
216#3079—611no—
217#3038D6111yescharacteristic
218C8×#41—611no—
219#1678—614no—
220#1677—614no—
221#1566—611no—
222#1548—611no—
223#1545—611no—
224C2×C61—22no—
225C2×C61—42no—
226C2×C61—162no—
227C2×C61#304011yescharacteristic
228C2×C61C2×#46511yescharacteristic
229C2×C61C2×#47211yescharacteristic
230C2×AGL(1,ℤ/8)—21no—
231C2×C4⋊Q8—611no—
232C2×#127C2×D6111yescharacteristic
233C4×#41C2×D6111yescharacteristic
234#486—614no—
235#486—1221no—
236#486—1221no—
237#477—612no—
238#476—612no—
239#475—1221no—
240#474—1221no—
241#460—21no—
242#460C2×D6112yes—
243#459—21no—
244#459C2×D6112yes—
245C8×Q8—612no—
246C8×D4—612no—
247C8×D4—1222no—
248#431—611no—
249#431—1221no—
250#428—1221no—
251#423—611no—
252#423C2×D6111yescharacteristic
253#418—21no—
254#417—1221no—
255C2×#102—1222no—
256C2×C4×C8—611no—
257C2×C4×C8—1221no—
258#334—614no—
259#333—614no—
260#331—614no—
261C61#1927411yescharacteristic, complemented, Sylow
262C2×#41C22⋊D6111yescharacteristic
263C22×Q8—1221no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267AGL(1,ℤ/8)—44no—
268C2×SD16—22no—
269C2×D8—22no—
270C2×M16—21no—
271C2×M16—1221no—
272C22×C8—612no—
273C22×C8—1222no—
274C4⋊Q8—612no—
275C4⋊Q8—1221no—
276C4⋊Q8—1221no—
277C4⋊Q8—1221no—
278#127—21no—
279#127—21no—
280#127—21no—
281#127C22×D6112yes—
282C4×Q8C22⋊D6111yescharacteristic
283C4×Q8C22×D6111yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C22⋊D6111yescharacteristic
287C4×D4C22×D6111yescharacteristic
288#117—21no—
289#117C22⋊D6111yescharacteristic
290C2×C4⋊C4—1221no—
291C2×C4⋊C4—1221no—
292C2×C42—21no—
293C2×C42C22×D6111yescharacteristic
294#107—1222no—
295#106—1222no—
296C4⋊C8—21no—
297C4⋊C8—612no—
298C4⋊C8—612no—
299C4⋊C8—1222no—
300C4⋊C8—1221no—
301C4⋊C8C22×D6112yes—
302#103—1222no—
303#102—22no—
304#102—22no—
305#102—1224no—
306#102—1224no—
307#102—1222no—
308C22⋊C8—1222no—
309C22⋊C8—1221no—
310C4×C8—611no—
311C4×C8—611no—
312C4×C8—612no—
313C4×C8—1222no—
314C4×C8—1221no—
315C24—41no—
316#41—22no—
317#41#1175820014yes—
318C2×Q8—1222no—
319C2×Q8—4882no—
320C2×Q8—2441no—
321C2×Q8C2×C22⋊D6111yescharacteristic
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—44no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—82no—
333C2×D4C2×C22⋊D6111yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—2441no—
338C22×C4D4×D6111yescharacteristic
339C22×C4#1175822411yescharacteristic
340C22×C4C2×C22⋊D6111yescharacteristic
341SD16—44no—
342D8—44no—
343M16—42no—
344M16—2442no—
345C2×C8—22no—
346C2×C8—612no—
347C2×C8—612no—
348C2×C8—1222no—
349C2×C8—1222no—
350C2×C8—2442no—
351C2×C8—2444no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—1222no—
355C4⋊C4—1222no—
356C4⋊C4—2441no—
357C4⋊C4—2441no—
358C4⋊C4C2×C22⋊D6111yescharacteristic
359C4⋊C4C2×C22⋊D6111yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42D4×D6111yescharacteristic
367C42#1175822411yescharacteristic
368C42C2×C22⋊D6111yescharacteristic
369C42C23×D6111yescharacteristic
370C23—21no—
371C23—42no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23#1451974911yescharacteristic
376Q8—21no—
377Q8—4882no—
378Q8—4882no—
379Q8C2×#1175820012yes—
380D4—24no—
381D4—21no—
382D4—42no—
383D4—42no—
384D4—44no—
385D4—82no—
386D4—82no—
387D4—82no—
388D4—82no—
389D4C2×#1175820012yes—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—4882no—
399C2×C4—2442no—
400C2×C4#1451974911yescharacteristic
401C2×C4#1451974911yescharacteristic
402C2×C4#1451974911yescharacteristic
403C2×C4C2×#1175820012yes—
404C2×C4#1451979712yes—
405C2×C4C2×D4×D6111yescharacteristic
406C2×C4C22×C22⋊D6111yescharacteristic
407C2×C4C2×#1175822411yescharacteristic
408C8—42no—
409C8—1222no—
410C8—1222no—
411C8—2442no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—164no—
422C22#1683998912yes—
423C22C2×#1451974911yescharacteristic
424C4—22no—
425C4—21no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—4882no—
430C4#1683998912yes—
431C4#1684003012yes—
432C4C22×#1175820011yescharacteristic
433C4C2×#1451979711yescharacteristic
434C4#1684055812yes—
435C2—22no—
436C2—42no—
437C2—162no—
438C2#2145531111yescharacteristic
439C2C2×#1683998911yescharacteristic
440C2C2×#1684003011yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3038C2
socleC22×C61C2×#121
hypercenter#3038D61
hypocenterC61#19274
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. C2×#16840108C2semidirect
2. #21455634C2semidirect
3. #41×#92418C2semidirect
4. #21445233C2semidirect
5. #21445226C2semidirect
6. #21444574C2semidirect
7. #21444501C2semidirect
8. #21444487C2semidirect
14. #16840057C22semidirect
15. #16840052C22semidirect
18. Q8×#92418C22semidirect
20. D4×#92418C22semidirect
29. #16839512C22semidirect
30. #16839510C22semidirect
261. C61#19274semidirect

Maximal subgroups

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

subgroupindex
C2×#168401082
#214556342
#41×#924182
#214452332
#214452262
#214445742
#214445012
#214444872
C61×#30382
#1927461