Finite Group Databasebeta

← #29671566

Subgroups — #29671566

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
1D8×Dic122C211yesmaximal, characteristic, complemented
2#21458107C212yesmaximal, complemented
3#21445428C211yesmaximal, characteristic, complemented
4#21445396C212yesmaximal, complemented
5#21445220C211yesmaximal, characteristic, complemented
6#21445208C211yesmaximal, characteristic, complemented
7#21445047C212yesmaximal, complemented
8#21444983C212yesmaximal, complemented
9#21444704C211yesmaximal, characteristic, complemented
10C8×#11758200C211yesmaximal, characteristic, complemented
11C61×#3341C211yesmaximal, characteristic
12D4×Dic122C2212yes—
13#16840167C2211yescharacteristic
14#16840166—21no—
15#16840114C2212yes—
16#16840112C2212yes—
17#16840103—22no—
18#16840054C2212yes—
19#16840042C2212yes—
20#16840040—22no—
21#16840006—22no—
22#16839967C2212yes—
23#16839900—22no—
24C8×Dic122C2211yescharacteristic
25C4×#11758200C2211yescharacteristic
26D8×Dic61—21no—
27#16839840C2212yes—
28#16839770—22no—
29#16839515C2211yescharacteristic
30#16839512C2211yescharacteristic
31#16839511C2212yes—
32#16839507C2212yes—
33#16839492C2211yescharacteristic
34D4⋊#92418C2211yescharacteristic
35D4⋊#92418C2212yes—
36C8×#92418C2211yescharacteristic
37C61×#555C2212yes—
38C61×#500C2211yescharacteristic
39C61×#482—22no—
40C61×#476C2212yes—
41C61×#466—21no—
42C4×D8×C61C2211yescharacteristic
43C8×D4×C61C2211yescharacteristic
44C61×#427—21no—
45C22×Dic122—22no—
46C2×#11758200—22no—
47C2×#11758200D411yescharacteristic
48C2×#11758194—22no—
49#14519777C2312yes—
50#14519776—22no—
51#14519756—22no—
52#14519754—42no—
53C4×Dic122C2311yescharacteristic
54D4×Dic61—22no—
55#14519711—21no—
56#14519711—21no—
57#14519710—22no—
58#14519710—22no—
59#14519708D411yescharacteristic
60C4⋊#92418C2312yes—
61C4⋊#92418C2311yescharacteristic
62#14519706—21no—
63#14519701—42no—
64#14519700—22no—
65#14519700—22no—
66#14519700—22no—
67#14519700—22no—
68#14519700D411yescharacteristic
69#14519698—22no—
70#14519698D411yescharacteristic
71C22⋊#92418—22no—
72C4×#92418C2311yescharacteristic
73C4×#92418C2311yescharacteristic
74C8×Dic61—21no—
75C2×D8×C61D411yescharacteristic
76C22×C8×C61—21no—
77C4⋊Q8×C61C2312yes—
78#124×C61—42no—
79#121×C61—22no—
80#121×C61—22no—
81C4×D4×C61C2311yescharacteristic
82C4×D4×C61C2312yes—
83#117×C61—22no—
84#117×C61—22no—
85#107×C61—21no—
86#107×C61—21no—
87#107×C61D411yescharacteristic
88#106×C61—22no—
89C4⋊C8×C61C2311yescharacteristic
90#102×C61—22no—
91#102×C61D412yes—
92C22⋊C8×C61—21no—
93C4×C8×C61C2311yescharacteristic
94C2×Q8×C61—22no—
95C2×D4×C61—42no—
96C2×D4×C61C2×D411yescharacteristic
97C2×D4×C61C2×D412yes—
98C22×C4×C61—22no—
99C22×C4×C61—21no—
100C22×Dic61—42no—
101C2×Dic122—22no—
102C2×Dic122—44no—
103C2×Dic122—42no—
104C2×Dic122—82no—
105C2×Dic122C2×D411yescharacteristic
106D8×C61—22no—
107C2×C8×C61—21no—
108C2×C8×C61—42no—
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2×D411yescharacteristic
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61C2×D411yescharacteristic
114C4⋊C4×C61C2×D412yes—
115C22⋊C4×C61—21no—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—42no—
118C22⋊C4×C61—42no—
119C42×C61—42no—
120C42×C61C2411yescharacteristic
121C22⋊Dic61—42no—
122#11758200—22no—
123#11758200—82no—
124C4⋊Dic61—22no—
125C4⋊Dic61—22no—
126C4⋊Dic61C2×D411yescharacteristic
127#11758196—21no—
128#11758196—42no—
129C4×Dic61—21no—
130#11758194—82no—
131C2×#92418—21no—
132C2×#92418—22no—
133C2×#92418—21no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136C23×C61—22no—
137C23×C61—21no—
138Q8×C61—82no—
139D4×C61—21no—
140D4×C61—22no—
141D4×C61—21no—
142D4×C61—22no—
143D4×C61—162no—
144C2×C4×C61—22no—
145C2×C4×C61—21no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C2×C4×C61H(5,𝔽2)11yescharacteristic
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—42no—
157C2×Dic61—84no—
158Dic122—21no—
159Dic122—42no—
160Dic122—42no—
161Dic122—84no—
162Dic122#13512yes—
163C8×C61—21no—
164C8×C61—41no—
165C8×C61#13712yes—
166#92418—21no—
167#92418—21no—
168#92418—41no—
169#92418—42no—
170#92418—41no—
171#19905—611nomaximal, Sylow
172C22×C61—41no—
173C22×C61—42no—
174C22×C61—42no—
175C22×C61—41no—
176C22×C61—41no—
177C22×C61—42no—
178C22×C61D4211yescharacteristic
179C4×C61—21no—
180C4×C61—41no—
181C4×C61—42no—
182C4×C61—82no—
183C4×C61C2×#13511yescharacteristic
184C4×C61C2×#13711yescharacteristic
185C4×C61#57611yescharacteristic
186C4×C61#57811yescharacteristic
187Dic61—22no—
188Dic61—41no—
189Dic61—82no—
190Q8×D8—611no—
191#3341D6111yescharacteristic
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—81no—
202C2×C61—82no—
203C2×C61#327611yescharacteristic
204C2×C61#328011yescharacteristic
205C2×C61#329011yescharacteristic
206#555C2×D6112yes—
207D4×Q8—612no—
208#500—611no—
209#500—1221no—
210#500C2×D6111yescharacteristic
211#487—612no—
212#487—612no—
213#485—1222no—
214#482—22no—
215#477—612no—
216#476C2×D6112yes—
217#474—612no—
218#474—1222no—
219#468—1222no—
220#466—21no—
221#461—612no—
222#448—1222no—
223C8×Q8—611no—
224C4×SD16—611no—
225C4×D8—1221no—
226C4×D8C2×D6111yescharacteristic
227#436—612no—
228C8×D4C2×D6111yescharacteristic
229#427—21no—
230#419—1222no—
231#335—611no—
232#334—611no—
233#333—612no—
234#332—612no—
235Q8⋊C8—611no—
236D4⋊C8—611no—
237D4⋊C8—612no—
238C82—611no—
239C61#1990511yescharacteristic, complemented, Sylow
240C22×Q8—1222no—
241C2×SD16—611no—
242C2×SD16—1222no—
243C2×D8C22⋊D6111yescharacteristic
244C2×M16—1222no—
245C22×C8—21no—
246C4⋊Q8—612no—
247C4⋊Q8—1222no—
248C4⋊Q8C22×D6112yes—
249#124—42no—
250C22⋊Q8—1222no—
251C22⋊Q8—2442no—
252#121—22no—
253#121—22no—
254C4×Q8—611no—
255C4×D4—1222no—
256C4×D4C22×D6111yescharacteristic
257C4×D4C22×D6112yes—
258#117—22no—
259#117—22no—
260#107—21no—
261#107—21no—
262#107—1222no—
263#107—1221no—
264#107—1221no—
265#107—1222no—
266#107C22⋊D6111yescharacteristic
267#106—22no—
268#106—611no—
269C4⋊C8—612no—
270C4⋊C8—611no—
271C4⋊C8—1221no—
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—21no—
284C22⋊C8—1222no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—1221no—
288C4×C8C22×D6111yescharacteristic
289C2×Q8—22no—
290C2×Q8—611no—
291C2×Q8—1222no—
292C2×Q8—4882no—
293C2×Q8—2442no—
294C2×Q8—2444no—
295C2×D4—42no—
296C2×D4D4×D6111yescharacteristic
297C2×D4C2×C22⋊D6112yes—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—2442no—
301SD16—1222no—
302SD16—4882no—
303D8—22no—
304M16—4882no—
305C2×C8—21no—
306C2×C8—42no—
307C2×C8—611no—
308C2×C8—611no—
309C2×C8—1222no—
310C2×C8—1221no—
311C2×C8—1221no—
312C2×C8C2×C22⋊D6111yescharacteristic
313C2×C8C2×C22⋊D6111yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—611no—
317C4⋊C4—1221no—
318C4⋊C4—1222no—
319C4⋊C4—1222no—
320C4⋊C4—2442no—
321C4⋊C4D4×D6111yescharacteristic
322C4⋊C4C2×C22⋊D6112yes—
323C22⋊C4—22no—
324C22⋊C4—21no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—2442no—
328C42—42no—
329C42—1221no—
330C42C23×D6111yescharacteristic
331C23—21no—
332C23—22no—
333Q8—82no—
334Q8—612no—
335Q8—1221no—
336Q8—4884no—
337Q8—2442no—
338Q8—2442no—
339D4—21no—
340D4—22no—
341D4—21no—
342D4—22no—
343D4—162no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—4884no—
354C2×C4—2442no—
355C2×C4C2×D4×D6111yescharacteristic
356C2×C4C22×C22⋊D6111yescharacteristic
357C2×C4#1451982211yescharacteristic
358C8—21no—
359C8—41no—
360C8—1221no—
361C8—1221no—
362C8—2441no—
363C8—2442no—
364C8—2441no—
365C8#1451980712yes—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22D4×C22⋊D6111yescharacteristic
373C4—21no—
374C4—41no—
375C4—42no—
376C4—82no—
377C4—1222no—
378C4—4882no—
379C4—2441no—
380C4#135×D6111yescharacteristic
381C4C2×#1451980711yescharacteristic
382C4#1684056411yescharacteristic
383C4#1684057711yescharacteristic
384C2—81no—
385C2—82no—
386C2#2145746111yescharacteristic
387C2#2145749811yescharacteristic
388C2#2145760111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3341C2
socleC22×C61D42
hypercenter#3341D61
hypocenterC61#19905
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3341

Complements

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

NH ≅ G/Ntype
1. D8×Dic122C2semidirect
2. #21458107C2semidirect
3. #21445428C2semidirect
4. #21445396C2semidirect
5. #21445220C2semidirect
6. #21445208C2semidirect
7. #21445047C2semidirect
8. #21444983C2semidirect
9. #21444704C2semidirect
10. C8×#11758200C2semidirect
239. C61#19905semidirect

Maximal subgroups

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

subgroupindex
D8×Dic1222
#214581072
#214454282
#214453962
#214452202
#214452082
#214450472
#214449832
#214447042
C8×#117582002
C61×#33412
#1990561