Finite Group Databasebeta

← #29665289

Subgroups — #29665289

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456762C211yesmaximal, characteristic
2#21455624C211yesmaximal, characteristic, complemented
3C2×#16839962C211yesmaximal, characteristic
4#21445235C214yesmaximal, complemented
5#21445167C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444486C211yesmaximal, characteristic
8#21444474C211yesmaximal, characteristic, complemented
9C61×#2982C211yesmaximal, characteristic, complemented
10C22×#11758201—21no—
11#16840319C2211yescharacteristic
12C2×#14519774C2211yescharacteristic
13#16840135C2212yes—
14#16840109C2212yes—
15#16840069—22no—
16#16840057C2212yescomplemented
17#16840047C2212yescomplemented
18#16839963—22no—
19#16839962—22no—
20#16839962C2214yes—
21#16839915—22no—
22#16839802—21no—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839771—21no—
26C2×#14519697—21no—
27#16839513C2214yescomplemented
28#16839510C2212yescomplemented
29#16839509C2212yescomplemented
30#16839505C2214yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C8×Q8×C61—22no—
33C8×Q8×C61C2214yescomplemented
34C61×#431—22no—
35C61×#431C2211yescharacteristic
36C61×#431C2211yescharacteristic
37C22×D244—21no—
38C22×D244—21no—
39C22×D244—21no—
40C2×#11758190—24no—
41C2×#11758201—24no—
42C2×#11758201—44no—
43C2×#11758194—21no—
44C22×#92418—21no—
45#14519777C2311yescharacteristic
46#14519777C2311yescharacteristic
47#14519774—21no—
48#14519774—21no—
49#14519774—21no—
50#14519774C2312yes—
51#14519769—21no—
52#14519756—21no—
53#14519756—21no—
54C4×Dic122—21no—
55C4×Dic122—21no—
56#14519736—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519701—22no—
64#14519701—22no—
65#14519699—24no—
66#14519697—24no—
67C22×Q8×C61D411yescharacteristic
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61D412yes—
71C4×Q8×C61C2312yes—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2311yescharacteristic
76C4⋊C8×C61—24no—
77C4⋊C8×C61D412yes—
78C4⋊C8×C61C2312yes—
79C22⋊C8×C61—21no—
80C22⋊C8×C61D412yes—
81C4×C8×C61—24no—
82C4×C8×C61—21no—
83C4×C8×C61C2312yes—
84C23×D61—41no—
85C2×Q8×C61—21no—
86C2×Q8×C61—22no—
87C2×Q8×C61SD1614yescomplemented
88C2×Q8×C61C2×D412yes—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×D244—22no—
95C2×D244—22no—
96C2×D244—22no—
97C2×D244—42no—
98C2×D244—42no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244—41no—
102C2×D244—82no—
103C2×D244—82no—
104C2×Dic122—21no—
105C2×Dic122—21no—
106C2×C8×C61—22no—
107C2×C8×C61C2×D414yes—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61C2×D412yes—
112C42×C61—22no—
113C42×C61—22no—
114C42×C61C2×D411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61#4111yescharacteristic
117C42×C61C2411yescharacteristic
118C22⋊Dic61—41no—
119#11758201—48no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124#11758196—41no—
125#11758196—41no—
126C4×Dic61—41no—
127#11758194—42no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418—42no—
131#11758190—84no—
132C23×C61#12111yescharacteristic
133C22×D61—42no—
134C22×D61—82no—
135C22×D61—82no—
136C22×D61—82no—
137Q8×C61—22no—
138Q8×C61—24no—
139Q8×C61C2×SD1614yes—
140C2×C4×C61—22no—
141C2×C4×C61—24no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—41no—
145C2×C4×C61—81no—
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61C2×SD1612yes—
150C2×C4×C61#13512yes—
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C2×C4×C61C2×#4111yescharacteristic
154C2×Dic61—41no—
155C2×Dic61—41no—
156D244—42no—
157D244—42no—
158D244—44no—
159D244—82no—
160D244—82no—
161D244—82no—
162D244—82no—
163Dic122—42no—
164Dic122—81no—
165C8×C61—24no—
166C8×C61—42no—
167#92418—42no—
168#92418—42no—
169#19396—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#46512yes—
173C22×C61C2×#12111yescharacteristic
174C2×D61—82no—
175C2×D61—82no—
176C2×D61—82no—
177C2×D61—164no—
178C4×C61—21no—
179C4×C61—24no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—42no—
183C4×C61#46512yes—
184C4×C61C22×SD1611yescharacteristic
185C4×C61C2×#13511yescharacteristic
186C4×C61#57712yes—
187Dic61—81no—
188Dic61—81no—
189#3208—611no—
190#3078—611no—
191C2×#460—611no—
192#2982D6111yescharacteristic, complemented
193#1678—614no—
194#1666—614no—
195#1570—611no—
196#1545—611no—
197#1542—611no—
198C2×C61—22no—
199C2×C61#304311yescharacteristic
200C2×C61C2×#46511yescharacteristic
201C2×C61#306811yescharacteristic
202D61—162no—
203C22×SD16—1221no—
204#533—611no—
205C2×#127—611no—
206C2×C4×Q8C2×D6111yescharacteristic
207#492—612no—
208#486—612no—
209#479—1222no—
210#477—612no—
211#475—612no—
212#460—614no—
213#460—1222no—
214#460—1222no—
215#450—1222no—
216C8×Q8—22no—
217C8×Q8C2×D6114yescomplemented
218#431—22no—
219#431C2×D6111yescharacteristic
220#431C2×D6111yescharacteristic
221#428—1221no—
222#423—611no—
223C2×C4⋊C8—611no—
224#419—1221no—
225C2×#102—1221no—
226#334—614no—
227#333—612no—
228#332—612no—
229#331—614no—
230C61#1939611yescharacteristic, complemented, Sylow
231C22×Q8D24411yescharacteristic
232C22×D4—1221no—
233C22×D4—1221no—
234C22×D4—1221no—
235C2×SD16—1224no—
236C2×SD16—1224no—
237C2×SD16—2444no—
238C2×M16—1221no—
239C22×C8—1221no—
240C4⋊Q8—611no—
241C4⋊Q8—611no—
242#127—612no—
243#127—1221no—
244#127—1221no—
245#127—1221no—
246#125—1221no—
247C22⋊Q8—1221no—
248C22⋊Q8—1221no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—1221no—
252C4×Q8—1221no—
253C4×Q8D24412yes—
254C4×Q8C22×D6112yes—
255#117—1221no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D24411yescharacteristic
258C2×C42—21no—
259C2×C42C22×D6111yescharacteristic
260#107—1222no—
261#106—1222no—
262C4⋊C8—24no—
263C4⋊C8—612no—
264C4⋊C8—612no—
265C4⋊C8—1221no—
266C4⋊C8—1221no—
267C4⋊C8C22⋊D6112yes—
268C4⋊C8C22×D6112yes—
269#103—1222no—
270#103—1222no—
271#102—1224no—
272#102—1224no—
273C22⋊C8—21no—
274C22⋊C8C22⋊D6112yes—
275C4×C8—24no—
276C4×C8—21no—
277C4×C8C22×D6112yes—
278C24—2441no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—1221no—
282C2×Q8—1221no—
283C2×Q8#1175819014yescomplemented
284C2×Q8C2×D24412yes—
285C2×D4—1222no—
286C2×D4—1222no—
287C2×D4—1222no—
288C2×D4—4882no—
289C2×D4—4882no—
290C2×D4—2441no—
291C2×D4—2441no—
292C2×D4—2441no—
293C2×D4—2442no—
294C2×D4—2442no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4C2×D24411yescharacteristic
298C22×C4#1175822211yescharacteristic
299C22×C4C2×C22⋊D6111yescharacteristic
300SD16—4884no—
301SD16—2448no—
302M16—2442no—
303C2×C8—22no—
304C2×C8—1222no—
305C2×C8—1222no—
306C2×C8—2442no—
307C2×C8C2×C22⋊D6114yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—1221no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4C2×D24412yes—
318C22⋊C4—2441no—
319C42—22no—
320C42—22no—
321C42—2441no—
322C42C2×D24411yescharacteristic
323C42#1175822211yescharacteristic
324C42C2×C22⋊D6111yescharacteristic
325C42C23×D6111yescharacteristic
326C23—4882no—
327C23—4882no—
328C23—4882no—
329C23—2442no—
330C23#1451975111yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—4881no—
334Q8—2442no—
335Q8C2×#1175819014yes—
336D4—4882no—
337D4—4882no—
338D4—4882no—
339D4—4882no—
340D4—2442no—
341D4—2442no—
342D4—2444no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4#1451975111yescharacteristic
352C2×C4#1451975111yescharacteristic
353C2×C4#1451975111yescharacteristic
354C2×C4C2×#1175819012yes—
355C2×C4#1451979912yes—
356C2×C4C22×C22⋊D6111yescharacteristic
357C2×C4C22×D24411yescharacteristic
358C2×C4C2×#1175822211yescharacteristic
359C8—24no—
360C8—42no—
361C8—2442no—
362C8—2442no—
363C22—21no—
364C22—21no—
365C22—9764no—
366C22—4882no—
367C22—4882no—
368C22—4882no—
369C22#1683999112yes—
370C22C2×#1451975111yescharacteristic
371C4—24no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—4881no—
377C4—4881no—
378C4#1683999112yes—
379C4C22×#1175819011yescharacteristic
380C4C2×#1451979911yescharacteristic
381C4#1684057012yes—
382C2—22no—
383C2—9762no—
384C2#2145534011yescharacteristic
385C2C2×#1683999111yescharacteristic
386C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2982C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#2982D61
hypocenterC61#19396
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #21455624C2semidirect
4. #21445235C2semidirect
5. #21445167C2semidirect
6. #21444590C2semidirect
8. #21444474C2semidirect
9. C61×#2982C2semidirect
16. #16840057C22semidirect
17. #16840047C22semidirect
27. #16839513C22semidirect
28. #16839510C22semidirect
29. #16839509C22semidirect
33. C8×Q8×C61C22semidirect
87. C2×Q8×C61SD16semidirect
192. #2982D61semidirect
217. C8×Q8C2×D61semidirect
230. C61#19396semidirect
283. C2×Q8#11758190semidirect

Maximal subgroups

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

subgroupindex
#214567622
#214556242
C2×#168399622
#214452352
#214451672
#214445902
#214444862
#214444742
C61×#29822
#1939661