Finite Group Databasebeta

← #29662656

Subgroups — #29662656

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456330C211yesmaximal, characteristic
2#21455860C212yesmaximal
3#446×D61C211yesmaximal, characteristic
4#21445344C211yesmaximal, characteristic, complemented
5#21445338C211yesmaximal, characteristic, complemented
6#21445337C211yesmaximal, characteristic, complemented
7#21445328C212yesmaximal, complemented
8#21445327C212yesmaximal, complemented
9#21444564C211yesmaximal, characteristic
10#21444473C212yesmaximal
11C61×#1696C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13C4×Q8×D61C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840163—21no—
16#16840163—21no—
17#16840158C2212yes—
18#16840099—22no—
19#16840054C2212yes—
20#16839973C2212yes—
21#16839972C2212yes—
22#16839919—22no—
23#16839893C2211yescharacteristic
24#16839892C2211yescharacteristic
25#16839891—21no—
26#16839890—21no—
27#16839889C2211yescharacteristic
28#16839856—22no—
29#16839854—22no—
30#16839814C2212yes—
31#16839799—21no—
32#16839799—21no—
33C4⋊C8×D61—21no—
34C4⋊C8×D61C2211yescharacteristic
35#97×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839514C2212yes—
38#16839513C2211yescharacteristic
39#16839511C2212yes—
40#16839509C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#477C2212yes—
44C61×#446C2211yescharacteristic
45C61×#334C2211yescharacteristic
46C61×#333C2212yes—
47C2×Q8×D61D411yescharacteristic
48C2×Dic244—22no—
49C2×#11758202—22no—
50C2×C8×D61—21no—
51C2×C8×D61D411yescharacteristic
52C2×C8×D61D411yescharacteristic
53#14519777C2312yes—
54#14519768—22no—
55#14519757—22no—
56#14519753—22no—
57Q8×Dic61—22no—
58Q8×Dic61D411yescharacteristic
59C4×Dic122—21no—
60C4×Dic122C2311yescharacteristic
61#14519742—21no—
62#14519742—22no—
63#14519742D411yescharacteristic
64#14519735—22no—
65C4⋊C4×D61—21no—
66C4⋊C4×D61D411yescharacteristic
67C42×D61—21no—
68C42×D61C2311yescharacteristic
69#14519711—21no—
70#14519711—21no—
71#14519710—22no—
72#14519708—22no—
73#14519708—22no—
74C4⋊#92418—21no—
75C4⋊#92418C2312yes—
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519706—21no—
80#14519702—22no—
81#14519700—22no—
82#14519700—22no—
83#14519689—22no—
84#14519688—21no—
85#14519687—21no—
86#14519687—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—22no—
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×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—42no—
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×C4×D61C2×D411yescharacteristic
108C2×C4×D61C2×D411yescharacteristic
109C2×C4×D61#4111yescharacteristic
110C2×Dic122—21no—
111C2×Dic122—22no—
112C2×Dic122C2×D411yescharacteristic
113C2×C8×C61—21no—
114C2×C8×C61—22no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C4⋊C4×C61—21no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61C2×D411yescharacteristic
121C42×C61—21no—
122C42×C61C2411yescharacteristic
123#11758202—44no—
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—22no—
142C2×#92418—21no—
143C2×#92418C2×D411yescharacteristic
144C2×#92418C2×D411yescharacteristic
145Dic244—44no—
146C8×D61—22no—
147C8×D61—42no—
148C8×D61Q1614yescomplemented
149C22×D61#12111yescharacteristic
150Q8×C61—22no—
151Q8×C61—82no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—42no—
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C2×#4111yescharacteristic
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—42no—
161C2×Dic61#12111yescharacteristic
162C2×Dic61#12111yescharacteristic
163C2×Dic61#12111yescharacteristic
164C4×D61—22no—
165C4×D61—41no—
166C4×D61—42no—
167C4×D61—42no—
168C4×D61C2×Q1612yes—
169C4×D61#13712yes—
170Dic122—22no—
171Dic122—42no—
172Dic122—42no—
173Dic122—42no—
174C8×C61—21no—
175C8×C61—41no—
176C8×C61—42no—
177C8×C61C2×Q1612yes—
178#92418—21no—
179#92418—41no—
180#92418—42no—
181#92418C2×Q1612yes—
182#19150—611nomaximal, Sylow
183C22×C61C2×#12111yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61#46212yes—
187C4×C61—21no—
188C4×C61—41no—
189C4×C61—41no—
190C4×C61—82no—
191C4×C61C22×Q1611yescharacteristic
192C4×C61C2×#13711yescharacteristic
193C4×C61#57512yes—
194Dic61—21no—
195Dic61—21no—
196Dic61—41no—
197Dic61—41no—
198Dic61—82no—
199Dic61#46212yes—
200#3160—611no—
201#3104—612no—
202C2×#446—611no—
203#1696—611no—
204#1696D6111yescharacteristic, complemented
205#1694—611no—
206#1694—611no—
207#1692—612no—
208#1692—612no—
209#1564—611no—
210#1542—612no—
211C2×C61C2×#46211yescharacteristic
212C2×C61#305411yescharacteristic
213C2×C61#306711yescharacteristic
214D61—22no—
215#533—612no—
216C2×C4×Q8—611no—
217#500—611no—
218#500C2×D6111yescharacteristic
219#499—1221no—
220#499—1221no—
221#498—612no—
222#484—1222no—
223#477—612no—
224#477C2×D6112yes—
225#462—612no—
226#462—612no—
227#451—1222no—
228#446—611no—
229#446—611no—
230#446—611no—
231#446—1221no—
232#446—1221no—
233#446C2×D6111yescharacteristic
234C4×Q16—1222no—
235C4×Q16—1222no—
236#431—612no—
237#427—1221no—
238#427—1221no—
239C2×C4⋊C8—611no—
240C2×C4⋊C8—1221no—
241C2×#97—1221no—
242C2×C4×C8—611no—
243#335—612no—
244#334—611no—
245#334C2×D6111yescharacteristic
246#333—612no—
247#333C2×D6112yes—
248#332—612no—
249#332—612no—
250C61#1915011yescharacteristic, complemented, Sylow
251C22×Q8—611no—
252C2×Q16—1222no—
253C2×Q16—1222no—
254C22×C8—611no—
255C22×C8—611no—
256C22×C8—1221no—
257C4⋊Q8—612no—
258C4⋊Q8C22×D6112yes—
259#125—1222no—
260C22⋊Q8—1222no—
261C22⋊Q8—1222no—
262C4×Q8—611no—
263C4×Q8—611no—
264C4×Q8—611no—
265C4×Q8—1221no—
266C4×Q8—1221no—
267C4×Q8—1222no—
268C4×Q8—1222no—
269C4×Q8C22×D6111yescharacteristic
270#117—1222no—
271C2×C4⋊C4—611no—
272C2×C4⋊C4—1221no—
273C2×C42—611no—
274C2×C42—1221no—
275#107—21no—
276#107—21no—
277#107—22no—
278#107—1222no—
279#107—1221no—
280#107—1221no—
281#106—1222no—
282#106—1222no—
283C4⋊C8—21no—
284C4⋊C8—611no—
285C4⋊C8—612no—
286C4⋊C8—1221no—
287C4⋊C8—1221no—
288C4⋊C8—1221no—
289C4⋊C8—1221no—
290C4⋊C8C22×D6111yescharacteristic
291C4⋊C8C22×D6112yes—
292#103—22no—
293#103—1222no—
294#103—1222no—
295#103—1222no—
296C22⋊C8—1222no—
297#97—21no—
298#97—1221no—
299#97—1221no—
300#97—1221no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—611no—
304C4×C8—1222no—
305C4×C8C22×D6111yescharacteristic
306C2×Q8—22no—
307C2×Q8—611no—
308C2×Q8—1221no—
309C2×Q8—1222no—
310C2×Q8—2442no—
311C2×Q8D4×D6111yescharacteristic
312C22×C4—611no—
313C22×C4—611no—
314C22×C4—611no—
315C22×C4—1221no—
316C22×C4—1221no—
317Q16—2444no—
318Q16—2444no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—611no—
322C2×C8—614no—
323C2×C8—611no—
324C2×C8—1222no—
325C2×C8—1222no—
326C2×C8—1221no—
327C2×C8—2442no—
328C2×C8D4×D6111yescharacteristic
329C2×C8D4×D6111yescharacteristic
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—611no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—1222no—
339C4⋊C4—1222no—
340C4⋊C4—2442no—
341C4⋊C4—2442no—
342C4⋊C4D4×D6111yescharacteristic
343C22⋊C4—2442no—
344C42—21no—
345C42—611no—
346C42—611no—
347C42—611no—
348C42—1221no—
349C42—1221no—
350C42—1221no—
351C42—2442no—
352C42C23×D6111yescharacteristic
353C23—611no—
354Q8—22no—
355Q8—82no—
356Q8—1222no—
357Q8—2442no—
358Q8—2442no—
359Q8—2442no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—611no—
364C2×C4—612no—
365C2×C4—612no—
366C2×C4—611no—
367C2×C4—611no—
368C2×C4—1222no—
369C2×C4—1221no—
370C2×C4—1221no—
371C2×C4—2441no—
372C2×C4—2442no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4C2×D4×D6111yescharacteristic
376C2×C4C2×D4×D6111yescharacteristic
377C2×C4#41×D6111yescharacteristic
378C8—21no—
379C8—41no—
380C8—42no—
381C8—612no—
382C8—1221no—
383C8—2441no—
384C8—2442no—
385C8Q16×D6112yes—
386C22—612no—
387C22—1221no—
388C22—1221no—
389C22#121×D6111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—82no—
394C4—612no—
395C4—1221no—
396C4—1221no—
397C4—4882no—
398C4—2441no—
399C4—2441no—
400C4C2×Q16×D6111yescharacteristic
401C4#137×D6111yescharacteristic
402C4#1684056112yes—
403C2—1222no—
404C2#462×D6111yescharacteristic
405C2#2145543011yescharacteristic
406C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1696C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1696D61
hypocenterC61#19150
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1696

Complements

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

NH ≅ G/Ntype
4. #21445344C2semidirect
5. #21445338C2semidirect
6. #21445337C2semidirect
7. #21445328C2semidirect
8. #21445327C2semidirect
11. C61×#1696C2semidirect
148. C8×D61Q16semidirect
204. #1696D61semidirect
250. C61#19150semidirect

Maximal subgroups

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

subgroupindex
#214563302
#214558602
#446×D612
#214453442
#214453382
#214453372
#214453282
#214453272
#214445642
#214444732
C61×#16962
#1915061