Finite Group Databasebeta

← #29663865

Subgroups — #29663865

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456406C211yesmaximal, characteristic
2#21455625C212yesmaximal
3C8×Q8×D61C211yesmaximal, characteristic
4#21445280C212yesmaximal, complemented
5#21445279C212yesmaximal, complemented
6#21445220C211yesmaximal, characteristic, complemented
7#21445190C211yesmaximal, characteristic, complemented
8#21445189C211yesmaximal, characteristic, complemented
9#21444576C211yesmaximal, characteristic
10#21444506C212yesmaximal
11C61×#1676C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13C4×Q8×D61C2211yescharacteristic
14#16840142C2211yescharacteristic
15#16840114C2212yes—
16#16840068—22no—
17#16840067—22no—
18#16840054C2212yes—
19#16840048C2212yes—
20#16840047C2212yes—
21#16840004—21no—
22#16840004—21no—
23Q8×#92418C2211yescharacteristic
24C8×Dic122—21no—
25C8×Dic122C2211yescharacteristic
26#16839886—21no—
27#16839886C2211yescharacteristic
28#16839856—21no—
29#16839856—21no—
30#16839803—22no—
31#16839784C2212yes—
32C4⋊C8×D61—21no—
33C4⋊C8×D61C2211yescharacteristic
34#16839769—22no—
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839514C2212yes—
39#16839511C2212yes—
40#16839509C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#477C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#333C2212yes—
47C2×Q8×D61D411yescharacteristic
48C2×Dic244—21no—
49C2×Dic244—21no—
50C2×#11758189—22no—
51C2×C8×D61—21no—
52C2×C8×D61D411yescharacteristic
53C2×C8×D61D411yescharacteristic
54#14519777C2312yes—
55#14519768—22no—
56#14519757—22no—
57#14519753—22no—
58Q8×Dic61—22no—
59Q8×Dic61D411yescharacteristic
60C4×Dic122—21no—
61C4×Dic122C2311yescharacteristic
62#14519742—21no—
63#14519742—22no—
64#14519742D411yescharacteristic
65#14519735—22no—
66C4⋊C4×D61—21no—
67C4⋊C4×D61D411yescharacteristic
68C42×D61—21no—
69C42×D61C2311yescharacteristic
70#14519710—22no—
71#14519709—22no—
72#14519708—22no—
73C4⋊#92418—21no—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2312yes—
76#14519706—21no—
77#14519706—21no—
78#14519706—22no—
79#14519706—21no—
80#14519702—22no—
81#14519701—22no—
82#14519700—22no—
83#14519700—22no—
84#14519700—22no—
85C4×#92418—21no—
86C4×#92418C2311yescharacteristic
87C8×Dic61—21no—
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C4⋊Q8×C61C2312yes—
92C4×Q8×C61C2311yescharacteristic
93#107×C61—21no—
94#107×C61—21no—
95#107×C61—22no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#103×C61—22no—
100C4×C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104Q8×D61—22no—
105Q8×D61Q1614yescomplemented
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61C2×D411yescharacteristic
109C2×C4×D61C2×D411yescharacteristic
110C2×C4×D61#4111yescharacteristic
111C2×Dic122—21no—
112C2×Dic122—22no—
113C2×Dic122C2×D411yescharacteristic
114C2×C8×C61—21no—
115C2×C8×C61—22no—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C4⋊C4×C61—21no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61C2×D411yescharacteristic
122C42×C61—21no—
123C42×C61C2411yescharacteristic
124#11758198—42no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61—42no—
129C4⋊Dic61C2×D411yescharacteristic
130#11758196—21no—
131#11758196—21no—
132#11758196—21no—
133#11758196—42no—
134C4×Dic61—21no—
135C4×Dic61—21no—
136C4×Dic61—21no—
137C4×Dic61—42no—
138C4×Dic61C2×D411yescharacteristic
139C4×Dic61C2×D411yescharacteristic
140C4×Dic61#4111yescharacteristic
141C2×#92418—21no—
142C2×#92418—22no—
143C2×#92418C2×D411yescharacteristic
144C2×#92418C2×D411yescharacteristic
145Dic244—42no—
146Dic244—42no—
147#11758189—44no—
148C8×D61—22no—
149C8×D61—41no—
150C8×D61—42no—
151C22×D61#12111yescharacteristic
152Q8×C61—21no—
153Q8×C61—82no—
154Q8×C61C2×Q1612yes—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—42no—
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—42no—
164C2×Dic61#12111yescharacteristic
165C2×Dic61#12111yescharacteristic
166C2×Dic61#12111yescharacteristic
167C4×D61—24no—
168C4×D61—21no—
169C4×D61—21no—
170C4×D61—41no—
171C4×D61—42no—
172C4×D61C2×Q1612yes—
173C4×D61#13512yes—
174Dic122—21no—
175Dic122—24no—
176Dic122—42no—
177Dic122—42no—
178Dic122C2×Q1612yes—
179C8×C61—21no—
180C8×C61—42no—
181C8×C61—41no—
182C8×C61#13712yes—
183#92418—21no—
184#92418—21no—
185#92418—42no—
186#92418—41no—
187#19262—611nomaximal, Sylow
188C22×C61C2×#12111yescharacteristic
189C2×D61—21no—
190C2×D61—21no—
191C2×D61#46712yes—
192C4×C61—22no—
193C4×C61—21no—
194C4×C61—41no—
195C4×C61—82no—
196C4×C61C22×Q1611yescharacteristic
197C4×C61C2×#13511yescharacteristic
198C4×C61C2×#13711yescharacteristic
199C4×C61#57511yescharacteristic
200Dic61—21no—
201Dic61—22no—
202Dic61—21no—
203Dic61—41no—
204Dic61—82no—
205Dic61#46712yes—
206#3170—611no—
207#3078—612no—
208C2×C8×Q8—611no—
209#1684—612no—
210#1684—612no—
211#1676—611no—
212#1676D6111yescharacteristic, complemented
213#1670—611no—
214#1670—611no—
215#1567—611no—
216#1550—612no—
217C2×C61#303711yescharacteristic
218C2×C61C2×#46711yescharacteristic
219C2×C61#306711yescharacteristic
220D61—22no—
221#533—612no—
222C2×C4×Q8—611no—
223#500C2×D6111yescharacteristic
224#494—611no—
225#487—612no—
226#479—1222no—
227#479—1222no—
228#477—612no—
229#477C2×D6112yes—
230#475—612no—
231#475—612no—
232#467—1221no—
233#467—1221no—
234C8×Q8—611no—
235C8×Q8—611no—
236C8×Q8—611no—
237C8×Q8—1221no—
238C8×Q8—1221no—
239C8×Q8C2×D6111yescharacteristic
240C4×Q16—1221no—
241C4×Q16—1221no—
242#428—1222no—
243#423—612no—
244C2×C4⋊C8—611no—
245C2×C4⋊C8—1221no—
246#419—1222no—
247C2×C4×C8—611no—
248C2×C4×C8—1221no—
249#335—612no—
250#335—611no—
251#335C2×D6111yescharacteristic
252#333—612no—
253#333C2×D6112yes—
254#332—612no—
255#332—612no—
256C61#1926211yescharacteristic, complemented, Sylow
257C22×Q8—611no—
258C2×Q16—1221no—
259C2×Q16—1221no—
260C2×M16—1222no—
261C22×C8—611no—
262C22×C8—611no—
263C22×C8—1221no—
264C4⋊Q8—612no—
265C4⋊Q8C22×D6112yes—
266#125—1222no—
267C22⋊Q8—1222no—
268C22⋊Q8—1222no—
269C4×Q8—611no—
270C4×Q8—611no—
271C4×Q8—611no—
272C4×Q8—1221no—
273C4×Q8—1221no—
274C4×Q8—1222no—
275C4×Q8—1222no—
276C4×Q8C22×D6111yescharacteristic
277#117—1222no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—1221no—
280C2×C42—611no—
281C2×C42—1221no—
282#107—21no—
283#107—21no—
284#107—22no—
285#107—1222no—
286#106—1222no—
287#106—1222no—
288C4⋊C8—21no—
289C4⋊C8—611no—
290C4⋊C8—612no—
291C4⋊C8—1221no—
292C4⋊C8—1221no—
293C4⋊C8—1222no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8C22×D6111yescharacteristic
297C4⋊C8C22×D6112yes—
298#103—22no—
299#103—1222no—
300#103—1222no—
301#103—1222no—
302#103—1222no—
303#103—1222no—
304C4×C8—21no—
305C4×C8—611no—
306C4×C8—611no—
307C4×C8—611no—
308C4×C8—1221no—
309C4×C8—1221no—
310C4×C8—1221no—
311C4×C8C22×D6111yescharacteristic
312C2×Q8—22no—
313C2×Q8—614no—
314C2×Q8—611no—
315C2×Q8—1221no—
316C2×Q8—1222no—
317C2×Q8—1222no—
318C2×Q8D4×D6111yescharacteristic
319C22×C4—611no—
320C22×C4—611no—
321C22×C4—611no—
322C22×C4—1221no—
323C22×C4—1221no—
324Q16—2442no—
325Q16—2442no—
326M16—2444no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—1222no—
332C2×C8—1221no—
333C2×C8—1222no—
334C2×C8—2441no—
335C2×C8—2442no—
336C2×C8D4×D6111yescharacteristic
337C2×C8D4×D6111yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—611no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—1222no—
347C4⋊C4—1222no—
348C4⋊C4—2442no—
349C4⋊C4—2442no—
350C4⋊C4D4×D6111yescharacteristic
351C22⋊C4—2442no—
352C42—21no—
353C42—611no—
354C42—611no—
355C42—611no—
356C42—1221no—
357C42—1221no—
358C42—1221no—
359C42—2442no—
360C42C23×D6111yescharacteristic
361C23—611no—
362Q8—21no—
363Q8—82no—
364Q8—612no—
365Q8—1224no—
366Q8—1221no—
367Q8—2442no—
368Q8—2442no—
369Q8Q16×D6112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—611no—
374C2×C4—612no—
375C2×C4—612no—
376C2×C4—611no—
377C2×C4—611no—
378C2×C4—1224no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—2441no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4C2×D4×D6111yescharacteristic
387C2×C4C2×D4×D6111yescharacteristic
388C2×C4#41×D6111yescharacteristic
389C8—21no—
390C8—42no—
391C8—41no—
392C8—1221no—
393C8—1221no—
394C8—2442no—
395C8—2441no—
396C8#1451980912yes—
397C22—612no—
398C22—1221no—
399C22—1221no—
400C22#121×D6111yescharacteristic
401C4—22no—
402C4—21no—
403C4—41no—
404C4—82no—
405C4—612no—
406C4—1222no—
407C4—1221no—
408C4—1221no—
409C4—4882no—
410C4—2441no—
411C4C2×Q16×D6111yescharacteristic
412C4#135×D6111yescharacteristic
413C4C2×#1451980911yescharacteristic
414C4#1684056111yescharacteristic
415C2—1222no—
416C2#2145529211yescharacteristic
417C2#467×D6111yescharacteristic
418C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1676C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1676D61
hypocenterC61#19262
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1676

Complements

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

NH ≅ G/Ntype
4. #21445280C2semidirect
5. #21445279C2semidirect
6. #21445220C2semidirect
7. #21445190C2semidirect
8. #21445189C2semidirect
11. C61×#1676C2semidirect
105. Q8×D61Q16semidirect
212. #1676D61semidirect
256. C61#19262semidirect

Maximal subgroups

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

subgroupindex
#214564062
#214556252
C8×Q8×D612
#214452802
#214452792
#214452202
#214451902
#214451892
#214445762
#214445062
C61×#16762
#1926261