Finite Group Databasebeta

← #29671525

Subgroups — #29671525

Order 15616.

672 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458124C212yesmaximal, complemented
2#21457540C211yesmaximal, characteristic, complemented
3#21445417C211yesmaximal, characteristic, complemented
4#21445364C212yesmaximal, complemented
5#21445205C211yesmaximal, characteristic, complemented
6#21445184C211yesmaximal, characteristic, complemented
7#21445103C212yesmaximal, complemented
8#21445102C212yesmaximal, complemented
9#21444704C211yesmaximal, characteristic, complemented
10C8×#11758199C211yesmaximal, characteristic, complemented
11C61×#3334C211yesmaximal, characteristic, complemented
12C2×#14519796—21no—
13#16840416C2212yes—
14#16840158C2212yes—
15#16840142C2211yescharacteristic
16#16840112C2212yes—
17#16840095—22no—
18#16840042C2212yes—
19#16840010—22no—
20#16840004—21no—
21#16839967C2212yes—
22#16839964C2212yes—
23#16839925—22no—
24#16839907—22no—
25C4×#11758199C2211yescharacteristic
26#16839840C2212yes—
27C8×D244C2211yescharacteristic
28#16839763—22no—
29#16839515C2211yescharacteristic
30#16839514C2211yescharacteristic
31#16839509C2212yes—
32#16839508C2212yes—
33D4⋊#92418C2211yescharacteristic
34D4⋊#92418C2212yes—
35#16839489C2211yescharacteristic
36C8×#92418C2211yescharacteristic
37C61×#547C2212yes—
38C61×#500C2211yescharacteristic
39C61×#480—22no—
40C61×#474C2212yes—
41C61×#466—21no—
42C4×D8×C61C2211yescharacteristic
43C8×D4×C61C2211yescharacteristic
44C2×#107×C61—21no—
45C2×#11758224—22no—
46#14519796—44no—
47C2×Dic244—21no—
48C2×Dic244—21no—
49C2×#11758200—22no—
50C2×#11758201—22no—
51C2×#11758199D411yescharacteristic
52C2×#11758194—22no—
53C2×C8×D61—21no—
54#14519777C2312yes—
55#14519761—42no—
56#14519757—22no—
57#14519751—22no—
58C4×D244C2311yescharacteristic
59C2×C4⋊Dic61—22no—
60#14519710—22no—
61#14519710D411yescharacteristic
62#14519708—42no—
63C4⋊#92418C2312yes—
64C4⋊#92418C2311yescharacteristic
65#14519701—22no—
66#14519701—22no—
67#14519701—22no—
68#14519700—22no—
69#14519698—22no—
70#14519698D411yescharacteristic
71#14519697D411yescharacteristic
72C22⋊#92418—22no—
73#14519689—21no—
74C4×#92418C2311yescharacteristic
75C4×#92418C2311yescharacteristic
76C2×#41×C61—22no—
77C2×D8×C61D411yescharacteristic
78C22×C8×C61—21no—
79C4⋊Q8×C61C2312yes—
80#123×C61—42no—
81C22⋊Q8×C61—22no—
82#121×C61—22no—
83C4×D4×C61C2311yescharacteristic
84C4×D4×C61C2312yes—
85C2×C4⋊C4×C61—22no—
86#107×C61—21no—
87#107×C61—21no—
88#107×C61—22no—
89#107×C61D411yescharacteristic
90C4⋊C8×C61C2311yescharacteristic
91#102×C61—22no—
92#102×C61D412yes—
93C22⋊C8×C61—21no—
94C4×C8×C61C2311yescharacteristic
95#41×C61—44no—
96#41×C61—82no—
97C2×Q8×C61—22no—
98C2×D4×C61—42no—
99C2×D4×C61C2×D412yes—
100C2×D4×C61C2×D411yescharacteristic
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—42no—
104C2×C22⋊D61—42no—
105C22×Dic61—42no—
106#11758224—44no—
107#11758224—82no—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×Dic122—22no—
111D8×C61—22no—
112C2×C8×C61—21no—
113C2×C8×C61—42no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D411yescharacteristic
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—42no—
119C4⋊C4×C61C2×D411yescharacteristic
120C4⋊C4×C61C2×D412yes—
121C22⋊C4×C61—21no—
122C22⋊C4×C61—22no—
123C22⋊C4×C61—42no—
124C42×C61C2411yescharacteristic
125#11758201—82no—
126#11758200—44no—
127#11758199—22no—
128#11758198—21no—
129#11758198—42no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—42no—
133C4⋊Dic61C2×D411yescharacteristic
134#11758194—82no—
135C2×#92418—21no—
136C2×#92418—21no—
137C2×#92418—22no—
138C2×#92418C2×D411yescharacteristic
139C2×#92418C2×D411yescharacteristic
140Dic244—42no—
141Dic244—81no—
142C8×D61—42no—
143C23×C61—21no—
144C23×C61—22no—
145C22×D61—21no—
146Q8×C61—42no—
147Q8×C61—42no—
148D4×C61—22no—
149D4×C61—21no—
150D4×C61—22no—
151D4×C61—84no—
152D4×C61#13512yescomplemented
153C2×C4×C61—22no—
154C2×C4×C61—21no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—84no—
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61H(5,𝔽2)11yescharacteristic
164C22⋊D61—84no—
165C2×Dic61—21no—
166C2×Dic61—42no—
167C2×Dic61—84no—
168D244—21no—
169D244—21no—
170C4×D61—41no—
171C4×D61—41no—
172Dic122—42no—
173Dic122—42no—
174C8×C61—21no—
175C8×C61—41no—
176C8×C61AGL(1,ℤ/8)12yes—
177#92418—22no—
178#92418—21no—
179#92418—21no—
180#92418—41no—
181#92418—42no—
182#19902—611nomaximal, Sylow
183C22×C61—22no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61D4211yescharacteristic
190C2×D61—41no—
191C2×D61—41no—
192C2×D61—41no—
193C4×C61—21no—
194C4×C61—41no—
195C4×C61—42no—
196C4×C61—82no—
197C4×C61C2×#13511yescharacteristic
198C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
199C4×C61#57611yescharacteristic
200C4×C61#57711yescharacteristic
201Dic61—41no—
202Dic61—82no—
203#3334D6111yescharacteristic, complemented
204#3330—612no—
205#3283—611no—
206#1707—611no—
207#1699—612no—
208#1673—611no—
209#1669—611no—
210#1654—612no—
211#1654—612no—
212#1589—611no—
213C8×D8—611no—
214C2×C61—42no—
215C2×C61—82no—
216C2×C61#327211yescharacteristic
217C2×C61#328011yescharacteristic
218C2×C61#329311yescharacteristic
219D61—81no—
220C2×#135—1221no—
221#547—612no—
222#547C2×D6112yes—
223#500C2×D6111yescharacteristic
224#498—612no—
225#494—611no—
226#487—612no—
227#484—1222no—
228#480—22no—
229#474—612no—
230#474C2×D6112yes—
231#468—1222no—
232#467—1221no—
233#466—21no—
234#461—612no—
235#461—612no—
236#452—1222no—
237#449—1222no—
238C4×D8—611no—
239C4×D8C2×D6111yescharacteristic
240#436—612no—
241C8×D4—611no—
242C8×D4C2×D6111yescharacteristic
243C2×#107—21no—
244#417—1222no—
245#335—611no—
246#335—611no—
247#332—612no—
248#332—612no—
249D4⋊C8—611no—
250D4⋊C8—612no—
251D4⋊C8—611no—
252C82—611no—
253C61#1990211yescharacteristic, complemented, Sylow
254C2×#41—22no—
255C2×#41—1222no—
256#135—2444no—
257C2×Q16—1221no—
258C2×Q16—1221no—
259C2×SD16—1222no—
260C2×SD16—1222no—
261C2×D8—611no—
262C2×D8C22⋊D6111yescharacteristic
263C2×M16—1222no—
264C22×C8—21no—
265C22×C8—1221no—
266C4⋊Q8—612no—
267C4⋊Q8C22×D6112yes—
268#123—42no—
269#123—2442no—
270C22⋊Q8—22no—
271C22⋊Q8—1222no—
272#121—22no—
273#121—1222no—
274C4×D4—611no—
275C4×D4C22×D6111yescharacteristic
276C4×D4C22×D6112yes—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—1222no—
279#107—21no—
280#107—21no—
281#107—22no—
282#107—611no—
283#107—1222no—
284#107C22⋊D6111yescharacteristic
285#106—2442no—
286C4⋊C8—611no—
287C4⋊C8—612no—
288C4⋊C8C22×D6111yescharacteristic
289#103—1222no—
290#103—1222no—
291#103—1222no—
292#103—1222no—
293#102—22no—
294#102—611no—
295#102—611no—
296#102—1222no—
297#102C22⋊D6112yes—
298C22⋊C8—21no—
299C22⋊C8—1222no—
300C22⋊C8—1221no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8C22×D6111yescharacteristic
304#41—44no—
305#41—82no—
306#41—4882no—
307#41—2444no—
308C2×Q8—22no—
309C2×Q8—1222no—
310C2×D4—42no—
311C2×D4—611no—
312C2×D4—2442no—
313C2×D4D4×D6111yescharacteristic
314C2×D4C2×C22⋊D6112yes—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—1221no—
319C22×C4—2442no—
320Q16—4881no—
321Q16—2442no—
322SD16—4882no—
323SD16—2444no—
324D8—22no—
325D8—1222no—
326M16—4882no—
327C2×C8—21no—
328C2×C8—42no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—1221no—
332C2×C8—1222no—
333C2×C8—1221no—
334C2×C8—2442no—
335C2×C8C2×C22⋊D6111yescharacteristic
336C2×C8C2×C22⋊D6111yescharacteristic
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—611no—
341C4⋊C4—1222no—
342C4⋊C4—1222no—
343C4⋊C4—2442no—
344C4⋊C4D4×D6111yescharacteristic
345C4⋊C4C2×C22⋊D6112yes—
346C22⋊C4—21no—
347C22⋊C4—22no—
348C22⋊C4—42no—
349C22⋊C4—1221no—
350C22⋊C4—2442no—
351C42C23×D6111yescharacteristic
352C23—21no—
353C23—22no—
354C23—1221no—
355Q8—42no—
356Q8—42no—
357Q8—2442no—
358Q8—2442no—
359D4—22no—
360D4—21no—
361D4—22no—
362D4—84no—
363D4—1221no—
364D4—1221no—
365D4—4884no—
366D4#1451979612yescomplemented
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4—1221no—
376C2×C4—4884no—
377C2×C4—2442no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4C2×D4×D6111yescharacteristic
381C2×C4C22×C22⋊D6111yescharacteristic
382C2×C4#1451982211yescharacteristic
383C8—21no—
384C8—41no—
385C8—1222no—
386C8—1221no—
387C8—1221no—
388C8—2441no—
389C8—2442no—
390C8#1451980212yes—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—2441no—
398C22—2441no—
399C22—2441no—
400C22D4×C22⋊D6111yescharacteristic
401C4—21no—
402C4—41no—
403C4—42no—
404C4—82no—
405C4—4882no—
406C4—2441no—
407C4C2×#1451979611yescharacteristic
408C4C2×#1451980211yescharacteristic
409C4#1684056311yescharacteristic
410C4#1684057011yescharacteristic
411C2—42no—
412C2—82no—
413C2—4881no—
414C2#2145741911yescharacteristic
415C2#2145750411yescharacteristic
416C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3334C2
socleC22×C61D42
hypercenter#3334D61
hypocenterC61#19902
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #21458124C2semidirect
2. #21457540C2semidirect
3. #21445417C2semidirect
4. #21445364C2semidirect
5. #21445205C2semidirect
6. #21445184C2semidirect
7. #21445103C2semidirect
8. #21445102C2semidirect
9. #21444704C2semidirect
10. C8×#11758199C2semidirect
11. C61×#3334C2semidirect
152. D4×C61#135semidirect
203. #3334D61semidirect
253. C61#19902semidirect
366. D4#14519796semidirect

Maximal subgroups

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

subgroupindex
#214581242
#214575402
#214454172
#214453642
#214452052
#214451842
#214451032
#214451022
#214447042
C8×#117581992
C61×#33342
#1990261