Finite Group Databasebeta

← #29665307

Subgroups — #29665307

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456739C211yesmaximal, characteristic
2#21455291C211yesmaximal, characteristic, complemented
3C2×#16839962C211yesmaximal, characteristic, complemented
4#21445263C214yesmaximal, complemented
5#21445179C214yesmaximal, complemented
6#21444594C211yesmaximal, characteristic, complemented
7#21444486C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic, complemented
9C61×#2982C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11C22×#11758201—21no—
12#16840312C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14#16840139C2212yes—
15#16840109C2212yes—
16#16839984—22no—
17#16839972C2212yescomplemented
18#16839962—22no—
19#16839962C2214yes—
20#16839962C2212yescomplemented
21#16839958—22no—
22#16839898—22no—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839761—21no—
26C2×#14519697—21no—
27#16839515C2214yescomplemented
28#16839507C2212yescomplemented
29#16839505C2214yes—
30#16839505C2212yescomplemented
31C2×C4×Q8×C61C2211yescharacteristic
32C8×Q8×C61—22no—
33C8×Q8×C61C2214yescomplemented
34C61×#431—22no—
35C61×#431C2211yescharacteristic
36C61×#431C2211yescharacteristic
37C2×#11758222—21no—
38C22×D244—21no—
39C22×D244—21no—
40C22×D244—21no—
41#14519808—44no—
42C2×#11758202—22no—
43C2×#11758201—24no—
44C2×#11758201—22no—
45C2×#11758201—44no—
46C2×D488—24no—
47C2×#11758194—21no—
48C22×#92418—21no—
49#14519777C2311yescharacteristic
50#14519774—21no—
51#14519774—21no—
52#14519774—21no—
53#14519774C2312yes—
54#14519774C2311yescharacteristic
55#14519765—21no—
56#14519751—21no—
57#14519751—21no—
58C4×D244—21no—
59C4×D244—21no—
60C4⋊#92418—21no—
61C4⋊#92418—21no—
62C4⋊#92418C2312yes—
63C4⋊#92418C2312yes—
64#14519700—22no—
65#14519699—24no—
66#14519697—24no—
67#14519697—22no—
68C22×Q8×C61D411yescharacteristic
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71C4×Q8×C61D412yes—
72C4×Q8×C61C2312yes—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C42×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—24no—
78C4⋊C8×C61D412yes—
79C4⋊C8×C61C2312yes—
80C22⋊C8×C61—21no—
81C22⋊C8×C61D412yes—
82C4×C8×C61—24no—
83C4×C8×C61—21no—
84C4×C8×C61C2312yes—
85C23×D61—41no—
86C2×Q8×C61—21no—
87C2×Q8×C61—22no—
88C2×Q8×C61D814yescomplemented
89C2×Q8×C61C2×D412yes—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C2×C22⋊D61—41no—
96#11758222—42no—
97#11758222—42no—
98C2×D244—22no—
99C2×D244—22no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—22no—
103C2×D244—21no—
104C2×D244—42no—
105C2×D244—42no—
106C2×D244—41no—
107C2×D244—41no—
108C2×D244—41no—
109C2×D244—82no—
110C2×D244—82no—
111C2×C4×D61—41no—
112C2×Dic122—21no—
113C2×C8×C61—22no—
114C2×C8×C61C2×D414yes—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61C2×D412yes—
119C42×C61—22no—
120C42×C61—22no—
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124C42×C61C2411yescharacteristic
125#11758202—44no—
126#11758201—44no—
127#11758201—48no—
128#11758198—41no—
129#11758198—41no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132#11758194—42no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418—42no—
136D488—84no—
137C23×C61#12111yescharacteristic
138C22×D61—41no—
139C22×D61—42no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143Q8×C61—22no—
144Q8×C61—24no—
145Q8×C61C2×D814yes—
146C2×C4×C61—22no—
147C2×C4×C61—24no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—81no—
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61C2×D812yes—
156C2×C4×C61#13512yes—
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C22⋊D61—82no—
161C2×Dic61—41no—
162D244—42no—
163D244—41no—
164D244—42no—
165D244—41no—
166D244—42no—
167D244—44no—
168D244—81no—
169D244—82no—
170D244—82no—
171D244—82no—
172D244—82no—
173C4×D61—82no—
174Dic122—41no—
175Dic122—41no—
176C8×C61—24no—
177C8×C61—42no—
178#92418—42no—
179#92418—42no—
180#19398—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61#46612yes—
184C22×C61C2×#12111yescharacteristic
185C2×D61—82no—
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—82no—
189C2×D61—82no—
190C2×D61—81no—
191C2×D61—164no—
192C4×C61—21no—
193C4×C61—24no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—42no—
197C4×C61#46612yes—
198C4×C61C22×D811yescharacteristic
199C4×C61C2×#13511yescharacteristic
200C4×C61#57712yes—
201Dic61—81no—
202#3205—611no—
203#3037—611no—
204C2×#460—611no—
205#2982D6111yescharacteristic, complemented
206#1682—614no—
207#1668—614no—
208#1571—611no—
209#1545—611no—
210#1540—611no—
211C2×C61—22no—
212C2×C61#304311yescharacteristic
213C2×C61C2×#46611yescharacteristic
214C2×C61#306811yescharacteristic
215D61—161no—
216D61—162no—
217C2×#137—1221no—
218C22×SD16—1221no—
219#532—611no—
220C2×#127—611no—
221C2×C4×Q8C2×D6111yescharacteristic
222#493—612no—
223#486—612no—
224#464—1222no—
225#462—612no—
226#460—614no—
227#460—612no—
228#460—1222no—
229#459—1222no—
230#447—1222no—
231C8×Q8—22no—
232C8×Q8C2×D6114yescomplemented
233#431—22no—
234#431C2×D6111yescharacteristic
235#431C2×D6111yescharacteristic
236#423—611no—
237C2×C4⋊C8—611no—
238#417—1221no—
239C2×#102—1221no—
240#335—614no—
241#332—612no—
242#331—612no—
243#331—614no—
244C61#1939811yescharacteristic, complemented, Sylow
245C2×#41—1221no—
246C22×Q8D24411yescharacteristic
247C22×D4—1221no—
248C22×D4—1221no—
249C22×D4—1221no—
250#137—2444no—
251C2×Q16—1222no—
252C2×SD16—1224no—
253C2×SD16—1222no—
254C2×SD16—2444no—
255C2×D8—1224no—
256C2×M16—1221no—
257C22×C8—1221no—
258C4⋊Q8—611no—
259#127—612no—
260#127—611no—
261#127—1221no—
262#127—1221no—
263#127—1221no—
264#124—1221no—
265#121—1221no—
266#121—1221no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8D24412yes—
270C4×Q8C22×D6112yes—
271C4×D4—1221no—
272C4×D4—1221no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4D24411yescharacteristic
275C2×C42—21no—
276C2×C42C22×D6111yescharacteristic
277C4⋊C8—24no—
278C4⋊C8—612no—
279C4⋊C8—612no—
280C4⋊C8—1221no—
281C4⋊C8—1221no—
282C4⋊C8C22⋊D6112yes—
283C4⋊C8C22×D6112yes—
284#103—1222no—
285#102—1224no—
286#102—1224no—
287#102—1222no—
288C22⋊C8—21no—
289C22⋊C8C22⋊D6112yes—
290C4×C8—24no—
291C4×C8—21no—
292C4×C8C22×D6112yes—
293C24—2441no—
294#41—2442no—
295#41—2442no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—1221no—
299C2×Q8D48814yescomplemented
300C2×Q8C2×D24412yes—
301C2×D4—1222no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—1222no—
305C2×D4—1221no—
306C2×D4—1222no—
307C2×D4—4882no—
308C2×D4—4882no—
309C2×D4—2441no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2442no—
314C2×D4—2442no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2441no—
318C22×C4C2×D24411yescharacteristic
319C22×C4#1175822211yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321Q16—2444no—
322SD16—2448no—
323SD16—2444no—
324D8—4884no—
325M16—2442no—
326C2×C8—22no—
327C2×C8—1222no—
328C2×C8—1222no—
329C2×C8—2442no—
330C2×C8C2×C22⋊D6114yes—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4C2×D24412yes—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—22no—
340C42—22no—
341C42C2×D24411yescharacteristic
342C42#1175822211yescharacteristic
343C42C2×C22⋊D6111yescharacteristic
344C42C23×D6111yescharacteristic
345C23—4882no—
346C23—4882no—
347C23—4882no—
348C23—2441no—
349C23—2442no—
350C23#1451975111yescharacteristic
351Q8—24no—
352Q8—22no—
353Q8—2441no—
354Q8—2441no—
355Q8C2×D48814yes—
356D4—4881no—
357D4—4882no—
358D4—4882no—
359D4—4882no—
360D4—4882no—
361D4—4882no—
362D4—2442no—
363D4—2441no—
364D4—2442no—
365D4—2441no—
366D4—2442no—
367D4—2444no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—4882no—
375C2×C4—2441no—
376C2×C4#1451975111yescharacteristic
377C2×C4#1451975111yescharacteristic
378C2×C4#1451975111yescharacteristic
379C2×C4C2×D48812yes—
380C2×C4#1451979912yes—
381C2×C4C22×C22⋊D6111yescharacteristic
382C2×C4C22×D24411yescharacteristic
383C2×C4C2×#1175822211yescharacteristic
384C8—24no—
385C8—42no—
386C8—2442no—
387C8—2442no—
388C22—21no—
389C22—21no—
390C22—9764no—
391C22—4881no—
392C22—4882no—
393C22—4882no—
394C22—4881no—
395C22—4882no—
396C22—4881no—
397C22#1683999812yes—
398C22C2×#1451975111yescharacteristic
399C4—24no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—4881no—
405C4#1683999812yes—
406C4C22×D48811yescharacteristic
407C4C2×#1451979911yescharacteristic
408C4#1684057012yes—
409C2—22no—
410C2—9762no—
411C2—9761no—
412C2#2145534011yescharacteristic
413C2C2×#1683999811yescharacteristic
414C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2982C2
socleC22×C61C2×#121
prime radical#21456739C2
hypercenter#2982D61
hypocenterC61#19398
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. #21455291C2semidirect
3. C2×#16839962C2semidirect
4. #21445263C2semidirect
5. #21445179C2semidirect
6. #21444594C2semidirect
7. #21444486C2semidirect
8. #21444466C2semidirect
9. C61×#2982C2semidirect
17. #16839972C22semidirect
20. #16839962C22semidirect
27. #16839515C22semidirect
28. #16839507C22semidirect
30. #16839505C22semidirect
33. C8×Q8×C61C22semidirect
88. C2×Q8×C61D8semidirect
205. #2982D61semidirect
232. C8×Q8C2×D61semidirect
244. C61#19398semidirect
299. C2×Q8D488semidirect

Maximal subgroups

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

subgroupindex
#214567392
#214552912
C2×#168399622
#214452632
#214451792
#214445942
#214444862
#214444662
C61×#29822
#1939861