Finite Group Databasebeta

← #30599044

Subgroups — #30599044

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#21461699C211yesmaximal, characteristic, complemented
2#21461696C211yesmaximal, characteristic, complemented
3#21461236C212yesmaximal
4#21460719C212yesmaximal, complemented
5#21454515C211yesmaximal, characteristic
6C2×#14519848—21no—
7C2×#14519846—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#14519848—42no—
29#14519846—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—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—41no—
119C22×D61—41no—
120C22×D61#12111yescharacteristic
121C22×D61C22⋊Q811yescharacteristic
122C2×#6923—82no—
123C2×#6923—82no—
124C2×#6923—82no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C2×C4⋊C411yescharacteristic
131C22⋊D61—42no—
132C2×Dic61—21no—
133C2×Dic61—41no—
134C2×Dic61#12111yescharacteristic
135C2×Dic61C22⋊Q811yescharacteristic
136D244—41no—
137D244—42no—
138D244—41no—
139D244—41no—
140C4×D61—21no—
141C4×D61—21no—
142C4×D61—21no—
143C4×D61—21no—
144C4×D61—41no—
145C4×D61—41no—
146C4×D61—41no—
147C4×D61—41no—
148Dic122—41no—
149#92420—82no—
150C8×C61—21no—
151C8×C61—41no—
152C8×C61#9912yes—
153#92418—21no—
154#92418—21no—
155#92418—41no—
156#8627—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38011yescharacteristic
161C2×D61—21no—
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—21no—
165C2×D61—21no—
166C2×D61—42no—
167C2×D61—42no—
168C2×D61—41no—
169C2×D61—41no—
170C2×D61—41no—
171C2×D61#47211yescharacteristic
172C2×D61#47311yescharacteristic
173#6923—162no—
174C4×C61—21no—
175C4×C61—41no—
176C4×C61—41no—
177C4×C61C2×#9911yescharacteristic
178C4×C61#41011yescharacteristic
179Dic61—21no—
180Dic61—21no—
181Dic61—41no—
182#2929—611no—
183#1810—611no—
184#1808—611no—
185#1513—612no—
186#1305—612no—
187C2×C61—41no—
188C2×C61#179911yescharacteristic
189C2×C61#181411yescharacteristic
190C2×C61#181611yescharacteristic
191D61—21no—
192D61—21no—
193D61—41no—
194D61—81no—
195C2×#131—611no—
196C23×C8—611no—
197#525—611no—
198#435—1221no—
199#435—1221no—
200C8×D4—1221no—
201C8×D4—1221no—
202C2×#108—1221no—
203C2×#107—1221no—
204#424—611no—
205#408—611no—
206#407—611no—
207#407—612no—
208#405#692311yescharacteristic, complemented
209#380—612no—
210#341—1221no—
211#340—1221no—
212#328—1222no—
213#327—1222no—
214C61#862711yescharacteristic, complemented, Sylow
215C2×#41—611no—
216C23×C4—611no—
217#131—2442no—
218C2×M16—611no—
219C2×M16—1221no—
220C2×M16—1222no—
221C2×M16C2×#692311yescharacteristic
222C22×C8—611no—
223C22×C8—611no—
224C22×C8—611no—
225C22×C8—611no—
226C22×C8—611no—
227C22×C8—1221no—
228C22×C8—2442no—
229C22×C8C2×#692311yescharacteristic
230#123—1221no—
231C22⋊Q8—611no—
232C22⋊Q8—611no—
233#121—611no—
234#121—611no—
235#120—1221no—
236C4×D4—1221no—
237C4×D4—1221no—
238#117C2×#692311yescharacteristic
239C2×C4⋊C4—1222no—
240C2×C22⋊C4—1222no—
241#108—2442no—
242#107—2442no—
243C4⋊C8—1221no—
244C4⋊C8—1221no—
245C22⋊C8—1221no—
246C22⋊C8—1221no—
247C22⋊C8—1221no—
248C22⋊C8—1221no—
249C22⋊C8—1222no—
250C22⋊C8—1222no—
251#97—21no—
252C4×C8—21no—
253#95—1222no—
254C24—611no—
255#41—2441no—
256#41—2441no—
257C2×Q8—611no—
258C2×D4—611no—
259C2×D4—1221no—
260C2×D4—1221no—
261C2×D4—1221no—
262C22×C4—611no—
263C22×C4—611no—
264C22×C4—611no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—1221no—
268C22×C4—2441no—
269C22×C4—2441no—
270C22×C4—2442no—
271C22×C4C22×#692311yescharacteristic
272M16—41no—
273M16—2442no—
274M16—2442no—
275M16—2442no—
276M16—2441no—
277C2×C8—22no—
278C2×C8—21no—
279C2×C8—611no—
280C2×C8—611no—
281C2×C8—1222no—
282C2×C8—1222no—
283C2×C8—1221no—
284C2×C8—2441no—
285C2×C8—2441no—
286C2×C8—2441no—
287C2×C8—2442no—
288C2×C8—2442no—
289C2×C8C22⋊#692311yescharacteristic
290C2×C8C22⋊#692311yescharacteristic
291C4⋊C4—1221no—
292C4⋊C4—1221no—
293C4⋊C4—2442no—
294C4⋊C4—2442no—
295C4⋊C4C4⋊#692311yescharacteristic
296C4⋊C4C4⋊#692311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—1221no—
299C22⋊C4—1221no—
300C22⋊C4—1221no—
301C22⋊C4—1221no—
302C22⋊C4—4882no—
303C42—21no—
304C23—611no—
305C23—611no—
306C23—1221no—
307C23—1221no—
308C23—1221no—
309C23—2441no—
310C23—2441no—
311C23#1451985611yescharacteristic
312Q8—2441no—
313D4—2441no—
314D4—2441no—
315D4—2441no—
316D4—2442no—
317D4—2442no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—611no—
323C2×C4—611no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—1221no—
329C2×C4—4882no—
330C2×C4—4882no—
331C2×C4—4882no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4C2×C22⋊#692311yescharacteristic
338C2×C4C2×C4⋊#692311yescharacteristic
339C8—21no—
340C8—41no—
341C8—1221no—
342C8—1221no—
343C8—4882no—
344C8—2441no—
345C8#1451983612yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—611no—
350C22—611no—
351C22—1221no—
352C22—1221no—
353C22—1221no—
354C22—1221no—
355C22—1221no—
356C22—2442no—
357C22—2442no—
358C22—2441no—
359C22—2441no—
360C22—2441no—
361C22#1684070411yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—1221no—
366C4—1221no—
367C4—9762no—
368C4—2441no—
369C4C2×#1451983611yescharacteristic
370C4#1684074111yescharacteristic
371C2—41no—
372C2—1221no—
373C2—1221no—
374C2—4881no—
375C2—2441no—
376C2#2146168011yescharacteristic
377C2#2146170611yescharacteristic
378C2#2146170811yescharacteristic
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#8627
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. #21461699C2semidirect
2. #21461696C2semidirect
4. #21460719C2semidirect
14. C2×#14519783C4semidirect
21. #16839789C4semidirect
22. #16839730C4semidirect
24. C61×#405C4semidirect
208. #405#6923semidirect
214. C61#8627semidirect

Maximal subgroups

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

subgroupindex
#214616992
#214616962
#214612362
#214607192
#214545152
#862761