Finite Group Databasebeta

← #29671562

Subgroups — #29671562

Order 15616.

604 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458243C211yesmaximal, characteristic
2#21458113C212yesmaximal, complemented
3#21445428C211yesmaximal, characteristic, complemented
4#21445392C212yesmaximal, complemented
5#21445223C211yesmaximal, characteristic, complemented
6#21445205C211yesmaximal, characteristic, complemented
7#21445046C212yesmaximal, complemented
8#21444984C212yesmaximal, complemented
9#21444704C211yesmaximal, characteristic, complemented
10C8×#11758201C211yesmaximal, characteristic, complemented
11Q8×D8×C61C211yesmaximal, characteristic, complemented
12#16840477C2212yes—
13#16840167C2211yescharacteristic
14#16840114C2212yes—
15#16840112C2212yes—
16#16840100—22no—
17#16840084—22no—
18#16840057C2212yes—
19#16840052C2212yes—
20#16840010—22no—
21#16839995—21no—
22#16839964C2212yes—
23#16839901—22no—
24C4×#11758201C2211yescharacteristic
25#16839840C2212yes—
26C8×D244C2211yescharacteristic
27#16839799—21no—
28#16839771—22no—
29#16839515C2211yescharacteristic
30#16839512C2211yescharacteristic
31#16839510C2212yes—
32#16839508C2212yes—
33Q8⋊#92418C2211yescharacteristic
34D4⋊#92418C2212yes—
35#16839489C2211yescharacteristic
36C8×#92418C2211yescharacteristic
37D4×Q8×C61C2212yes—
38C61×#500—21no—
39C61×#500C2211yescharacteristic
40C61×#474—22no—
41C61×#474C2212yes—
42C8×Q8×C61C2211yescharacteristic
43C4×D8×C61—21no—
44C4×D8×C61C2211yescharacteristic
45C2×#11758201—22no—
46C2×#11758201D411yescharacteristic
47C2×#11758194—22no—
48C2×C8×D61—21no—
49#14519777C2312yes—
50#14519763—42no—
51#14519751—22no—
52#14519748—22no—
53C4×D244C2311yescharacteristic
54#14519736—22no—
55#14519735—22no—
56#14519711—21no—
57#14519711—21no—
58#14519710—22no—
59#14519708—22no—
60#14519708D411yescharacteristic
61C4⋊#92418C2312yes—
62C4⋊#92418C2311yescharacteristic
63#14519701—22no—
64#14519701—22no—
65#14519701—22no—
66#14519701—22no—
67#14519701D411yescharacteristic
68#14519700—42no—
69#14519698—22no—
70#14519697D411yescharacteristic
71C22⋊#92418—22no—
72#14519689—21no—
73C4×#92418C2311yescharacteristic
74C4×#92418C2311yescharacteristic
75C22×Q8×C61—22no—
76C2×D8×C61D411yescharacteristic
77C4⋊Q8×C61—22no—
78C4⋊Q8×C61C2312yes—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—42no—
81C4×Q8×C61C2311yescharacteristic
82C4×D4×C61—22no—
83C4×D4×C61C2312yes—
84#107×C61—21no—
85#107×C61—21no—
86#107×C61—22no—
87#107×C61D411yescharacteristic
88C4⋊C8×C61—21no—
89C4⋊C8×C61C2311yescharacteristic
90#102×C61—22no—
91#102×C61D412yes—
92C4×C8×C61—21no—
93C4×C8×C61C2311yescharacteristic
94C2×Q8×C61—22no—
95C2×Q8×C61—42no—
96C2×Q8×C61—44no—
97C2×Q8×C61—82no—
98C2×Q8×C61C2×D411yescharacteristic
99C2×D4×C61C2×D412yes—
100C22×C4×C61—22no—
101C22×C4×C61—42no—
102C2×C22⋊D61—42no—
103C2×D244C2×D411yescharacteristic
104C2×C4×D61—21no—
105C2×Dic122—22no—
106D8×C61—22no—
107C2×C8×C61—21no—
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61C2×D411yescharacteristic
110C4⋊C4×C61—21no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—42no—
114C4⋊C4×C61C2×D411yescharacteristic
115C4⋊C4×C61C2×D412yes—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—42no—
118C42×C61—21no—
119C42×C61C2411yescharacteristic
120C22⋊Dic61—42no—
121#11758201—22no—
122#11758201—82no—
123#11758198—21no—
124#11758198—42no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127C4⋊Dic61C2×D411yescharacteristic
128C4×Dic61—42no—
129#11758194—82no—
130C2×#92418—21no—
131C2×#92418—22no—
132C2×#92418—21no—
133C2×#92418C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135C8×D61—42no—
136C23×C61—22no—
137C22×D61—21no—
138Q8×C61—21no—
139Q8×C61—42no—
140Q8×C61—42no—
141Q8×C61—84no—
142Q8×C61#13512yes—
143D4×C61—22no—
144D4×C61—22no—
145C2×C4×C61—21no—
146C2×C4×C61—22no—
147C2×C4×C61—21no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—84no—
152C2×C4×C61C22×D411yescharacteristic
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61H(5,𝔽2)11yescharacteristic
155C22⋊D61—162no—
156C2×Dic61—21no—
157C2×Dic61—42no—
158D244—21no—
159D244—21no—
160C4×D61—41no—
161C4×D61—41no—
162Dic122—82no—
163C8×C61—21no—
164C8×C61—41no—
165C8×C61#13712yes—
166#92418—21no—
167#92418—21no—
168#92418—41no—
169#92418—41no—
170#92418—42no—
171#19905—611nomaximal, Sylow
172C22×C61—42no—
173C22×C61—42no—
174C22×C61—42no—
175C22×C61D4211yescharacteristic
176C2×D61—41no—
177C2×D61—41no—
178C2×D61—41no—
179C4×C61—22no—
180C4×C61—21no—
181C4×C61—42no—
182C4×C61—41no—
183C4×C61—82no—
184C4×C61C2×#13511yescharacteristic
185C4×C61C2×#13711yescharacteristic
186C4×C61#57611yescharacteristic
187C4×C61#57811yescharacteristic
188Dic61—41no—
189Dic61—82no—
190Q8×D8D6111yescharacteristic, complemented
191#3341—611no—
192#3329—612no—
193#1710—611no—
194#1703—612no—
195#1676—611no—
196#1673—611no—
197#1646—612no—
198#1637—612no—
199#1589—611no—
200C8×SD16—611no—
201C2×C61—82no—
202C2×C61#327611yescharacteristic
203C2×C61#328011yescharacteristic
204C2×C61#329011yescharacteristic
205D61—81no—
206#555—612no—
207D4×Q8C2×D6112yes—
208#500—21no—
209#500—611no—
210#500C2×D6111yescharacteristic
211#487—612no—
212#487—612no—
213#485—1222no—
214#482—1222no—
215#477—612no—
216#476—612no—
217#474—22no—
218#474C2×D6112yes—
219#468—1222no—
220#466—1221no—
221#461—612no—
222#448—1222no—
223C8×Q8C2×D6111yescharacteristic
224C4×SD16—611no—
225C4×D8—21no—
226C4×D8C2×D6111yescharacteristic
227#436—612no—
228C8×D4—611no—
229#427—1221no—
230#419—1222no—
231#335—611no—
232#334—611no—
233#333—612no—
234#332—612no—
235Q8⋊C8—611no—
236D4⋊C8—612no—
237D4⋊C8—611no—
238C82—611no—
239C61#1990511yescharacteristic, complemented, Sylow
240C22×Q8—22no—
241C2×SD16—611no—
242C2×SD16—1222no—
243C2×D8C22⋊D6111yescharacteristic
244C2×M16—1222no—
245C22×C8—1221no—
246C4⋊Q8—22no—
247C4⋊Q8—612no—
248C4⋊Q8C22×D6112yes—
249#124—2442no—
250C22⋊Q8—22no—
251C22⋊Q8—42no—
252#121—1222no—
253#121—1222no—
254C4×Q8C22×D6111yescharacteristic
255C4×D4—22no—
256C4×D4—611no—
257C4×D4C22×D6112yes—
258#117—1222no—
259#117—1222no—
260#107—21no—
261#107—21no—
262#107—22no—
263#107—1221no—
264#107—1221no—
265#107—1222no—
266#107C22⋊D6111yescharacteristic
267#106—611no—
268#106—1222no—
269C4⋊C8—21no—
270C4⋊C8—612no—
271C4⋊C8—611no—
272C4⋊C8C22×D6111yescharacteristic
273#103—611no—
274#103—1222no—
275#103—1222no—
276#103—1222no—
277#103—1222no—
278#103—2442no—
279#102—22no—
280#102—611no—
281#102—1222no—
282#102C22⋊D6112yes—
283C22⋊C8—1222no—
284C22⋊C8—1221no—
285C4×C8—21no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8C22×D6111yescharacteristic
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—44no—
292C2×Q8—82no—
293C2×Q8—1222no—
294C2×Q8D4×D6111yescharacteristic
295C2×D4—611no—
296C2×D4—2442no—
297C2×D4C2×C22⋊D6112yes—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—1221no—
301SD16—1222no—
302SD16—4882no—
303D8—22no—
304M16—4882no—
305C2×C8—21no—
306C2×C8—611no—
307C2×C8—611no—
308C2×C8—1222no—
309C2×C8—1221no—
310C2×C8—1221no—
311C2×C8—2442no—
312C2×C8C2×C22⋊D6111yescharacteristic
313C2×C8C2×C22⋊D6111yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—611no—
319C4⋊C4—1222no—
320C4⋊C4—1222no—
321C4⋊C4D4×D6111yescharacteristic
322C4⋊C4C2×C22⋊D6112yes—
323C22⋊C4—22no—
324C22⋊C4—42no—
325C22⋊C4—1221no—
326C22⋊C4—2442no—
327C22⋊C4—2442no—
328C42—21no—
329C42—2442no—
330C42C23×D6111yescharacteristic
331C23—22no—
332C23—1221no—
333Q8—21no—
334Q8—42no—
335Q8—42no—
336Q8—84no—
337Q8—4882no—
338Q8#1451979612yes—
339D4—22no—
340D4—22no—
341D4—1221no—
342D4—1221no—
343D4—9762no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—84no—
351C2×C4—1221no—
352C2×C4—2442no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4C2×D4×D6111yescharacteristic
356C2×C4C22×C22⋊D6111yescharacteristic
357C2×C4#1451982211yescharacteristic
358C8—21no—
359C8—41no—
360C8—1221no—
361C8—1221no—
362C8—2441no—
363C8—2441no—
364C8—2442no—
365C8#1451980812yes—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—2441no—
370C22—2441no—
371C22—2441no—
372C22D4×C22⋊D6111yescharacteristic
373C4—22no—
374C4—21no—
375C4—42no—
376C4—41no—
377C4—82no—
378C4—4882no—
379C4—2441no—
380C4C2×#1451979611yescharacteristic
381C4C2×#1451980811yescharacteristic
382C4#1684056311yescharacteristic
383C4#1684057711yescharacteristic
384C2—82no—
385C2—4881no—
386C2#2145746511yescharacteristic
387C2#2145750411yescharacteristic
388C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×D8×C61C2
socleC22×C61D42
prime radical#21458243C2
hypercenterQ8×D8D61
hypocenterC61#19905
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×D8

Complements

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

NH ≅ G/Ntype
2. #21458113C2semidirect
3. #21445428C2semidirect
4. #21445392C2semidirect
5. #21445223C2semidirect
6. #21445205C2semidirect
7. #21445046C2semidirect
8. #21444984C2semidirect
9. #21444704C2semidirect
10. C8×#11758201C2semidirect
11. Q8×D8×C61C2semidirect
190. Q8×D8D61semidirect
239. C61#19905semidirect

Maximal subgroups

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

subgroupindex
#214582432
#214581132
#214454282
#214453922
#214452232
#214452052
#214450462
#214449842
#214447042
C8×#117582012
Q8×D8×C612
#1990561