Finite Group Databasebeta

← #29663967

Subgroups — #29663967

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×#16840109C211yesmaximal, characteristic, complemented
2#21455634C211yesmaximal, characteristic, complemented
3#21455299C211yesmaximal, characteristic
4#21445235C214yesmaximal, complemented
5#21445228C214yesmaximal, complemented
6#21444575C211yesmaximal, characteristic, complemented
7#21444499C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C8×#41×C61C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C2×#14519777C2211yescharacteristic
12C2×#14519774C2211yescharacteristic
13#16840109—21no—
14#16840109—21no—
15#16840109C2214yes—
16#16840057C2212yescomplemented
17#16840052C2212yescomplemented
18#16840047—21no—
19#16840042—21no—
20#16839962—21no—
21#16839962C2212yes—
22#16839957—21no—
23#16839957C2212yes—
24#16839805—21no—
25#16839786C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839766—21no—
28#16839763—21no—
29C2×#14519699—22no—
30#16839513C2214yescomplemented
31#16839510C2214yescomplemented
32#16839505C2214yes—
33C4×#41×C61C2211yescharacteristic
34C8×Q8×C61C2212yescomplemented
35C8×D4×C61—22no—
36C8×D4×C61C2212yescomplemented
37C61×#431—21no—
38C61×#431C2211yescharacteristic
39C2×C4×C8×C61—21no—
40C2×C4×C8×C61C2211yescharacteristic
41C22×Dic122—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45#14519802—44no—
46C2×#11758201—22no—
47C2×#11758199—22no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777C2312yes—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774C2312yes—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60#14519710—22no—
61#14519708—22no—
62C4⋊#92418—21no—
63C4⋊#92418—21no—
64C4⋊#92418C2312yes—
65C4⋊#92418C2312yes—
66#14519701—22no—
67#14519699—24no—
68#14519699—24no—
69#14519698—22no—
70#14519697—22no—
71#14519697—22no—
72C2×#41×C61D411yescharacteristic
73C22×C8×C61—22no—
74C22×C8×C61D412yes—
75C4×Q8×C61D411yescharacteristic
76C4×Q8×C61C2311yescharacteristic
77C4×D4×C61—21no—
78C4×D4×C61—21no—
79C4×D4×C61D411yescharacteristic
80C4×D4×C61C2311yescharacteristic
81#117×C61—21no—
82#117×C61D411yescharacteristic
83C2×C42×C61—21no—
84C2×C42×C61C2311yescharacteristic
85C4⋊C8×C61—22no—
86C4⋊C8×C61C2312yes—
87C22⋊C8×C61—21no—
88C22⋊C8×C61—22no—
89C4×C8×C61—22no—
90C4×C8×C61—21no—
91C4×C8×C61D411yescharacteristic
92C4×C8×C61D411yescharacteristic
93C4×C8×C61C2312yes—
94C23×D61—41no—
95#41×C61—22no—
96#41×C61SD1614yes—
97C2×Q8×C61C2×D411yescharacteristic
98C2×D4×C61—21no—
99C2×D4×C61C2×D411yescharacteristic
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C22×Dic61—41no—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—44no—
111C2×D244—42no—
112C2×D244—42no—
113C2×D244—41no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—82no—
117C2×Dic122—22no—
118C2×Dic122—41no—
119C2×Dic122—82no—
120C2×C8×C61—22no—
121C2×C8×C61—42no—
122C2×C8×C61—44no—
123C2×C8×C61C2×D412yes—
124C2×C8×C61C2×D412yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61C2×D411yescharacteristic
128C4⋊C4×C61C2×D411yescharacteristic
129C22⋊C4×C61—21no—
130C22⋊C4×C61—21no—
131C22⋊C4×C61—21no—
132C42×C61—21no—
133C42×C61—21no—
134C42×C61—21no—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61#4111yescharacteristic
138C42×C61C2411yescharacteristic
139#11758201—44no—
140#11758199—44no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—41no—
144C4⋊Dic61—41no—
145#11758194—42no—
146#11758194—42no—
147C2×#92418—22no—
148C2×#92418—22no—
149C23×C61—21no—
150C23×C61#12111yescharacteristic
151C22×D61—42no—
152C22×D61—82no—
153C22×D61—82no—
154C22×D61—82no—
155Q8×C61—21no—
156Q8×C61C2×SD1612yes—
157D4×C61—24no—
158D4×C61—21no—
159D4×C61C2×SD1612yes—
160C2×C4×C61—21no—
161C2×C4×C61—22no—
162C2×C4×C61—24no—
163C2×C4×C61—22no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C2×SD1612yes—
172C2×C4×C61#13512yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C2×Dic61—42no—
177C2×Dic61—82no—
178D244—42no—
179D244—42no—
180D244—44no—
181D244—82no—
182D244—82no—
183D244—82no—
184D244—82no—
185Dic122—82no—
186Dic122—82no—
187C8×C61—22no—
188C8×C61—22no—
189C8×C61—42no—
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#46512yes—
199C22×C61C2×#12111yescharacteristic
200C2×D61—82no—
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—164no—
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#3038—611no—
218C8×#41D6111yescharacteristic, complemented
219#1678—614no—
220#1677—614no—
221#1566—611no—
222#1548—611no—
223#1545—611no—
224C2×C61—22no—
225C2×C61—42no—
226C2×C61#304011yescharacteristic
227C2×C61C2×#46511yescharacteristic
228C2×C61C2×#47211yescharacteristic
229D61—162no—
230C2×AGL(1,ℤ/8)—1221no—
231C2×C4⋊Q8—611no—
232C2×#127—611no—
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—612no—
242#460—1221no—
243#459—612no—
244#459—1221no—
245C8×Q8C2×D6112yescomplemented
246C8×D4—22no—
247C8×D4C2×D6112yescomplemented
248#431—21no—
249#431C2×D6111yescharacteristic
250#428—1221no—
251#423—611no—
252#423—611no—
253#418—1221no—
254#417—1221no—
255C2×#102—1222no—
256C2×C4×C8—21no—
257C2×C4×C8C2×D6111yescharacteristic
258#334—614no—
259#333—614no—
260#331—614no—
261C61#1927411yescharacteristic, complemented, Sylow
262C2×#41D24411yescharacteristic
263C22×Q8—1221no—
264C22×D4—1221no—
265C22×D4—1221no—
266C22×D4—1221no—
267AGL(1,ℤ/8)—2444no—
268C2×SD16—1222no—
269C2×D8—1222no—
270C2×M16—1221no—
271C2×M16—1221no—
272C22×C8—22no—
273C22×C8C22⋊D6112yes—
274C4⋊Q8—612no—
275C4⋊Q8—1221no—
276C4⋊Q8—1221no—
277C4⋊Q8—1221no—
278#127—612no—
279#127—1221no—
280#127—1221no—
281#127—1221no—
282C4×Q8D24411yescharacteristic
283C4×Q8C22×D6111yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4D24411yescharacteristic
287C4×D4C22×D6111yescharacteristic
288#117—21no—
289#117D24411yescharacteristic
290C2×C4⋊C4—1221no—
291C2×C4⋊C4—1221no—
292C2×C42—21no—
293C2×C42C22×D6111yescharacteristic
294#107—1222no—
295#106—1222no—
296C4⋊C8—22no—
297C4⋊C8—612no—
298C4⋊C8—612no—
299C4⋊C8—1221no—
300C4⋊C8—1221no—
301C4⋊C8C22×D6112yes—
302#103—1222no—
303#102—1222no—
304#102—1224no—
305#102—1224no—
306#102—1222no—
307#102—1222no—
308C22⋊C8—21no—
309C22⋊C8—22no—
310C4×C8—22no—
311C4×C8—21no—
312C4×C8C22⋊D6111yescharacteristic
313C4×C8C22⋊D6111yescharacteristic
314C4×C8C22×D6112yes—
315C24—2441no—
316#41—22no—
317#41#1175819014yes—
318C2×Q8—1222no—
319C2×Q8—4882no—
320C2×Q8—2441no—
321C2×Q8C2×D24411yescharacteristic
322C2×D4—21no—
323C2×D4—1222no—
324C2×D4—1222no—
325C2×D4—1222no—
326C2×D4—4882no—
327C2×D4—2444no—
328C2×D4—2441no—
329C2×D4—2441no—
330C2×D4—2441no—
331C2×D4—2442no—
332C2×D4—2442no—
333C2×D4C2×D24411yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—2441no—
338C22×C4C2×D24411yescharacteristic
339C22×C4#1175822211yescharacteristic
340C22×C4C2×C22⋊D6111yescharacteristic
341SD16—2444no—
342D8—2444no—
343M16—2442no—
344M16—2442no—
345C2×C8—22no—
346C2×C8—42no—
347C2×C8—44no—
348C2×C8—1222no—
349C2×C8—1222no—
350C2×C8C2×C22⋊D6112yes—
351C2×C8C2×C22⋊D6112yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—1222no—
355C4⋊C4—1222no—
356C4⋊C4—2441no—
357C4⋊C4—2441no—
358C4⋊C4C2×D24411yescharacteristic
359C4⋊C4C2×D24411yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42C2×D24411yescharacteristic
367C42#1175822211yescharacteristic
368C42C2×C22⋊D6111yescharacteristic
369C42C23×D6111yescharacteristic
370C23—21no—
371C23—4882no—
372C23—4882no—
373C23—4882no—
374C23—2442no—
375C23#1451975111yescharacteristic
376Q8—21no—
377Q8—4882no—
378Q8—4882no—
379Q8C2×#1175819012yes—
380D4—24no—
381D4—21no—
382D4—4882no—
383D4—4882no—
384D4—4882no—
385D4—4882no—
386D4—2442no—
387D4—2442no—
388D4—2444no—
389D4C2×#1175819012yes—
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#1451975111yescharacteristic
401C2×C4#1451975111yescharacteristic
402C2×C4#1451975111yescharacteristic
403C2×C4C2×#1175819012yes—
404C2×C4#1451979912yes—
405C2×C4C22×C22⋊D6111yescharacteristic
406C2×C4C22×D24411yescharacteristic
407C2×C4C2×#1175822211yescharacteristic
408C8—22no—
409C8—22no—
410C8—42no—
411C8—2442no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—9764no—
419C22—4882no—
420C22—4882no—
421C22—4882no—
422C22#1683999112yes—
423C22C2×#1451975111yescharacteristic
424C4—22no—
425C4—21no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—4882no—
430C4#1683999112yes—
431C4#1684003212yes—
432C4C22×#1175819011yescharacteristic
433C4C2×#1451979911yescharacteristic
434C4#1684056012yes—
435C2—22no—
436C2—42no—
437C2—9762no—
438C2#2145531311yescharacteristic
439C2C2×#1683999111yescharacteristic
440C2C2×#1684003211yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×#41×C61C2
socleC22×C61C2×#121
prime radical#21455299C2
hypercenterC8×#41D61
hypocenterC61#19274
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840109C2semidirect
2. #21455634C2semidirect
4. #21445235C2semidirect
5. #21445228C2semidirect
6. #21444575C2semidirect
7. #21444499C2semidirect
8. #21444486C2semidirect
9. C8×#41×C61C2semidirect
16. #16840057C22semidirect
17. #16840052C22semidirect
30. #16839513C22semidirect
31. #16839510C22semidirect
34. C8×Q8×C61C22semidirect
36. C8×D4×C61C22semidirect
218. C8×#41D61semidirect
245. C8×Q8C2×D61semidirect
247. C8×D4C2×D61semidirect
261. C61#19274semidirect

Maximal subgroups

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

subgroupindex
C2×#168401092
#214556342
#214552992
#214452352
#214452282
#214445752
#214444992
#214444862
C8×#41×C612
#1927461