Finite Group Databasebeta

← #29596287

Subgroups — #29596287

Order 15616.

614 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 87 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#21448410C211yesmaximal, characteristic
2#21445746C212yesmaximal
3#21443946C211yesmaximal, characteristic, complemented
4#21443945C212yesmaximal, complemented
5C61×#1394C211yesmaximal, characteristic, complemented
6C22×#11758226C2211yescharacteristic
7C2×#14519734C411yescharacteristic
8C2×C4⋊C4×D61C411yescharacteristic
9C2×#14519696C2212yes—
10C2×#14519693—22no—
11#16839725—22no—
12#16839669C2211yescharacteristic
13#16839664C411yescharacteristic
14#16839663C411yescharacteristic
15#16839640—22no—
16#16839621—22no—
17C2×#101×C61C2212yes—
18C61×#391C2211yescharacteristic
19C2×#11758226—22no—
20C2×#11758226—21no—
21C2×#11758226—21no—
22C2×#11758226—42no—
23C2×#11758226—42no—
24C22×D244C2×C411yescharacteristic
25C22×D244D412yes—
26C22×C4×D61—21no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D412yes—
29C2×#11758194—22no—
30#14519734—42no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758196—21no—
33C2×#11758196—21no—
34C4⋊C4×D61—42no—
35C2×#11758198—21no—
36C2×#11758198—22no—
37C2×#11758198—22no—
38C2×C4×Dic61C2×C411yescharacteristic
39#14519696—22no—
40#14519696—22no—
41#14519693—44no—
42#14519689—44no—
43#14519684—22no—
44C22×Q8×C61C2311yescharacteristic
45C2×M16×C61—22no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C42×C61C2×C411yescharacteristic
48#101×C61—22no—
49#101×C61—22no—
50#95×C61—22no—
51C23×D61C22⋊C412yes—
52C23×D61C22⋊C411yescharacteristic
53C2×Q8×C61—41no—
54C2×Q8×C61—41no—
55C2×Q8×C61#4112yes—
56C2×Q8×C61#4111yescharacteristic
57C2×Q8×C61#4111yescharacteristic
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D412yes—
62C22×Dic61—21no—
63C22×Dic61—21no—
64C22×Dic61C22⋊C411yescharacteristic
65#11758226—44no—
66#11758226—48no—
67#11758226—44no—
68C2×D244—24no—
69C2×D244—21no—
70C2×D244—21no—
71C2×D244—22no—
72C2×D244—22no—
73C2×D244—41no—
74C2×D244—41no—
75C2×D244—42no—
76C2×C4×D61—22no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—22no—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—24no—
83C2×C4×D61—42no—
84C2×C4×D61—42no—
85C2×C4×D61—82no—
86M16×C61—42no—
87M16×C61—42no—
88C2×C8×C61—42no—
89C4⋊C4×C61—41no—
90C42×C61—41no—
91#11758198—42no—
92#11758198—82no—
93#11758198—82no—
94C4⋊Dic61—41no—
95#11758196—42no—
96#11758196—42no—
97C4×Dic61—41no—
98#11758194—42no—
99#11758194—42no—
100C2×#92418—42no—
101C23×C61C2×C22⋊C411yescharacteristic
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—22no—
105C22×D61—22no—
106C22×D61—22no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—42no—
110C22×D61—42no—
111C22×D61#9912yes—
112Q8×C61—41no—
113Q8×C61—42no—
114Q8×C61—41no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—41no—
120C2×C4×C61—42no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61#11711yescharacteristic
125C2×C4×C61#11711yescharacteristic
126C2×C4×C61#12312yes—
127C2×C4×C61#12312yes—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61—42no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61#9912yes—
136D244—44no—
137D244—44no—
138D244—42no—
139D244—42no—
140C4×D61—24no—
141C4×D61—24no—
142C4×D61—44no—
143C4×D61—44no—
144C4×D61—84no—
145C8×C61—82no—
146#92418—82no—
147#12863—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#38111yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C22×C61C2×#9911yescharacteristic
153C2×D61—22no—
154C2×D61—42no—
155C2×D61—41no—
156C2×D61—41no—
157C2×D61—42no—
158C2×D61—44no—
159C2×D61—42no—
160C2×D61—42no—
161C2×D61—42no—
162C2×D61—82no—
163C4×C61—22no—
164C4×C61—22no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—81no—
168C4×C61—81no—
169Dic61—22no—
170Dic61—81no—
171Dic61—81no—
172#2311—611no—
173#1785—612no—
174#1394—612no—
175#1394—611no—
176#1394D6111yescharacteristic, complemented
177C2×C61—21no—
178C2×C61—21no—
179C2×C61#181911yescharacteristic
180C2×C61#213411yescharacteristic
181C2×C61#213511yescharacteristic
182D61—42no—
183D61—82no—
184C22×#41—611no—
185C2×#117—611no—
186C22×C4⋊C4—611no—
187C2×#101—612no—
188C2×#101C2×D6112yes—
189C2×#100—1222no—
190#407—1222no—
191#391—611no—
192#391C2×D6111yescharacteristic
193#390—611no—
194#390—611no—
195#386—1222no—
196#380—1222no—
197C61#1286311yescharacteristic, complemented, Sylow
198C2×#41—1222no—
199C2×#41—1221no—
200C2×#41—1221no—
201C2×#41—2442no—
202C2×#41—2442no—
203C22×Q8C22×D6111yescharacteristic
204C22×D4—611no—
205C22×D4—612no—
206C23×C4—611no—
207C23×C4—612no—
208C23×C4—1221no—
209C2×M16—22no—
210C2×M16—1222no—
211#117—2442no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—2442no—
216C2×C4⋊C4C4×D6111yescharacteristic
217C2×C22⋊C4—1222no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4—1221no—
220C2×C42—611no—
221C2×C42C4×D6111yescharacteristic
222#101—22no—
223#101—22no—
224#101—1222no—
225#101—1222no—
226#100—2444no—
227C22⋊C8—2444no—
228#95—22no—
229#95—1222no—
230C24—611no—
231C24—612no—
232#41—2448no—
233#41—2444no—
234#41—2444no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8#1175822212yes—
238C2×Q8#1175822411yescharacteristic
239C2×Q8#1175822611yescharacteristic
240C2×D4—1224no—
241C2×D4—1221no—
242C2×D4—1221no—
243C2×D4—1222no—
244C2×D4—1222no—
245C2×D4—2441no—
246C2×D4—2441no—
247C2×D4—2442no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—611no—
251C22×C4—1222no—
252C22×C4—1224no—
253C22×C4—1222no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—1222no—
257C22×C4—1222no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—4882no—
261C22×C4—2442no—
262C22×C4—2442no—
263C22×C4C2×C4×D6111yescharacteristic
264C22×C4D4×D6112yes—
265M16—42no—
266M16—42no—
267M16—2442no—
268M16—2442no—
269C2×C8—42no—
270C2×C8—2442no—
271C4⋊C4—41no—
272C4⋊C4—2441no—
273C4⋊C4—2442no—
274C4⋊C4—2442no—
275C22⋊C4—4882no—
276C22⋊C4—4882no—
277C22⋊C4—2442no—
278C42—41no—
279C42—2441no—
280C23—612no—
281C23—1221no—
282C23—1221no—
283C23—1222no—
284C23—1222no—
285C23—1222no—
286C23—2442no—
287C23—2442no—
288C23—2441no—
289C23—2441no—
290C23C22⋊C4×D6111yescharacteristic
291Q8—42no—
292Q8—41no—
293Q8—41no—
294D4—2444no—
295D4—2444no—
296D4—2442no—
297D4—2442no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—612no—
308C2×C4—1224no—
309C2×C4—1224no—
310C2×C4—1221no—
311C2×C4—1221no—
312C2×C4—4884no—
313C2×C4—2444no—
314C2×C4—2444no—
315C2×C4—2441no—
316C2×C4—2442no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4#1451973411yescharacteristic
321C2×C4#1451973511yescharacteristic
322C2×C4#1451975912yes—
323C2×C4#1451976212yes—
324C8—82no—
325C8—4882no—
326C22—21no—
327C22—21no—
328C22—1222no—
329C22—4882no—
330C22—2444no—
331C22—2442no—
332C22—2442no—
333C22—2442no—
334C22—2442no—
335C22—2442no—
336C22—2441no—
337C22—2441no—
338C22#1683962411yescharacteristic
339C22#99×D6111yescharacteristic
340C22#99×D6111yescharacteristic
341C4—22no—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—1222no—
348C4—4881no—
349C4—4881no—
350C2—21no—
351C2—21no—
352C2—4882no—
353C2—2442no—
354C2#2144589211yescharacteristic
355C2#2144750911yescharacteristic
356C2#2144751211yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1394C2
socleC22×C61C2×#99
prime radicalC22×#11758226C22
cosocleC22×Q8×C61C23
hypercenter#1394D61
hypocenterC61#12863
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #21443946C2semidirect
4. #21443945C2semidirect
5. C61×#1394C2semidirect
176. #1394D61semidirect
197. C61#12863semidirect

Maximal subgroups

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

subgroupindex
#214484102
#214457462
#214439462
#214439452
C61×#13942
#1286361