Finite Group Databasebeta

← #30604987

Subgroups — #30604987

Order 15616.

622 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840642C211yesmaximal, characteristic, complemented
2#21461069C211yesmaximal, characteristic, complemented
3#21460820C212yesmaximal, complemented
4#21460801C212yesmaximal, complemented
5#21449150C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840732—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608—21no—
11#16840608C2212yes—
12C2×#14519758—21no—
13C22×#11758196C2211yescharacteristic
14#16839681C411yescharacteristic
15#16839681C2211yescharacteristic
16#16839676C411yescharacteristic
17#16839676C2211yescharacteristic
18#16839642—21no—
19#16839635—21no—
20#16839635—21no—
21#16839625C411yescharacteristic
22C22×#41×C61C411yescharacteristic
23C2×#11758217—24no—
24C2×#11758217—22no—
25C2×#11758217—41no—
26C2×#11758217—42no—
27C2×#11758217—41no—
28C22×#92420—41no—
29C22⋊#92420—22no—
30C22⋊#92420—41no—
31C23×Dic61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33#14519758—44no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196—41no—
37C2×#11758196—41no—
38C2×#11758196C2×C411yescharacteristic
39C2×#11758196D411yescharacteristic
40C2×#11758196D411yescharacteristic
41C2×#11758196C2311yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684C2×C411yescharacteristic
51#14519684C2×C411yescharacteristic
52C2×#41×C61—41no—
53C2×#41×C61—41no—
54C2×#41×C61—44no—
55C22×Q8×C61C2×C411yescharacteristic
56C22×D4×C61—21no—
57C22×D4×C61C2×C411yescharacteristic
58C23×C4×C61—21no—
59C23×C4×C61C2×C411yescharacteristic
60C24×C61—21no—
61C24×C61C22⋊C411yescharacteristic
62#41×C61—44no—
63#41×C61—44no—
64#41×C61—44no—
65#41×C61—44no—
66C2×Q8×C61—42no—
67C2×Q8×C61—41no—
68C2×D4×C61—24no—
69C2×D4×C61—41no—
70C2×D4×C61—42no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—42no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61—42no—
80C22×C4×C61—42no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—22no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—42no—
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C2×D411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97#11758217—24no—
98#11758217—44no—
99#11758217—42no—
100#11758217—82no—
101C2×#92420—44no—
102C2×#92420—42no—
103C2×#92420—42no—
104C22⋊Dic61—42no—
105C22⋊Dic61—42no—
106C22⋊Dic61—42no—
107C22⋊Dic61—42no—
108C22⋊Dic61—41no—
109C22⋊Dic61—41no—
110#11758196—41no—
111#11758196—41no—
112#11758196—41no—
113#11758196—41no—
114#11758196—42no—
115#11758196—42no—
116C23×C61—22no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—22no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61#9911yescharacteristic
128C23×C61#9911yescharacteristic
129C23×C61C2×C22⋊C411yescharacteristic
130Q8×C61—42no—
131Q8×C61—42no—
132D4×C61—44no—
133D4×C61—42no—
134D4×C61—42no—
135D4×C61—44no—
136C2×C4×C61—22no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—48no—
142C2×C4×C61—42no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—22no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—81no—
166C2×Dic61#10412yes—
167C2×Dic61AGL(1,ℤ/8)11yescharacteristic
168C2×Dic61#13711yescharacteristic
169#92420—44no—
170#92420—82no—
171#11762—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—42no—
182C22×C61—42no—
183C22×C61—42no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61AΣL(1,𝔽16)12yes—
189C22×C61C2×#9911yescharacteristic
190C22×C61#41711yescharacteristic
191C22×C61C2×#10411yescharacteristic
192C4×C61—41no—
193C4×C61—42no—
194C4×C61—41no—
195Dic61—21no—
196Dic61—22no—
197Dic61—21no—
198Dic61—41no—
199Dic61—41no—
200Dic61—81no—
201Dic61—81no—
202#2406—611no—
203C2×#343—611no—
204#1463—611no—
205#1354—612no—
206#1345—612no—
207C2×C61—21no—
208C2×C61—21no—
209C2×C61—22no—
210C2×C61—41no—
211C2×C61—82no—
212C2×C61#150411yescharacteristic
213C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
214C2×C61#213311yescharacteristic
215C22×#41#692311yescharacteristic
216C22×M16—1221no—
217C2×#123—1221no—
218C22×C4⋊C4—611no—
219#407—1221no—
220#395—611no—
221#395—611no—
222#394—611no—
223#394—611no—
224#386—1221no—
225#385—1221no—
226#385—1221no—
227#381—611no—
228#343—612no—
229#343—1221no—
230#324—612no—
231#324—1221no—
232C61#1176211yescharacteristic, complemented, Sylow
233C2×#41—41no—
234C2×#41—41no—
235C2×#41—44no—
236C22×Q8C2×#692311yescharacteristic
237C22×D4—21no—
238C22×D4C2×#692311yescharacteristic
239C23×C4—21no—
240C23×C4—611no—
241C23×C4—611no—
242C23×C4C2×#692311yescharacteristic
243C2×M16—1222no—
244C2×M16—1224no—
245C2×M16—2441no—
246C2×M16—2442no—
247C2×M16—2441no—
248C22×C8—2441no—
249#123—2444no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—611no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—2441no—
257C2×C4⋊C4—2441no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C22⋊C8—1222no—
264C22⋊C8—2441no—
265#95—611no—
266#95—611no—
267#95—1221no—
268#95—1221no—
269#95—1221no—
270C24—21no—
271C24C22⋊#692311yescharacteristic
272#41—44no—
273#41—44no—
274#41—44no—
275#41—44no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×D4—24no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—611no—
292C22×C4—611no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—1222no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—2442no—
305C22×C4C22⋊#692311yescharacteristic
306C22×C4C22×#692311yescharacteristic
307M16—1224no—
308M16—4882no—
309M16—2442no—
310M16—2444no—
311C2×C8—2442no—
312C2×C8—2444no—
313C2×C8—2442no—
314C4⋊C4—2441no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C4⋊C4—2442no—
319C4⋊C4—2442no—
320C22⋊C4—2441no—
321C22⋊C4—2441no—
322C22⋊C4—2442no—
323C22⋊C4—2442no—
324C22⋊C4—2442no—
325C22⋊C4—2442no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#1451983511yescharacteristic
338C23#1451983511yescharacteristic
339C23C2×C22⋊#692311yescharacteristic
340Q8—42no—
341Q8—42no—
342D4—44no—
343D4—42no—
344D4—42no—
345D4—44no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—611no—
357C2×C4—611no—
358C2×C4—612no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—1222no—
362C2×C4—1222no—
363C2×C4—1221no—
364C2×C4—1221no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—1221no—
369C2×C4—4881no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2442no—
377C2×C4—2441no—
378C2×C4—2441no—
379C8—4882no—
380C8—2444no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22#1684066212yes—
398C22C2×#1451983511yescharacteristic
399C22#1684075311yescharacteristic
400C22D61⋊#10411yescharacteristic
401C4—41no—
402C4—42no—
403C4—41no—
404C4—1221no—
405C4—1222no—
406C4—1221no—
407C4—4881no—
408C4—4881no—
409C4—2441no—
410C4—2441no—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2—82no—
416C2#2146120511yescharacteristic
417C2C2×#1684066211yescharacteristic
418C2#2146211411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
Fitting, prime radicalC22×#41×C61C4
socleC22×C61C2×#99
cosocleC2×#11758196C23
hypercenterC22×#41#6923
hypocenterC61#11762
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840642C2semidirect
2. #21461069C2semidirect
3. #21460820C2semidirect
4. #21460801C2semidirect
232. C61#11762semidirect

Maximal subgroups

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

subgroupindex
C2×#168406422
#214610692
#214608202
#214608012
#214491502
#1176261