Finite Group Databasebeta

← #29665303

Subgroups — #29665303

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#21455287C211yesmaximal, characteristic, complemented
2C2×#16839959C211yesmaximal, characteristic, complemented
3#21454899C211yesmaximal, characteristic, complemented
4#21445260C214yesmaximal, complemented
5#21445176C214yesmaximal, complemented
6#21444594C211yesmaximal, characteristic, complemented
7#21444487C211yesmaximal, characteristic, complemented
8#21444468C211yesmaximal, characteristic, complemented
9C61×#3205C211yesmaximal, characteristic
10C2×#14519807—21no—
11C22×#11758200—21no—
12C2×C4×Dic122C2211yescharacteristic
13#16839981—22no—
14#16839969C2212yescomplemented
15#16839959—22no—
16#16839959C2214yes—
17#16839959C2212yescomplemented
18C8×Dic122C2214yescomplemented
19#16839886—22no—
20#16839815C2211yescharacteristic
21#16839812—22no—
22#16839786C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24#16839763—21no—
25C2×#14519698—21no—
26#16839515C2214yescomplemented
27#16839508C2212yescomplemented
28#16839506C2214yes—
29#16839506C2212yescomplemented
30C61×#532C2211yescharacteristic
31C2×#127×C61C2211yescharacteristic
32C61×#493C2212yes—
33C61×#486C2212yes—
34C61×#459—22no—
35C61×#447—22no—
36C61×#431C2211yescharacteristic
37C22×Dic122D411yescharacteristic
38#14519807—44no—
39C2×#11758202—22no—
40C2×#11758200—24no—
41C2×#11758200—22no—
42C2×#11758200—44no—
43C2×#11758194—21no—
44C22×#92418—21no—
45C4×Dic122—21no—
46C4×Dic122—21no—
47C4×Dic122D412yes—
48C4×Dic122C2312yes—
49C2×C4⋊Dic61D411yescharacteristic
50C2×#11758196—21no—
51C2×C4×Dic61—21no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418C2312yes—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57#14519706—24no—
58#14519701—22no—
59#14519698—24no—
60#14519698—22no—
61C22⋊#92418—21no—
62C4×#92418—21no—
63C8×Dic61—24no—
64C2×#41×C61—21no—
65C22×D4×C61—21no—
66C22×D4×C61—21no—
67C22×D4×C61—21no—
68C2×D8×C61—24no—
69C4⋊Q8×C61C2311yescharacteristic
70#127×C61—21no—
71#127×C61—21no—
72#127×C61—21no—
73#127×C61C2312yes—
74#127×C61C2311yescharacteristic
75#124×C61—21no—
76#121×C61—21no—
77#121×C61—21no—
78C4×D4×C61—21no—
79C4×D4×C61—21no—
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61D412yes—
82#102×C61—24no—
83C22⋊C8×C61D412yes—
84C4×C8×C61C2312yes—
85C24×C61—41no—
86#41×C61—42no—
87#41×C61—42no—
88C2×Q8×C61—21no—
89C2×D4×C61—22no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—22no—
93C2×D4×C61—21no—
94C2×D4×C61—22no—
95C2×D4×C61—41no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—42no—
100C2×D4×C61—42no—
101C2×D4×C61—82no—
102C2×D4×C61—82no—
103C22×C4×C61—41no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—21no—
109C2×Dic122—21no—
110C2×Dic122—22no—
111C2×Dic122D814yescomplemented
112C2×Dic122C2×D412yes—
113D8×C61—84no—
114C2×C8×C61C2×D414yes—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—21no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122C42×C61C2411yescharacteristic
123#11758202—44no—
124#11758200—48no—
125#11758200—44no—
126C4⋊Dic61—21no—
127C4⋊Dic61C2×D412yes—
128#11758196—22no—
129#11758196—22no—
130C4×Dic61—22no—
131C4×Dic61—22no—
132#11758194—42no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418—42no—
137C23×C61—41no—
138C23×C61—42no—
139C23×C61—82no—
140C23×C61—82no—
141C23×C61—82no—
142C23×C61#12111yescharacteristic
143Q8×C61—41no—
144Q8×C61—41no—
145D4×C61—42no—
146D4×C61—41no—
147D4×C61—42no—
148D4×C61—41no—
149D4×C61—42no—
150D4×C61—44no—
151D4×C61—81no—
152D4×C61—82no—
153D4×C61—82no—
154D4×C61—82no—
155D4×C61—82no—
156D4×C61—82no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—82no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61C2×D812yes—
165C2×C4×C61#13512yes—
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C2×Dic61—22no—
170C2×Dic61—24no—
171C2×Dic61—22no—
172C2×Dic61—81no—
173Dic122—22no—
174Dic122—24no—
175Dic122C2×D814yes—
176C8×C61—24no—
177#92418—42no—
178#92418—42no—
179#92418—42no—
180#19398—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—82no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—82no—
187C22×C61—82no—
188C22×C61—81no—
189C22×C61—164no—
190C22×C61#46612yes—
191C22×C61C2×#12111yescharacteristic
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—81no—
196C4×C61#46612yes—
197C4×C61C22×D811yescharacteristic
198C4×C61C2×#13511yescharacteristic
199C4×C61#57712yes—
200Dic61—24no—
201Dic61—42no—
202#3205D6111yescharacteristic
203#3037—611no—
204C2×#460—611no—
205#2982—611no—
206#1682—614no—
207#1668—614no—
208#1571—611no—
209#1545—611no—
210#1540—611no—
211C2×C61—22no—
212C2×C61—161no—
213C2×C61—162no—
214C2×C61#304311yescharacteristic
215C2×C61C2×#46611yescharacteristic
216C2×C61#306811yescharacteristic
217C2×#137—1221no—
218C22×SD16—1221no—
219#532C2×D6111yescharacteristic
220C2×#127C2×D6111yescharacteristic
221C2×C4×Q8—611no—
222#493C2×D6112yes—
223#486C2×D6112yes—
224#464—1222no—
225#462—612no—
226#460—614no—
227#460—612no—
228#460—1222no—
229#459—22no—
230#447—22no—
231C8×Q8—614no—
232C8×Q8—1222no—
233#431—611no—
234#431—1222no—
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—21no—
246C22×Q8—611no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250#137—2444no—
251C2×Q16—1222no—
252C2×SD16—1224no—
253C2×SD16—1222no—
254C2×SD16—2444no—
255C2×D8—24no—
256C2×M16—1221no—
257C22×C8—1221no—
258C4⋊Q8C22×D6111yescharacteristic
259#127—21no—
260#127—21no—
261#127—21no—
262#127C22×D6112yes—
263#127C22×D6111yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—612no—
268C4×Q8—612no—
269C4×Q8—1221no—
270C4×Q8—1221no—
271C4×D4—21no—
272C4×D4—21no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—1221no—
275C2×C42—1221no—
276C2×C42C22×D6111yescharacteristic
277C4⋊C8—612no—
278C4⋊C8—612no—
279C4⋊C8—612no—
280C4⋊C8—1224no—
281C4⋊C8—1221no—
282C4⋊C8—1221no—
283C4⋊C8C22⋊D6112yes—
284#103—1222no—
285#102—24no—
286#102—1224no—
287#102—1222no—
288C22⋊C8—1221no—
289C22⋊C8C22⋊D6112yes—
290C4×C8—1224no—
291C4×C8—1221no—
292C4×C8C22×D6112yes—
293C24—41no—
294#41—42no—
295#41—42no—
296C2×Q8—21no—
297C2×Q8—612no—
298C2×Q8—614no—
299C2×Q8—1221no—
300C2×Q8—1222no—
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—82no—
314C2×D4—82no—
315C22×C4—41no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4D4×D6111yescharacteristic
319C22×C4#1175822411yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321Q16—2444no—
322SD16—2448no—
323SD16—2444no—
324D8—84no—
325M16—2442no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8—1222no—
329C2×C8—2442no—
330C2×C8C2×C22⋊D6114yes—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—612no—
334C4⋊C4—1222no—
335C4⋊C4—1221no—
336C4⋊C4—1222no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—1222no—
340C42—1222no—
341C42D4×D6111yescharacteristic
342C42#1175822411yescharacteristic
343C42C2×C22⋊D6111yescharacteristic
344C42C23×D6111yescharacteristic
345C23—41no—
346C23—42no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23#1451974811yescharacteristic
351Q8—41no—
352Q8—41no—
353Q8—614no—
354Q8—1224no—
355Q8—1222no—
356D4—42no—
357D4—41no—
358D4—42no—
359D4—41no—
360D4—42no—
361D4—44no—
362D4—81no—
363D4—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367D4—82no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—1224no—
373C2×C4—1222no—
374C2×C4—1222no—
375C2×C4—4881no—
376C2×C4#1451974811yescharacteristic
377C2×C4#1451974811yescharacteristic
378C2×C4#1451974811yescharacteristic
379C2×C4D8×D6112yes—
380C2×C4#1451979612yes—
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C22×C22⋊D6111yescharacteristic
383C2×C4C2×#1175822411yescharacteristic
384C8—24no—
385C8—2442no—
386C8—2442no—
387C8—2442no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—82no—
395C22—81no—
396C22—164no—
397C22#1683999512yes—
398C22C2×#1451974811yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—1224no—
404C4—2442no—
405C4#1683999512yes—
406C4C2×D8×D6111yescharacteristic
407C4C2×#1451979611yescharacteristic
408C4#1684057312yes—
409C2—22no—
410C2—162no—
411C2—161no—
412C2#2145533611yescharacteristic
413C2C2×#1683999511yescharacteristic
414C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3205C2
socleC22×C61C2×#121
hypercenter#3205D61
hypocenterC61#19398
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21455287C2semidirect
2. C2×#16839959C2semidirect
3. #21454899C2semidirect
4. #21445260C2semidirect
5. #21445176C2semidirect
6. #21444594C2semidirect
7. #21444487C2semidirect
8. #21444468C2semidirect
14. #16839969C22semidirect
17. #16839959C22semidirect
18. C8×Dic122C22semidirect
26. #16839515C22semidirect
27. #16839508C22semidirect
29. #16839506C22semidirect
111. C2×Dic122D8semidirect
244. C61#19398semidirect

Maximal subgroups

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

subgroupindex
#214552872
C2×#168399592
#214548992
#214452602
#214451762
#214445942
#214444872
#214444682
C61×#32052
#1939861