Finite Group Databasebeta

← #29663257

Subgroups — #29663257

Order 15616.

606 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#21456766C212yesmaximal
2#500×D61C211yesmaximal, characteristic
3#21454750C211yesmaximal, characteristic
4#21445338C211yesmaximal, characteristic, complemented
5#21445337C211yesmaximal, characteristic, complemented
6#21445334C211yesmaximal, characteristic, complemented
7#21445212C212yesmaximal, complemented
8#21445211C212yesmaximal, complemented
9#21444564C211yesmaximal, characteristic
10#21444502C212yesmaximal
11C61×#1694C211yesmaximal, characteristic, complemented
12#16840316C2211yescharacteristic
13C4⋊Q8×D61C2212yes—
14#16840167C2211yescharacteristic
15#16840166—21no—
16#16840166—21no—
17#16840165C2212yes—
18#16840140—22no—
19#16840111—22no—
20#16839973C2212yes—
21#16839972C2212yes—
22#16839969C2212yes—
23#16839919—22no—
24#16839916—22no—
25#16839892C2211yescharacteristic
26#16839890—21no—
27Q8×#92418C2211yescharacteristic
28#16839886—21no—
29#16839886C2211yescharacteristic
30#16839814C2212yes—
31#107×D61—21no—
32#107×D61—21no—
33#16839787—21no—
34#16839784C2211yescharacteristic
35#16839719—21no—
36C4×C8×D61C2211yescharacteristic
37#16839514C2212yes—
38#16839513C2211yescharacteristic
39#16839509C2212yes—
40#16839508C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#462C2212yes—
44C61×#446C2211yescharacteristic
45C61×#334C2211yescharacteristic
46C61×#332C2212yes—
47C2×Q8×D61—22no—
48C2×Dic244—22no—
49C2×#11758202—22no—
50C2×#11758202—22no—
51C2×#11758189—21no—
52C2×C8×D61D411yescharacteristic
53C2×C8×D61D411yescharacteristic
54#14519777C2312yes—
55#14519776—22no—
56#14519776—22no—
57#14519776D411yescharacteristic
58#14519775—22no—
59#14519775D411yescharacteristic
60#14519767—21no—
61#14519757D411yescharacteristic
62#14519755—21no—
63#14519753D411yescharacteristic
64C4×Dic122—21no—
65C4×Dic122C2311yescharacteristic
66#14519734—21no—
67C4⋊C4×D61—22no—
68C4⋊C4×D61—22no—
69C42×D61C2311yescharacteristic
70#14519711—21no—
71#14519711—21no—
72#14519710—22no—
73#14519710—22no—
74C4⋊#92418—21no—
75C4⋊#92418C2312yes—
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519706—21no—
80#14519702—22no—
81#14519701—22no—
82#14519700—22no—
83#14519689—22no—
84#14519687—21no—
85C4×#92418—21no—
86C4×#92418C2311yescharacteristic
87C8×Dic61—22no—
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×Q16×C61—22no—
92C4⋊Q8×C61C2312yes—
93C4×Q8×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2312yes—
98C4⋊C8×C61C2311yescharacteristic
99#103×C61—22no—
100#97×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104Q8×D61—82no—
105Q8×D61—82no—
106C2×C4×D61—42no—
107C2×C4×D61C2×D411yescharacteristic
108C2×C4×D61C2×D411yescharacteristic
109C2×C4×D61#4111yescharacteristic
110C2×Dic122—21no—
111C2×Dic122—22no—
112C2×Dic122—42no—
113C2×Dic122C2×D411yescharacteristic
114Q16×C61—82no—
115C2×C8×C61—21no—
116C2×C8×C61—22no—
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61C2×D411yescharacteristic
123C42×C61—21no—
124C42×C61C2411yescharacteristic
125#11758202—82no—
126#11758202—82no—
127#11758198—21no—
128#11758198—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61C2×D411yescharacteristic
133#11758196—21no—
134#11758196—21no—
135#11758196—21no—
136#11758196—21no—
137#11758196—21no—
138#11758196—42no—
139#11758196—42no—
140C4×Dic61—21no—
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61#4111yescharacteristic
144C2×#92418—22no—
145C2×#92418—21no—
146C2×#92418C2×D411yescharacteristic
147C2×#92418C2×D411yescharacteristic
148Dic244—82no—
149#11758189—42no—
150C8×D61—22no—
151C8×D61Q1614yescomplemented
152C22×D61#12111yescharacteristic
153Q8×C61—22no—
154Q8×C61—42no—
155Q8×C61—42no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—42no—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C2×#4111yescharacteristic
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—42no—
165C2×Dic61#12111yescharacteristic
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12111yescharacteristic
168C4×D61—22no—
169C4×D61—22no—
170C4×D61—84no—
171C4×D61C2×Q1612yes—
172C4×D61#13712yes—
173Dic122—22no—
174Dic122—42no—
175Dic122—42no—
176Dic122—42no—
177Dic122—84no—
178C8×C61—21no—
179C8×C61—41no—
180C8×C61—42no—
181C8×C61C2×Q1612yes—
182#92418—21no—
183#92418—41no—
184#92418—42no—
185#92418C2×Q1612yes—
186#19208—611nomaximal, Sylow
187C22×C61C2×#12111yescharacteristic
188C2×D61—21no—
189C2×D61—21no—
190C2×D61#46212yes—
191C4×C61—21no—
192C4×C61—41no—
193C4×C61—41no—
194C4×C61—82no—
195C4×C61C22×Q1611yescharacteristic
196C4×C61C2×#13711yescharacteristic
197C4×C61#57612yes—
198Dic61—21no—
199Dic61—41no—
200Dic61—41no—
201Dic61—82no—
202Dic61#46212yes—
203Dic61#47312yes—
204#3209—612no—
205C2×#500—611no—
206#2960—611no—
207#1694—611no—
208#1694—611no—
209#1694—611no—
210#1694D6111yescharacteristic, complemented
211#1674—612no—
212#1674—612no—
213#1564—611no—
214#1549—612no—
215C2×C61C2×#46211yescharacteristic
216C2×C61#305811yescharacteristic
217C2×C61C2×#47311yescharacteristic
218D61—22no—
219#533—611no—
220C2×C4⋊Q8—612no—
221#500—612no—
222#500—611no—
223#500—1221no—
224#500—1221no—
225#500C2×D6111yescharacteristic
226#494—1222no—
227#487—1222no—
228#462—612no—
229#462—612no—
230#462—612no—
231#462C2×D6112yes—
232#451—1222no—
233#451—1222no—
234#446—611no—
235#446—1221no—
236#446C2×D6111yescharacteristic
237C8×Q8—611no—
238C8×Q8—611no—
239C8×Q8—1221no—
240#431—612no—
241C2×#107—1221no—
242C2×#107—1221no—
243#424—1221no—
244#423—611no—
245#405—1221no—
246C2×C4×C8—611no—
247#335—612no—
248#334—611no—
249#334C2×D6111yescharacteristic
250#332—612no—
251#332—612no—
252#332—612no—
253#332C2×D6112yes—
254C61#1920811yescharacteristic, complemented, Sylow
255C22×Q8—1222no—
256C2×Q16—22no—
257C2×Q16—1222no—
258C2×Q16—1222no—
259C2×Q16—1222no—
260C2×M16—1221no—
261C22×C8—611no—
262C22×C8—611no—
263C4⋊Q8—611no—
264C4⋊Q8—611no—
265C4⋊Q8—612no—
266C4⋊Q8—1222no—
267C4⋊Q8—1222no—
268C4⋊Q8—1222no—
269C4⋊Q8C22×D6112yes—
270#125—1221no—
271C22⋊Q8—611no—
272C22⋊Q8—611no—
273C22⋊Q8—1221no—
274C4×Q8—611no—
275C4×Q8—1221no—
276C4×Q8C22×D6111yescharacteristic
277#117—1221no—
278C2×C4⋊C4—1222no—
279C2×C4⋊C4—1222no—
280C2×C42—611no—
281#107—21no—
282#107—21no—
283#107—1222no—
284#107—1222no—
285#107—1221no—
286#107—1221no—
287C4⋊C8—21no—
288C4⋊C8—611no—
289C4⋊C8—612no—
290C4⋊C8—1221no—
291C4⋊C8—1221no—
292C4⋊C8—1221no—
293C4⋊C8—1221no—
294C4⋊C8C22×D6111yescharacteristic
295C4⋊C8C22×D6112yes—
296#103—22no—
297#103—1222no—
298#103—1222no—
299#103—1222no—
300C22⋊C8—1222no—
301#97—21no—
302#97—1221no—
303C4×C8—611no—
304C4×C8—611no—
305C4×C8—611no—
306C4×C8—1221no—
307C4×C8—1222no—
308C4×C8—1221no—
309C4×C8C22×D6111yescharacteristic
310C2×Q8—22no—
311C2×Q8—611no—
312C2×Q8—1221no—
313C2×Q8—1222no—
314C2×Q8—4882no—
315C2×Q8—4882no—
316C2×Q8—2442no—
317C2×Q8D4×D6111yescharacteristic
318C22×C4—611no—
319C22×C4—611no—
320C22×C4—611no—
321C22×C4—2442no—
322Q16—82no—
323Q16—4882no—
324Q16—4882no—
325Q16—4882no—
326M16—2442no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—611no—
330C2×C8—614no—
331C2×C8—611no—
332C2×C8—1222no—
333C2×C8—1221no—
334C2×C8—1222no—
335C2×C8D4×D6111yescharacteristic
336C2×C8D4×D6111yescharacteristic
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—611no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—2442no—
350C4⋊C4—2442no—
351C4⋊C4D4×D6111yescharacteristic
352C22⋊C4—1221no—
353C22⋊C4—1221no—
354C42—21no—
355C42—611no—
356C42—611no—
357C42—611no—
358C42—1221no—
359C42C23×D6111yescharacteristic
360C23—611no—
361Q8—22no—
362Q8—42no—
363Q8—42no—
364Q8—1222no—
365Q8—4884no—
366Q8—2442no—
367Q8—2442no—
368Q8—2442no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—42no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—612no—
375C2×C4—612no—
376C2×C4—611no—
377C2×C4—1222no—
378C2×C4—1222no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—4884no—
382C2×C4—2442no—
383C2×C4C2×D4×D6111yescharacteristic
384C2×C4C2×D4×D6111yescharacteristic
385C2×C4#41×D6111yescharacteristic
386C8—21no—
387C8—41no—
388C8—42no—
389C8—612no—
390C8—1221no—
391C8—2441no—
392C8—2442no—
393C8Q16×D6112yes—
394C22—612no—
395C22—1221no—
396C22—1221no—
397C22#121×D6111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—612no—
403C4—612no—
404C4—1221no—
405C4—4882no—
406C4—2441no—
407C4—2441no—
408C4C2×Q16×D6111yescharacteristic
409C4#137×D6111yescharacteristic
410C4#1684056612yes—
411C2—1222no—
412C2#462×D6111yescharacteristic
413C2#2145546511yescharacteristic
414C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1694C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1694D61
hypocenterC61#19208
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1694

Complements

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

NH ≅ G/Ntype
4. #21445338C2semidirect
5. #21445337C2semidirect
6. #21445334C2semidirect
7. #21445212C2semidirect
8. #21445211C2semidirect
11. C61×#1694C2semidirect
151. C8×D61Q16semidirect
210. #1694D61semidirect
254. C61#19208semidirect

Maximal subgroups

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

subgroupindex
#214567662
#500×D612
#214547502
#214453382
#214453372
#214453342
#214452122
#214452112
#214445642
#214445022
C61×#16942
#1920861