Finite Group Databasebeta

← #29665298

Subgroups — #29665298

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456746C211yesmaximal, characteristic
2C2×#16840047C211yesmaximal, characteristic, complemented
3#21455291C211yesmaximal, characteristic
4#21445249C214yesmaximal, complemented
5#21445167C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444503C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic
9C61×#2982C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840135C2212yes—
14#16840113C2212yes—
15#16840047—22no—
16#16840047C2214yes—
17#16839984—22no—
18#16839972C2212yes—
19#16839968—22no—
20#16839962C2212yes—
21#16839903—22no—
22C2×#14519708—21no—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839761—21no—
26C2×#14519701—21no—
27#16839513C2214yes—
28#16839509C2214yes—
29#16839507C2212yes—
30#16839505C2212yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C8×Q8×C61—22no—
33C8×Q8×C61C2214yes—
34C61×#431—22no—
35C61×#431C2211yescharacteristic
36C61×#431C2211yescharacteristic
37C2×#11758222—21no—
38C22×Dic122—21no—
39#14519808—44no—
40C2×#11758202—22no—
41C2×#11758190—24no—
42C2×#11758201—22no—
43C2×#11758194—21no—
44C22×#92418—21no—
45#14519777—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777C2311yescharacteristic
49#14519777C2312yes—
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58#14519708—24no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519702—24no—
64#14519701—24no—
65#14519700—22no—
66#14519697—22no—
67C22×Q8×C61D411yescharacteristic
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61D412yes—
71C4×Q8×C61C2312yes—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2311yescharacteristic
76C4⋊C8×C61—24no—
77C4⋊C8×C61D412yes—
78C4⋊C8×C61C2312yes—
79C22⋊C8×C61—21no—
80C22⋊C8×C61D412yes—
81C4×C8×C61—24no—
82C4×C8×C61—21no—
83C4×C8×C61C2312yes—
84C2×Q8×C61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61SD1614yescomplemented
87C2×Q8×C61C2×D412yes—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C2×C22⋊D61—41no—
94C22×Dic61—41no—
95#11758222—42no—
96#11758222—42no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×C4×D61—41no—
101C2×Dic122—22no—
102C2×Dic122—21no—
103C2×Dic122—41no—
104C2×Dic122—82no—
105C2×C8×C61—22no—
106C2×C8×C61C2×D414yes—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61C2×D412yes—
111C42×C61—22no—
112C42×C61—22no—
113C42×C61C2×D411yescharacteristic
114C42×C61C2×D411yescharacteristic
115C42×C61#4111yescharacteristic
116C42×C61C2411yescharacteristic
117#11758202—44no—
118#11758201—44no—
119#11758198—41no—
120#11758198—41no—
121C4⋊Dic61—22no—
122C4⋊Dic61—21no—
123C4⋊Dic61—22no—
124C4⋊Dic61—21no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127#11758194—42no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418—42no—
131#11758190—84no—
132C23×C61#12111yescharacteristic
133C22×D61—41no—
134Q8×C61—22no—
135Q8×C61—24no—
136Q8×C61C2×SD1614yes—
137C2×C4×C61—22no—
138C2×C4×C61—24no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—41no—
142C2×C4×C61—81no—
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61C2×SD1612yes—
147C2×C4×C61#13512yes—
148C2×C4×C61C22×D411yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C2×#4111yescharacteristic
151C22⋊D61—82no—
152C2×Dic61—42no—
153C2×Dic61—41no—
154C2×Dic61—82no—
155D244—41no—
156D244—41no—
157D244—42no—
158D244—81no—
159C4×D61—82no—
160Dic122—41no—
161Dic122—41no—
162Dic122—44no—
163Dic122—82no—
164C8×C61—24no—
165C8×C61—42no—
166#92418—42no—
167#92418—42no—
168#19397—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61#46512yes—
172C22×C61C2×#12111yescharacteristic
173C2×D61—81no—
174C2×D61—81no—
175C2×D61—81no—
176C4×C61—21no—
177C4×C61—24no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—42no—
181C4×C61#46512yes—
182C4×C61C22×SD1611yescharacteristic
183C4×C61C2×#13511yescharacteristic
184C4×C61#57712yes—
185Dic61—82no—
186Dic61—81no—
187#3206—611no—
188C2×#475—611no—
189#3037—611no—
190#2982D6111yescharacteristic, complemented
191#1680—614no—
192#1666—614no—
193#1570—611no—
194#1549—611no—
195#1540—611no—
196C2×C61—22no—
197C2×C61#304311yescharacteristic
198C2×C61C2×#46511yescharacteristic
199C2×C61#306911yescharacteristic
200D61—161no—
201C2×#137—1221no—
202#532—611no—
203C2×C4⋊Q8—611no—
204C2×C4×Q8C2×D6111yescharacteristic
205#492—612no—
206#487—612no—
207#475—614no—
208#475—1222no—
209#464—1222no—
210#462—612no—
211#461—1222no—
212#460—612no—
213#448—1222no—
214C8×Q8—22no—
215C8×Q8C2×D6114yes—
216#431—22no—
217#431C2×D6111yescharacteristic
218#431C2×D6111yescharacteristic
219C2×#106—1221no—
220#423—611no—
221C2×C4⋊C8—611no—
222#417—1221no—
223C2×#103—1221no—
224#334—614no—
225#332—612no—
226#332—614no—
227#331—612no—
228C61#1939711yescharacteristic, complemented, Sylow
229C2×#41—1221no—
230C22×Q8—1221no—
231C22×Q8D24411yescharacteristic
232#137—2444no—
233C2×Q16—1222no—
234C2×SD16—1222no—
235C2×SD16—1224no—
236C2×M16—1221no—
237C22×C8—1221no—
238C4⋊Q8—612no—
239C4⋊Q8—611no—
240C4⋊Q8—1221no—
241C4⋊Q8—1221no—
242C4⋊Q8—1221no—
243#127—611no—
244#124—1221no—
245#121—1221no—
246#121—1221no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D24412yes—
250C4×Q8C22×D6112yes—
251C4×D4—1221no—
252C4×D4—1221no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4D24411yescharacteristic
257C2×C42—21no—
258C2×C42C22×D6111yescharacteristic
259#106—1224no—
260C4⋊C8—24no—
261C4⋊C8—612no—
262C4⋊C8—612no—
263C4⋊C8—1221no—
264C4⋊C8—1221no—
265C4⋊C8C22⋊D6112yes—
266C4⋊C8C22×D6112yes—
267#103—1224no—
268#103—1224no—
269#103—1222no—
270#102—1222no—
271C22⋊C8—21no—
272C22⋊C8C22⋊D6112yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C22×D6112yes—
276#41—2442no—
277#41—2442no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—1222no—
281C2×Q8—1221no—
282C2×Q8—4882no—
283C2×Q8—2441no—
284C2×Q8#1175819014yescomplemented
285C2×Q8C2×D24412yes—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—1221no—
289C2×D4—2441no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4C2×D24411yescharacteristic
295C22×C4#1175822211yescharacteristic
296C22×C4C2×C22⋊D6111yescharacteristic
297Q16—2444no—
298SD16—4884no—
299SD16—2444no—
300M16—2442no—
301C2×C8—22no—
302C2×C8—1222no—
303C2×C8—1222no—
304C2×C8—2442no—
305C2×C8C2×C22⋊D6114yes—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—1221no—
310C4⋊C4—1222no—
311C4⋊C4—1222no—
312C4⋊C4—1221no—
313C4⋊C4—2441no—
314C4⋊C4—2441no—
315C4⋊C4C2×D24412yes—
316C22⋊C4—2441no—
317C22⋊C4—2441no—
318C42—22no—
319C42—22no—
320C42C2×D24411yescharacteristic
321C42#1175822211yescharacteristic
322C42C2×C22⋊D6111yescharacteristic
323C42C23×D6111yescharacteristic
324C23—2441no—
325C23#1451975111yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—4882no—
329Q8—2441no—
330Q8—2441no—
331Q8—2444no—
332Q8C2×#1175819014yes—
333D4—4881no—
334D4—4882no—
335D4—2441no—
336D4—2441no—
337D4—2442no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—4882no—
345C2×C4—4882no—
346C2×C4—2441no—
347C2×C4—2442no—
348C2×C4#1451975111yescharacteristic
349C2×C4#1451975111yescharacteristic
350C2×C4#1451975111yescharacteristic
351C2×C4C2×#1175819012yes—
352C2×C4#1451979912yes—
353C2×C4C22×C22⋊D6111yescharacteristic
354C2×C4C22×D24411yescharacteristic
355C2×C4C2×#1175822211yescharacteristic
356C8—24no—
357C8—42no—
358C8—2442no—
359C8—2442no—
360C22—21no—
361C22—21no—
362C22—4881no—
363C22—4881no—
364C22—4881no—
365C22#1683999112yes—
366C22C2×#1451975111yescharacteristic
367C4—24no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—4881no—
373C4—4882no—
374C4#1683999112yes—
375C4C22×#1175819011yescharacteristic
376C4C2×#1451979911yescharacteristic
377C4#1684057012yes—
378C2—22no—
379C2—9761no—
380C2#2145534011yescharacteristic
381C2C2×#1683999111yescharacteristic
382C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2982C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#2982D61
hypocenterC61#19397
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#16840047C2semidirect
4. #21445249C2semidirect
5. #21445167C2semidirect
6. #21444590C2semidirect
7. #21444503C2semidirect
9. C61×#2982C2semidirect
86. C2×Q8×C61SD16semidirect
190. #2982D61semidirect
228. C61#19397semidirect
284. C2×Q8#11758190semidirect

Maximal subgroups

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

subgroupindex
#214567462
C2×#168400472
#214552912
#214452492
#214451672
#214445902
#214445032
#214444662
C61×#29822
#1939761