Finite Group Databasebeta

← #29665308

Subgroups — #29665308

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#21455293C211yesmaximal, characteristic, complemented
3C2×#16839963C211yesmaximal, characteristic, complemented
4#21454899C211yesmaximal, characteristic, complemented
5#21445264C214yesmaximal, complemented
6#21445180C214yesmaximal, complemented
7#21444488C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic, complemented
9C61×#1571C211yesmaximal, characteristic, complemented
10C2×#14519810—21no—
11C22×#11758190—21no—
12#16840312C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15#16840139C2212yes—
16#16840109C2212yes—
17#16839985—22no—
18#16839973C2212yescomplemented
19#16839963—22no—
20#16839963C2214yes—
21#16839963C2212yescomplemented
22#16839958—22no—
23#16839898—22no—
24C8×Dic122C2214yescomplemented
25#16839886—22no—
26#16839815C2211yescharacteristic
27#16839812—22no—
28#16839764—21no—
29C2×#14519699—21no—
30#16839509C2212yescomplemented
31#16839505C2214yes—
32#16839505C2212yescomplemented
33C61×#431C2211yescharacteristic
34C61×#423C2211yescharacteristic
35C2×C4⋊C8×C61C2211yescharacteristic
36C61×#335C2214yescomplemented
37C2×#11758222—21no—
38C22×Dic122D411yescharacteristic
39C22×D244—21no—
40C22×D244—21no—
41C22×D244—21no—
42#14519810—44no—
43C2×Dic244—22no—
44C2×#11758190—24no—
45C2×#11758190—22no—
46C2×#11758190—44no—
47C2×D488—24no—
48#14519777C2311yescharacteristic
49#14519774—21no—
50#14519774—21no—
51#14519774—21no—
52#14519774C2312yes—
53#14519774C2311yescharacteristic
54#14519765—21no—
55#14519751—21no—
56#14519751—21no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59C4×Dic122D412yes—
60C4×Dic122C2312yes—
61C4×D244—21no—
62C4×D244—21no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×#11758196—21no—
65C2×C4×Dic61—21no—
66C4⋊#92418C2312yes—
67#14519706—24no—
68#14519702—22no—
69#14519699—24no—
70#14519699—24no—
71#14519699—22no—
72C22⋊#92418—21no—
73C4×#92418—21no—
74C8×Dic61—24no—
75C2×M16×C61—21no—
76C22×C8×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—21no—
79C4⋊C8×C61—21no—
80C4⋊C8×C61D412yes—
81C4⋊C8×C61C2312yes—
82C4⋊C8×C61C2312yes—
83C22⋊C8×C61D412yes—
84C4×C8×C61C2312yes—
85C23×D61—41no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C2×C22⋊D61—41no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92#11758222—42no—
93#11758222—42no—
94C2×D244—22no—
95C2×D244—22no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244—22no—
99C2×D244—21no—
100C2×D244—42no—
101C2×D244—42no—
102C2×D244—41no—
103C2×D244—41no—
104C2×D244—41no—
105C2×D244—82no—
106C2×D244—82no—
107C2×C4×D61—41no—
108C2×Dic122—21no—
109C2×Dic122—22no—
110C2×Dic122—21no—
111C2×Dic122D814yescomplemented
112C2×Dic122C2×D412yes—
113M16×C61—42no—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C2×C8×C61—42no—
117C2×C8×C61C2×D414yes—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61C2411yescharacteristic
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61C2×D412yes—
128#11758196—22no—
129#11758196—22no—
130C4×Dic61—22no—
131C4×Dic61—22no—
132C2×#92418—22no—
133Dic244—44no—
134D488—84no—
135#11758190—44no—
136#11758190—48no—
137C23×C61#12111yescharacteristic
138C22×D61—41no—
139C22×D61—42no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143C2×C4×C61—22no—
144C2×C4×C61—41no—
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61C2×D812yes—
149C2×C4×C61#13512yes—
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C22⋊D61—82no—
154C2×Dic61—22no—
155C2×Dic61—24no—
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—81no—
159D244—42no—
160D244—41no—
161D244—42no—
162D244—41no—
163D244—42no—
164D244—44no—
165D244—81no—
166D244—82no—
167D244—82no—
168D244—82no—
169D244—82no—
170C4×D61—82no—
171Dic122—22no—
172Dic122—24no—
173Dic122—41no—
174Dic122—41no—
175Dic122C2×D814yes—
176C8×C61—24no—
177C8×C61—42no—
178C8×C61—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—21no—
194C4×C61—21no—
195C4×C61#46612yes—
196C4×C61C22×D811yescharacteristic
197C4×C61C2×#13511yescharacteristic
198C4×C61#57712yes—
199Dic61—24no—
200Dic61—42no—
201Dic61—81no—
202#3205—611no—
203#3037—611no—
204C2×#460—611no—
205#2982—611no—
206#1682—614no—
207#1668—614no—
208#1571D6111yescharacteristic, complemented
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×Q8—611no—
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—614no—
232C8×Q8—1222no—
233#431—611no—
234#431—1222no—
235#431C2×D6111yescharacteristic
236#423C2×D6111yescharacteristic
237C2×C4⋊C8C2×D6111yescharacteristic
238#417—1221no—
239C2×#102—1221no—
240#335C2×D6114yescomplemented
241#332—612no—
242#331—612no—
243#331—614no—
244C61#1939811yescharacteristic, complemented, Sylow
245C2×#41—1221no—
246C22×Q8—611no—
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—21no—
257C22×C8—21no—
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—612no—
268C4×Q8—612no—
269C4×Q8—1221no—
270C4×Q8—1221no—
271C4×D4—1221no—
272C4×D4—1221no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—1221no—
275C2×C42—1221no—
276C2×C42C22×D6111yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—612no—
280C4⋊C8—1224no—
281C4⋊C8D24412yes—
282C4⋊C8C22×D6112yes—
283C4⋊C8C22×D6112yes—
284#103—1222no—
285#102—1224no—
286#102—1222no—
287#102—1224no—
288C22⋊C8—1221no—
289C22⋊C8D24412yes—
290C4×C8—1224no—
291C4×C8—1221no—
292C4×C8C22×D6112yes—
293C24—2441no—
294#41—2442no—
295#41—2442no—
296C2×Q8—612no—
297C2×Q8—614no—
298C2×Q8—1221no—
299C2×Q8—1222no—
300C2×Q8—1221no—
301C2×D4—1221no—
302C2×D4—1222no—
303C2×D4—1222no—
304C2×D4—1221no—
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—1221no—
316C22×C4—1221no—
317C22×C4—2441no—
318C22×C4C2×D24411yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4#1175822611yescharacteristic
321Q16—2444no—
322SD16—2448no—
323SD16—2444no—
324D8—4884no—
325M16—42no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—42no—
329C2×C8—1222no—
330C2×C8C2×D24414yes—
331C4⋊C4—612no—
332C4⋊C4—1222no—
333C4⋊C4—1221no—
334C4⋊C4—1222no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—1222no—
340C42—1222no—
341C42C2×D24411yescharacteristic
342C42D4×D6111yescharacteristic
343C42#1175822611yescharacteristic
344C42C23×D6111yescharacteristic
345C23—4882no—
346C23—4882no—
347C23—4882no—
348C23—2441no—
349C23—2442no—
350C23#1451975211yescharacteristic
351Q8—614no—
352Q8—1224no—
353Q8—1222no—
354Q8—2441no—
355Q8—2441no—
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—22no—
369C2×C4—41no—
370C2×C4—1224no—
371C2×C4—1222no—
372C2×C4—1222no—
373C2×C4—4882no—
374C2×C4—4881no—
375C2×C4—2441no—
376C2×C4#1451975211yescharacteristic
377C2×C4#1451975211yescharacteristic
378C2×C4#1451975211yescharacteristic
379C2×C4D8×D6112yes—
380C2×C4#1451980012yes—
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C22×D24411yescharacteristic
383C2×C4C2×#1175822611yescharacteristic
384C8—24no—
385C8—42no—
386C8—42no—
387C8—2442no—
388C22—21no—
389C22—21no—
390C22—9764no—
391C22—4881no—
392C22—4882no—
393C22—4882no—
394C22—4881no—
395C22—4882no—
396C22—4881no—
397C22#1683999912yes—
398C22C2×#1451975211yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—1224no—
403C4—4881no—
404C4—2442no—
405C4#1683999912yes—
406C4C2×D8×D6111yescharacteristic
407C4C2×#1451980011yescharacteristic
408C4#1684057412yes—
409C2—22no—
410C2—9762no—
411C2—9761no—
412C2#2145534111yescharacteristic
413C2C2×#1683999911yescharacteristic
414C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1571C2
socleC22×C61C2×#121
prime radical#21456739C2
hypercenter#1571D61
hypocenterC61#19398
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
2. #21455293C2semidirect
3. C2×#16839963C2semidirect
4. #21454899C2semidirect
5. #21445264C2semidirect
6. #21445180C2semidirect
7. #21444488C2semidirect
8. #21444469C2semidirect
9. C61×#1571C2semidirect
18. #16839973C22semidirect
21. #16839963C22semidirect
24. C8×Dic122C22semidirect
30. #16839509C22semidirect
32. #16839505C22semidirect
36. C61×#335C22semidirect
111. C2×Dic122D8semidirect
208. #1571D61semidirect
240. #335C2×D61semidirect
244. C61#19398semidirect

Maximal subgroups

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

subgroupindex
#214567392
#214552932
C2×#168399632
#214548992
#214452642
#214451802
#214444882
#214444692
C61×#15712
#1939861