Finite Group Databasebeta

← #30599040

Subgroups — #30599040

Order 15616.

456 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 88 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
1#21461698C211yesmaximal, characteristic, complemented
2#21461695C211yesmaximal, characteristic, complemented
3#21461236C212yesmaximal
4#21460719C212yesmaximal, complemented
5#21454515C211yesmaximal, characteristic
6C2×#14519847—21no—
7C2×#14519845—21no—
8#16840733C2212yes—
9#16840704C2212yes—
10#16840638—21no—
11#16840636—21no—
12#16840615—22no—
13#16840614—22no—
14C2×#14519783C411yescharacteristic, complemented
15C22×C8×D61C2211yescharacteristic
16#16840274C2211yescharacteristic
17#16839835—21no—
18#16839835—21no—
19C8×D244—21no—
20C8×D244—21no—
21#16839789C411yescharacteristic, complemented
22#16839730C411yescharacteristic, complemented
23#16839725C2211yescharacteristic
24C61×#405C411yescharacteristic, complemented
25C2×#11758213—22no—
26C2×C4⋊#6923—22no—
27C2×C22⋊#6923—22no—
28#14519847—42no—
29#14519845—42no—
30#14519834—22no—
31C22⋊#92420—22no—
32(C2×#6923)⋊C4—22no—
33C2×#11758222C2×C411yescharacteristic
34C22×C4×D61C2311yescharacteristic
35#14519783—42no—
36C2×#11758194C2×C411yescharacteristic
37C2×#11758189—21no—
38C22×#92418C2×C411yescharacteristic
39C2×C8×D61—21no—
40C2×C8×D61—42no—
41C2×C8×D61D411yescharacteristic
42C2×C8×D61D411yescharacteristic
43C2×C8×D61D411yescharacteristic
44C2×C8×D61D411yescharacteristic
45#14519761—21no—
46#14519757D411yescharacteristic
47#14519757Q811yescharacteristic
48#14519752D411yescharacteristic
49#14519752Q811yescharacteristic
50#14519747—21no—
51C4×D244—21no—
52C4×D244—21no—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55#14519689—21no—
56#14519689—21no—
57#14519689—21no—
58#14519689—21no—
59C2×M16×C61C2×C411yescharacteristic
60C22×C8×C61C2×C411yescharacteristic
61#117×C61C2×C411yescharacteristic
62#97×C61—21no—
63C4×C8×C61—21no—
64C23×D61#4111yescharacteristic
65C22×#6923—42no—
66C22×C4×C61C22×C411yescharacteristic
67C2×C22⋊D61—21no—
68C22×Dic61#4111yescharacteristic
69#11758222—41no—
70#11758222—41no—
71C2×D244—21no—
72C2×D244—21no—
73C2×D244C4⋊C411yescharacteristic
74C2×C4×D61—21no—
75C2×C4×D61—41no—
76C2×C4×D61—41no—
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61C2×D411yescharacteristic
79C2×C4×D61C2×D411yescharacteristic
80C2×C4×D61C2×Q811yescharacteristic
81C2×Dic122C4⋊C411yescharacteristic
82C22⋊#6923—82no—
83C2×#92420—42no—
84C4⋊#6923—42no—
85C4⋊#6923—42no—
86#11758213—42no—
87#11758213—42no—
88M16×C61—41no—
89C2×C8×C61—22no—
90C2×C8×C61—21no—
91C2×C8×C61C22⋊C411yescharacteristic
92C2×C8×C61C22⋊C411yescharacteristic
93C4⋊C4×C61C4⋊C411yescharacteristic
94C4⋊C4×C61C4⋊C411yescharacteristic
95C22⋊C4×C61—21no—
96C42×C61—21no—
97#11758198—21no—
98#11758198—21no—
99#11758198—21no—
100#11758198—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103#11758194—41no—
104C2×#92418—21no—
105C2×#92418—41no—
106C2×#92418C22⋊C411yescharacteristic
107C2×#92418C22⋊C411yescharacteristic
108#11758189—42no—
109C8×D61—22no—
110C8×D61—22no—
111C8×D61—41no—
112C8×D61—41no—
113C8×D61—42no—
114C23×C61#11711yescharacteristic
115C22×D61—22no—
116C22×D61—21no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61#12111yescharacteristic
120C22×D61C22⋊Q811yescharacteristic
121C2×#6923—82no—
122C2×#6923—82no—
123C2×#6923—82no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C2×C4⋊C411yescharacteristic
130C22⋊D61—42no—
131C2×Dic61—21no—
132C2×Dic61—41no—
133C2×Dic61#12111yescharacteristic
134C2×Dic61C22⋊Q811yescharacteristic
135D244—41no—
136D244—42no—
137D244—42no—
138C4×D61—21no—
139C4×D61—21no—
140C4×D61—21no—
141C4×D61—21no—
142C4×D61—41no—
143C4×D61—41no—
144C4×D61—41no—
145C4×D61—41no—
146Dic122—41no—
147#92420—82no—
148C8×C61—21no—
149C8×C61—41no—
150C8×C61#9912yes—
151#92418—21no—
152#92418—21no—
153#92418—41no—
154#8626—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61#38011yescharacteristic
159C2×D61—21no—
160C2×D61—21no—
161C2×D61—21no—
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—44no—
165C2×D61—41no—
166C2×D61—41no—
167C2×D61—41no—
168C2×D61#47112yes—
169#6923—162no—
170C4×C61—21no—
171C4×C61—41no—
172C4×C61—41no—
173C4×C61C2×#9911yescharacteristic
174C4×C61#41011yescharacteristic
175Dic61—21no—
176Dic61—21no—
177Dic61—41no—
178#2929—611no—
179#1810—611no—
180#1807—611no—
181#1513—612no—
182#1305—612no—
183C2×C61—41no—
184C2×C61#179911yescharacteristic
185C2×C61#181411yescharacteristic
186C2×C61#181511yescharacteristic
187D61—22no—
188D61—41no—
189D61—81no—
190C2×#131—611no—
191C23×C8—611no—
192#525—611no—
193#435—1221no—
194#435—1221no—
195C8×D4—1221no—
196C8×D4—1221no—
197C2×#108—1221no—
198C2×#106—1221no—
199#424—611no—
200#408—611no—
201#407—611no—
202#407—612no—
203#405#692311yescharacteristic, complemented
204#380—612no—
205#341—1221no—
206#340—1221no—
207#328—1222no—
208#327—1222no—
209C61#862611yescharacteristic, complemented, Sylow
210C2×#41—611no—
211C23×C4—611no—
212#131—2442no—
213C2×M16—611no—
214C2×M16—1221no—
215C2×M16—1222no—
216C2×M16C2×#692311yescharacteristic
217C22×C8—611no—
218C22×C8—611no—
219C22×C8—611no—
220C22×C8—611no—
221C22×C8—611no—
222C22×C8—1221no—
223C22×C8—2442no—
224C22×C8C2×#692311yescharacteristic
225#123—1221no—
226C22⋊Q8—611no—
227C22⋊Q8—611no—
228#121—611no—
229#121—611no—
230#120—1221no—
231C4×D4—1221no—
232C4×D4—1221no—
233#117C2×#692311yescharacteristic
234C2×C4⋊C4—1222no—
235C2×C22⋊C4—1222no—
236#108—2442no—
237#106—2442no—
238C4⋊C8—1221no—
239C4⋊C8—1221no—
240C22⋊C8—1221no—
241C22⋊C8—1221no—
242C22⋊C8—1221no—
243C22⋊C8—1221no—
244C22⋊C8—1222no—
245C22⋊C8—1222no—
246#97—21no—
247C4×C8—21no—
248#95—1222no—
249C24—611no—
250#41—2441no—
251#41—2441no—
252C2×Q8—611no—
253C2×D4—611no—
254C2×D4—1221no—
255C2×D4—1221no—
256C2×D4—1221no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—611no—
261C22×C4—611no—
262C22×C4—1221no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4—2442no—
266C22×C4C22×#692311yescharacteristic
267M16—41no—
268M16—2442no—
269M16—2442no—
270M16—2442no—
271M16—2441no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—611no—
275C2×C8—611no—
276C2×C8—1222no—
277C2×C8—1222no—
278C2×C8—1221no—
279C2×C8—2441no—
280C2×C8—2441no—
281C2×C8—2441no—
282C2×C8—2442no—
283C2×C8—2442no—
284C2×C8C22⋊#692311yescharacteristic
285C2×C8C22⋊#692311yescharacteristic
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—2442no—
289C4⋊C4—2442no—
290C4⋊C4C4⋊#692311yescharacteristic
291C4⋊C4C4⋊#692311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—1221no—
294C22⋊C4—1221no—
295C22⋊C4—1221no—
296C22⋊C4—1221no—
297C22⋊C4—4882no—
298C42—21no—
299C23—611no—
300C23—611no—
301C23—1222no—
302C23—1221no—
303C23—2441no—
304C23—2441no—
305C23#1451985611yescharacteristic
306Q8—2441no—
307D4—2441no—
308D4—2442no—
309D4—2442no—
310D4—2442no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—611no—
316C2×C4—611no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—1221no—
322C2×C4—4882no—
323C2×C4—4882no—
324C2×C4—4882no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4C2×C22⋊#692311yescharacteristic
331C2×C4C2×C4⋊#692311yescharacteristic
332C8—21no—
333C8—41no—
334C8—1221no—
335C8—1221no—
336C8—4882no—
337C8—2441no—
338C8#1451983612yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—612no—
343C22—1221no—
344C22—1221no—
345C22—1221no—
346C22—1221no—
347C22—1221no—
348C22—2444no—
349C22—2441no—
350C22—2441no—
351C22—2441no—
352C22#1684070411yescharacteristic
353C4—21no—
354C4—41no—
355C4—41no—
356C4—1221no—
357C4—1221no—
358C4—9762no—
359C4—2441no—
360C4C2×#1451983611yescharacteristic
361C4#1684074111yescharacteristic
362C2—41no—
363C2—1222no—
364C2—4881no—
365C2—2441no—
366C2#2146168011yescharacteristic
367C2#2146170511yescharacteristic
368C2#2146170711yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#405C4
socleC22×C61#380
prime radical#16840274C22
cosocleC22×C4×D61C23
hypercenter#405#6923
hypocenterC61#8626
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405

Complements

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

NH ≅ G/Ntype
1. #21461698C2semidirect
2. #21461695C2semidirect
4. #21460719C2semidirect
14. C2×#14519783C4semidirect
21. #16839789C4semidirect
22. #16839730C4semidirect
24. C61×#405C4semidirect
203. #405#6923semidirect
209. C61#8626semidirect

Maximal subgroups

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

subgroupindex
#214616982
#214616952
#214612362
#214607192
#214545152
#862661