Finite Group Databasebeta

← #29596277

Subgroups — #29596277

Order 15616.

638 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16839669C211yesmaximal, characteristic
2#21445743C211yesmaximal, characteristic
3#21443945C214yesmaximal, complemented
4C61×#1784C211yesmaximal, characteristic
5C22×C4⋊Dic61C411yescharacteristic
6C22×C4×Dic61C411yescharacteristic
7C2×#14519696—21no—
8C2×#14519696C2212yes—
9#16839726—21no—
10#16839669—21no—
11#16839669—21no—
12#16839669C412yes—
13#16839669C2212yes—
14C2×#14519684—21no—
15C2×#14519684—21no—
16C23×Q8×C61C2211yescharacteristic
17C2×#101×C61—21no—
18C2×#101×C61C2212yes—
19C61×#407—21no—
20C23×Dic61—21no—
21C23×Dic61—21no—
22C2×#11758194—22no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—41no—
25C2×C4⋊Dic61—41no—
26C2×C4⋊Dic61C2×C412yes—
27C2×C4×Dic61—21no—
28C2×C4×Dic61—42no—
29C2×C4×Dic61C2×C412yes—
30#14519696—24no—
31#14519696—42no—
32C22⋊#92418—41no—
33C22⋊#92418—41no—
34#14519684—22no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38C22×Q8×C61—24no—
39C22×Q8×C61—41no—
40C22×Q8×C61—41no—
41C22×Q8×C61—42no—
42C22×Q8×C61C2×C411yescharacteristic
43C22×Q8×C61D411yescharacteristic
44C22×Q8×C61D411yescharacteristic
45C22×Q8×C61C2311yescharacteristic
46C23×C4×C61C2×C411yescharacteristic
47C23×C4×C61D411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C2×M16×C61—22no—
50#101×C61—24no—
51#101×C61—42no—
52C22⋊C8×C61—41no—
53C22⋊C8×C61—41no—
54C24×C61C22⋊C411yescharacteristic
55C2×Q8×C61—22no—
56C2×Q8×C61—22no—
57C2×Q8×C61—22no—
58C2×Q8×C61—22no—
59C2×Q8×C61—22no—
60C2×Q8×C61—41no—
61C2×Q8×C61—41no—
62C2×Q8×C61—41no—
63C2×Q8×C61—41no—
64C2×Q8×C61—41no—
65C2×Q8×C61—41no—
66C2×Q8×C61—48no—
67C2×Q8×C61—48no—
68C2×Q8×C61#4114yes—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—22no—
72C22×C4×C61—22no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—41no—
80C22×C4×C61—41no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×Dic61—24no—
88C22×Dic61—22no—
89C22×Dic61—22no—
90C22×Dic61—42no—
91C22×Dic61—42no—
92C22×Dic61—44no—
93C22×Dic61—41no—
94C22×Dic61—41no—
95M16×C61—42no—
96M16×C61—42no—
97C2×C8×C61—42no—
98C4⋊Dic61—42no—
99C4⋊Dic61—42no—
100C4×Dic61—42no—
101C4×Dic61—42no—
102#11758194—42no—
103#11758194—42no—
104C2×#92418—42no—
105C23×C61—22no—
106C23×C61—22no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61#9911yescharacteristic
110C23×C61#9911yescharacteristic
111C23×C61C2×C22⋊C411yescharacteristic
112Q8×C61—44no—
113Q8×C61—44no—
114Q8×C61—44no—
115Q8×C61—44no—
116C2×C4×C61—24no—
117C2×C4×C61—24no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61#9912yes—
137C2×C4×C61#11712yes—
138C2×C4×C61#12312yes—
139C2×C4×C61#12312yes—
140C2×Dic61—24no—
141C2×Dic61—48no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—44no—
145C2×Dic61—42no—
146C2×Dic61—82no—
147C2×Dic61—82no—
148C8×C61—82no—
149#92418—82no—
150#12860—611nomaximal, Sylow
151C22×C61—24no—
152C22×C61—24no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61#35512yes—
160C22×C61#35612yes—
161C22×C61#38111yescharacteristic
162C22×C61C2×#9911yescharacteristic
163C22×C61C2×#9911yescharacteristic
164C4×C61—22no—
165C4×C61—22no—
166C4×C61—22no—
167C4×C61—22no—
168C4×C61—41no—
169C4×C61—41no—
170Dic61—44no—
171Dic61—82no—
172C2×#391—611no—
173#1784—611no—
174#1784D6111yescharacteristic
175#1394—614no—
176C2×C61—21no—
177C2×C61—22no—
178C2×C61—22no—
179C2×C61—21no—
180C2×C61#181911yescharacteristic
181C2×C61C2×#35511yescharacteristic
182C2×C61C2×#35611yescharacteristic
183C23×Q8C2×D6111yescharacteristic
184C22×C4⋊C4—611no—
185C22×C42—611no—
186C2×#101—21no—
187C2×#101—612no—
188C2×#101—1221no—
189C2×#101C2×D6112yes—
190#407—21no—
191#407—1221no—
192#391—612no—
193#391—612no—
194#391—1221no—
195#391—1221no—
196C2×#95—1221no—
197C2×#95—1221no—
198C61#1286011yescharacteristic, complemented, Sylow
199C22×Q8—24no—
200C22×Q8—41no—
201C22×Q8—41no—
202C22×Q8—42no—
203C22×Q8C4×D6111yescharacteristic
204C22×Q8D24411yescharacteristic
205C22×Q8C22⋊D6111yescharacteristic
206C22×Q8C22×D6111yescharacteristic
207C23×C4—1221no—
208C23×C4—1221no—
209C23×C4C4×D6111yescharacteristic
210C23×C4D24411yescharacteristic
211C23×C4C22⋊D6111yescharacteristic
212C2×M16—22no—
213C2×M16—1222no—
214C2×C4⋊C4—612no—
215C2×C4⋊C4—1221no—
216C2×C4⋊C4—2441no—
217C2×C4⋊C4—2441no—
218C2×C42—612no—
219C2×C42—1221no—
220C2×C42—2442no—
221#101—24no—
222#101—42no—
223#101—1224no—
224#101—2442no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—2441no—
228C22⋊C8—2441no—
229#95—1222no—
230#95—1222no—
231#95—1222no—
232#95—1222no—
233C24#1175819811yescharacteristic
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8#1175822414yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—1224no—
261C22×C4—1222no—
262C22×C4—1222no—
263C22×C4—2444no—
264C22×C4—2441no—
265C22×C4—2441no—
266C22×C4—2442no—
267C22×C4—2442no—
268C22×C4#1175819811yescharacteristic
269C22×C4#1175819811yescharacteristic
270C22×C4#1175819811yescharacteristic
271C22×C4C2×C4×D6111yescharacteristic
272C22×C4C2×D24411yescharacteristic
273C22×C4C2×C22⋊D6111yescharacteristic
274M16—42no—
275M16—42no—
276M16—2442no—
277M16—2442no—
278C2×C8—42no—
279C2×C8—2442no—
280C4⋊C4—2442no—
281C4⋊C4—2442no—
282C42—2442no—
283C42—2442no—
284C23—21no—
285C23—22no—
286C23—22no—
287C23—21no—
288C23#1451969111yescharacteristic
289C23#1451969111yescharacteristic
290C23C2×#1175819811yescharacteristic
291Q8—44no—
292Q8—44no—
293Q8—44no—
294Q8—44no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—1224no—
316C2×C4—4882no—
317C2×C4—4882no—
318C2×C4—2444no—
319C2×C4—2442no—
320C2×C4—2448no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4#1451969112yes—
324C2×C4#1451973312yes—
325C2×C4#1451975812yes—
326C2×C4#1451976112yes—
327C8—82no—
328C8—4882no—
329C22—24no—
330C22—24no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22#1683955812yes—
338C22#1683956012yes—
339C22#1683962311yescharacteristic
340C22C2×#1451969111yescharacteristic
341C22C2×#1451969111yescharacteristic
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—4882no—
349C4—2444no—
350C2—21no—
351C2—22no—
352C2—22no—
353C2—21no—
354C2#2144589111yescharacteristic
355C2C2×#1683955811yescharacteristic
356C2C2×#1683956011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1784C2
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC22×Q8×C61C23
hypercenter#1784D61
hypocenterC61#12860
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #1784

Complements

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

NH ≅ G/Ntype
3. #21443945C2semidirect
198. C61#12860semidirect

Maximal subgroups

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

subgroupindex
C2×#168396692
#214457432
#214439452
C61×#17842
#1286061