Finite Group Databasebeta

← #29665304

Subgroups — #29665304

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#21456737C211yesmaximal, characteristic
2#21455291C211yesmaximal, characteristic, complemented
3#21454900C211yesmaximal, characteristic, complemented
4#21445261C214yesmaximal, complemented
5#21445177C214yesmaximal, complemented
6#21444593C211yesmaximal, characteristic, complemented
7#21444487C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic, complemented
9C2×C61×#460C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840138C2212yes—
13#16840108C2212yes—
14#16839984—22no—
15#16839972C2212yescomplemented
16#16839962C2212yescomplemented
17#16839954—22no—
18#16839896—22no—
19Q8×#92418—22no—
20Q8×#92418C2214yescomplemented
21#16839815—22no—
22#16839815C2211yescharacteristic
23#16839815C2211yescharacteristic
24#16839786C2211yescharacteristic
25#16839761—21no—
26#16839514C2214yescomplemented
27#16839509C2212yescomplemented
28#16839506C2214yes—
29#16839505C2212yescomplemented
30C22×SD16×C61—21no—
31C2×#127×C61C2211yescharacteristic
32C2×C4×Q8×C61C2211yescharacteristic
33C61×#460—22no—
34C61×#460C2214yes—
35C2×C4⋊C8×C61C2211yescharacteristic
36C2×#102×C61—21no—
37C2×#11758222—21no—
38#14519808—44no—
39C2×#11758202—22no—
40C2×#11758201—22no—
41C2×#11758199—24no—
42C2×#11758194—21no—
43#14519777C2311yescharacteristic
44#14519774C2311yescharacteristic
45#14519765—21no—
46#14519751—21no—
47#14519751—21no—
48C4×D244—21no—
49C4×D244—21no—
50C4⋊#92418—21no—
51C4⋊#92418—24no—
52C4⋊#92418D412yes—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519700—22no—
56#14519698—24no—
57#14519697—22no—
58C22⋊#92418—21no—
59C22⋊#92418D412yes—
60C4×#92418—21no—
61C4×#92418—24no—
62C4×#92418C2312yes—
63C22×Q8×C61D411yescharacteristic
64C22×D4×C61—21no—
65C22×D4×C61—21no—
66C22×D4×C61—21no—
67C2×SD16×C61—24no—
68C2×SD16×C61—44no—
69C22×C8×C61—21no—
70#127×C61—21no—
71#127×C61—21no—
72#127×C61—21no—
73#127×C61C2312yes—
74C4×Q8×C61—21no—
75C4×Q8×C61—21no—
76C4×Q8×C61D412yes—
77C4×Q8×C61C2312yes—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C42×C61—21no—
81C2×C42×C61C2311yescharacteristic
82C4⋊C8×C61—21no—
83C4⋊C8×C61C2312yes—
84#102×C61—24no—
85C24×C61—41no—
86C2×Q8×C61—21no—
87C2×Q8×C61—22no—
88C2×Q8×C61D814yescomplemented
89C2×Q8×C61C2×D412yes—
90C2×D4×C61—22no—
91C2×D4×C61—22no—
92C2×D4×C61—22no—
93C2×D4×C61—41no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C2×D4×C61—42no—
97C2×D4×C61—42no—
98C2×D4×C61—82no—
99C2×D4×C61—82no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—41no—
106#11758222—42no—
107#11758222—42no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×C4×D61—41no—
112C2×Dic122—21no—
113SD16×C61—48no—
114C2×C8×C61—22no—
115C2×C8×C61—42no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61C2×D412yes—
120C42×C61—22no—
121C42×C61—22no—
122C42×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61#4111yescharacteristic
125C42×C61C2411yescharacteristic
126#11758202—44no—
127#11758201—44no—
128#11758199—84no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133#11758194—42no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418C2×D414yes—
137C23×C61—42no—
138C23×C61—82no—
139C23×C61—82no—
140C23×C61—82no—
141C23×C61#12111yescharacteristic
142C22×D61—41no—
143Q8×C61—22no—
144Q8×C61—24no—
145Q8×C61C2×D814yes—
146D4×C61—42no—
147D4×C61—42no—
148D4×C61—44no—
149D4×C61—82no—
150D4×C61—82no—
151D4×C61—82no—
152D4×C61—82no—
153C2×C4×C61—22no—
154C2×C4×C61—24no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—81no—
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61C2×D812yes—
163C2×C4×C61#13512yes—
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C22⋊D61—82no—
168C2×Dic61—41no—
169D244—41no—
170D244—41no—
171D244—42no—
172D244—81no—
173C4×D61—82no—
174Dic122—41no—
175Dic122—41no—
176C8×C61—42no—
177#92418—24no—
178#92418—42no—
179#92418—42no—
180#19398—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—82no—
184C22×C61—82no—
185C22×C61—82no—
186C22×C61—164no—
187C22×C61#46612yes—
188C22×C61C2×#12111yescharacteristic
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—81no—
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×#460D6111yescharacteristic, complemented
205#2982—611no—
206#1682—614no—
207#1668—614no—
208#1571—611no—
209#1545—611no—
210#1540—611no—
211C2×C61—22no—
212C2×C61—162no—
213C2×C61#304311yescharacteristic
214C2×C61C2×#46611yescharacteristic
215C2×C61#306811yescharacteristic
216D61—161no—
217C2×#137—1221no—
218C22×SD16—21no—
219#532—611no—
220C2×#127C2×D6111yescharacteristic
221C2×C4×Q8C2×D6111yescharacteristic
222#493—612no—
223#486—612no—
224#464—1222no—
225#462—612no—
226#460—22no—
227#460—612no—
228#460C2×D6114yes—
229#459—1222no—
230#447—1222no—
231C8×Q8—614no—
232C8×Q8—1222no—
233#431—611no—
234#431—611no—
235#431—1222no—
236#423—611no—
237C2×C4⋊C8C2×D6111yescharacteristic
238#417—1221no—
239C2×#102—21no—
240#335—614no—
241#332—612no—
242#331—614no—
243#331—612no—
244C61#1939811yescharacteristic, complemented, Sylow
245C2×#41—1221no—
246C22×Q8C22⋊D6111yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250#137—2444no—
251C2×Q16—1222no—
252C2×SD16—24no—
253C2×SD16—44no—
254C2×SD16—1222no—
255C2×D8—1224no—
256C2×M16—1221no—
257C22×C8—21no—
258C4⋊Q8—611no—
259#127—21no—
260#127—21no—
261#127—21no—
262#127—611no—
263#127C22×D6112yes—
264#124—1221no—
265#121—1221no—
266#121—1221no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C22⋊D6112yes—
270C4×Q8C22×D6112yes—
271C4×D4—1221no—
272C4×D4—1221no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C22⋊D6111yescharacteristic
275C2×C42—21no—
276C2×C42C22×D6111yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—612no—
279C4⋊C8—612no—
280C4⋊C8—612no—
281C4⋊C8—1224no—
282C4⋊C8—1221no—
283C4⋊C8C22×D6112yes—
284#103—1222no—
285#102—24no—
286#102—1224no—
287#102—1222no—
288C22⋊C8—612no—
289C22⋊C8—1221no—
290C4×C8—612no—
291C4×C8—1224no—
292C4×C8—1221no—
293C24—41no—
294#41—2442no—
295#41—2442no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—1221no—
299C2×Q8#1175819914yescomplemented
300C2×Q8C2×C22⋊D6112yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—1221no—
314C2×D4—2441no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2441no—
318C22×C4D4×D6111yescharacteristic
319C22×C4#1175822411yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321Q16—2444no—
322SD16—48no—
323SD16—2444no—
324D8—4884no—
325M16—2442no—
326C2×C8—22no—
327C2×C8—42no—
328C2×C8—614no—
329C2×C8—1222no—
330C2×C8—1222no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4C2×C22⋊D6112yes—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—22no—
340C42—22no—
341C42D4×D6111yescharacteristic
342C42#1175822411yescharacteristic
343C42C2×C22⋊D6111yescharacteristic
344C42C23×D6111yescharacteristic
345C23—42no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—2441no—
350C23#1451974911yescharacteristic
351Q8—24no—
352Q8—22no—
353Q8—2441no—
354Q8—2441no—
355Q8C2×#1175819914yes—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—82no—
363D4—4881no—
364D4—4882no—
365D4—2441no—
366D4—2441no—
367D4—2442no—
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#1451974911yescharacteristic
377C2×C4#1451974911yescharacteristic
378C2×C4#1451974911yescharacteristic
379C2×C4C2×#1175819912yes—
380C2×C4#1451979712yes—
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C22×C22⋊D6111yescharacteristic
383C2×C4C2×#1175822411yescharacteristic
384C8—42no—
385C8—1224no—
386C8—2442no—
387C8—2442no—
388C22—21no—
389C22—21no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—164no—
394C22—4881no—
395C22—4881no—
396C22—4881no—
397C22#1683999612yes—
398C22C2×#1451974911yescharacteristic
399C4—24no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—4881no—
405C4#1683999612yes—
406C4C22×#1175819911yescharacteristic
407C4C2×#1451979711yescharacteristic
408C4#1684056812yes—
409C2—22no—
410C2—162no—
411C2—9761no—
412C2#2145533711yescharacteristic
413C2C2×#1683999611yescharacteristic
414C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#460C2
socleC22×C61C2×#121
prime radical#21456737C2
hypercenterC2×#460D61
hypocenterC61#19398
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #21455291C2semidirect
3. #21454900C2semidirect
4. #21445261C2semidirect
5. #21445177C2semidirect
6. #21444593C2semidirect
7. #21444487C2semidirect
8. #21444469C2semidirect
9. C2×C61×#460C2semidirect
15. #16839972C22semidirect
16. #16839962C22semidirect
20. Q8×#92418C22semidirect
26. #16839514C22semidirect
27. #16839509C22semidirect
29. #16839505C22semidirect
88. C2×Q8×C61D8semidirect
204. C2×#460D61semidirect
244. C61#19398semidirect
299. C2×Q8#11758199semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214552912
#214549002
#214452612
#214451772
#214445932
#214444872
#214444692
C2×C61×#4602
#1939861