Finite Group Databasebeta

← C22×#16840664 #30623899

Subgroups — C22×#16840664 #30623899

Order 15616.

1486 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1486 conjugacy classes of subgroups — too many to draw. Only its 240 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
1C22×#14519837C211yesmaximal, characteristic
2C22×#14519835C211yesmaximal, characteristic
3C2×#16840664C2112yesmaximal, direct product factor
4C22×#14519760C211yesmaximal, characteristic
5C22×#11758217—21no—
6C22×C22⋊#6923—21no—
7C2×#14519837—23no—
8C2×#14519837C2216yes—
9C2×#14519835—23no—
10C2×#14519835C2216yes—
11#16840664C22116yesdirect product factor
12C23×C22⋊D61C2211yescharacteristic
13C2×#14519760—23no—
14C2×#14519760—23no—
15C2×#14519760C416yescomplemented
16C2×#14519760C2216yes—
17C22×#11758196—21no—
18C22×C22⋊Dic61C411yescharacteristic, complemented
19C22×#11758198—21no—
20C24×C4×C61C411yescharacteristic, complemented
21C23×#6923—41no—
22C2×#11758217—26no—
23C2×#11758217—43no—
24C22×#92420—41no—
25C2×C22⋊#6923—26no—
26C2×C22⋊#6923—43no—
27#14519837—26no—
28#14519837C2314yes—
29#14519835—26no—
30#14519835C2314yes—
31C24×D61D411yescharacteristic
32C22×C22⋊D61—41no—
33C22×C22⋊D61—43no—
34C22×C22⋊D61C2×C413yes—
35C22×C22⋊D61D413yes—
36C22×C22⋊D61D413yes—
37C22×C22⋊D61C2313yes—
38C23×Dic61—21no—
39C23×Dic61D411yescharacteristic
40#14519760—212no—
41#14519760—26no—
42#14519760—26no—
43#14519760C2×C4112yescomplemented
44#14519760C2314yes—
45C2×#11758196—26no—
46C2×#11758196—26no—
47C2×C22⋊Dic61—23no—
48C2×C22⋊Dic61C2×C416yescomplemented
49C2×#11758198—26no—
50C2×#11758198—26no—
51C25×C61C2×C411yescharacteristic
52C23×C4×C61—21no—
53C23×C4×C61—23no—
54C23×C4×C61—43no—
55C23×C4×C61—41no—
56C23×C4×C61C2×C416yescomplemented
57C24×C61—23no—
58C24×C61—21no—
59C24×C61—44no—
60C24×C61C22⋊C411yescharacteristic
61C24×C61C22⋊C413yes—
62C24×C61C22×C413yes—
63C23×D61—21no—
64C23×D61—23no—
65C23×D61—44no—
66C23×D61C22⋊C413yes—
67C23×D61C2×D413yes—
68C22×#6923—46no—
69C22×#6923—84no—
70C22×C4×C61—26no—
71C22×C4×C61—26no—
72C22×C4×C61—46no—
73C22×C4×C61—46no—
74C22×C4×C61—43no—
75C22×C4×C61—41no—
76C22×C4×C61—46no—
77C22×C4×C61—46no—
78C22×C4×C61C22×C414yescomplemented
79C2×C22⋊D61—43no—
80C2×C22⋊D61—43no—
81C2×C22⋊D61—43no—
82C2×C22⋊D61—43no—
83C2×C22⋊D61—43no—
84C2×C22⋊D61—43no—
85C2×C22⋊D61—43no—
86C2×C22⋊D61—43no—
87C2×C22⋊D61C22⋊C416yes—
88C2×C22⋊D61C22×C413yes—
89C2×C22⋊D61C2×D413yes—
90C2×C22⋊D61C2×D413yes—
91C2×C22⋊D61C2411yescharacteristic
92C22×Dic61—21no—
93C22×Dic61—23no—
94C22×Dic61—26no—
95C22×Dic61—44no—
96C22×Dic61C22⋊C413yes—
97C22×Dic61C2×D413yes—
98C22⋊#6923—24no—
99C22⋊#6923—46no—
100#11758217—24no—
101#11758217—46no—
102C2×#92420—46no—
103C22⋊Dic61—26no—
104C22⋊Dic61C22×C414yescomplemented
105#11758198—212no—
106#11758198—24no—
107#11758196—212no—
108#11758196—24no—
109C23×C61—24no—
110C23×C61—23no—
111C23×C61—23no—
112C23×C61—23no—
113C23×C61—23no—
114C23×C61—412no—
115C23×C61—46no—
116C23×C61—46no—
117C23×C61—44no—
118C23×C61#9913yes—
119C23×C61#9911yescharacteristic
120C23×C61C2×C22⋊C413yes—
121C23×C61C2×C22⋊C413yes—
122C23×C61C23×C411yescharacteristic
123C22×D61—23no—
124C22×D61—23no—
125C22×D61—23no—
126C22×D61—23no—
127C22×D61—44no—
128C22×D61—46no—
129C22×D61—46no—
130C22×D61—86no—
131C22×D61C2×C22⋊C413yes—
132C22×D61C22×D411yescharacteristic
133C2×#6923—44no—
134C2×#6923—812no—
135C2×C4×C61—24no—
136C2×C4×C61—412no—
137C2×C4×C61—46no—
138C2×C4×C61—46no—
139C2×C4×C61—44no—
140C22⋊D61—46no—
141C22⋊D61—43no—
142C22⋊D61—43no—
143C22⋊D61—43no—
144C22⋊D61—41no—
145C2×Dic61—23no—
146C2×Dic61—23no—
147C2×Dic61—23no—
148C2×Dic61—23no—
149C2×Dic61—24no—
150C2×Dic61—412no—
151C2×Dic61C2×C22⋊C413yes—
152C2×Dic61C22×D411yescharacteristic
153#92420—44no—
154C22×#352—611nomaximal, Sylow
155C22×C61—26no—
156C22×C61—26no—
157C22×C61—23no—
158C22×C61—21no—
159C22×C61—43no—
160C22×C61—41no—
161C22×C61—46no—
162C22×C61—46no—
163C22×C61—46no—
164C22×C61—46no—
165C22×C61#35214yescomplemented
166C22×C61C2×#9913yes—
167C22×C61C2×#9913yes—
168C22×C61C22×C22⋊C411yescharacteristic
169C2×D61—21no—
170C2×D61—23no—
171C2×D61—46no—
172C2×D61—46no—
173C2×D61—43no—
174C2×D61—41no—
175C2×D61—83no—
176C2×D61—83no—
177#6923—84no—
178C4×C61—44no—
179Dic61—21no—
180Dic61—23no—
181Dic61—44no—
182C22×#123—611no—
183C22×#100—611no—
184C22×#99—611no—
185C2×#352—6112no—
186C2×C61—21no—
187C2×C61—23no—
188C2×C61—43no—
189C2×C61—41no—
190C2×C61C2×#35216yescomplemented
191C2×C61C22×#9911yescharacteristic
192D61—41no—
193D61—43no—
194C23×D4—611no—
195C24×C4#692311yescharacteristic, complemented
196C22×M16—1221no—
197C2×#123—616no—
198C2×#123—616no—
199C2×#123—1223no—
200C2×#123—1223no—
201C22×C4⋊C4—1221no—
202C22×C22⋊C4—611no—
203C22×C22⋊C4—1221no—
204C22×C22⋊C4—1221no—
205C2×#100—616no—
206C2×#100—1223no—
207C2×#99—616no—
208C2×#99—1223no—
209#352—6116no—
210C61C22×#35211yescharacteristic, complemented, Sylow
211C25—611no—
212C25C2×#692311yescharacteristic
213C22×D4—613no—
214C22×D4—613no—
215C22×D4—613no—
216C22×D4—613no—
217C22×D4—2443no—
218C22×D4—2441no—
219C23×C4—23no—
220C23×C4—21no—
221C23×C4—43no—
222C23×C4—41no—
223C23×C4—611no—
224C23×C4—1221no—
225C23×C4—2441no—
226C23×C4C2×#692316yescomplemented
227C2×M16—1226no—
228C2×M16—2443no—
229C22×C8—2441no—
230#123—614no—
231#123—6112no—
232#123—1226no—
233#123—1226no—
234#123—12212no—
235C2×C4⋊C4—1226no—
236C2×C4⋊C4—1226no—
237C2×C22⋊C4—616no—
238C2×C22⋊C4—1226no—
239C2×C22⋊C4—1226no—
240C2×C22⋊C4—1223no—
241C2×C22⋊C4—1226no—
242C2×C22⋊C4—2443no—
243#100—614no—
244#100—1226no—
245#99—614no—
246#99—1226no—
247C24—23no—
248C24—21no—
249C24—44no—
250C24—613no—
251C24—613no—
252C24—1223no—
253C24—1221no—
254C24—2444no—
255C24C22⋊#692313yes—
256C24C22⋊#692311yescharacteristic
257C24C22×#692313yes—
258C2×D4—613no—
259C2×D4—613no—
260C2×D4—616no—
261C2×D4—611no—
262C2×D4—613no—
263C2×D4—2443no—
264C2×D4—2443no—
265C2×D4—2443no—
266C2×D4—2443no—
267C2×D4—2443no—
268C2×D4—2443no—
269C2×D4—2443no—
270C2×D4—2443no—
271C22×C4—26no—
272C22×C4—26no—
273C22×C4—46no—
274C22×C4—46no—
275C22×C4—43no—
276C22×C4—41no—
277C22×C4—46no—
278C22×C4—46no—
279C22×C4—613no—
280C22×C4—613no—
281C22×C4—1223no—
282C22×C4—1221no—
283C22×C4—1226no—
284C22×C4—4884no—
285C22×C4—2446no—
286C22×C4—2444no—
287C22×C4C22×#692314yescomplemented
288M16—1224no—
289M16—2446no—
290C2×C8—2446no—
291C4⋊C4—12212no—
292C4⋊C4—1224no—
293C22⋊C4—614no—
294C22⋊C4—12212no—
295C22⋊C4—1224no—
296C22⋊C4—1226no—
297C22⋊C4—1224no—
298C22⋊C4—2446no—
299C23—24no—
300C23—23no—
301C23—23no—
302C23—23no—
303C23—23no—
304C23—46no—
305C23—46no—
306C23—412no—
307C23—44no—
308C23—613no—
309C23—611no—
310C23—1223no—
311C23—1223no—
312C23—1223no—
313C23—1223no—
314C23—4886no—
315C23—2446no—
316C23—2446no—
317C23—2444no—
318C23#1451983513yes—
319C23#1451983511yescharacteristic
320C23C2×C22⋊#692313yes—
321C23C2×C22⋊#692313yes—
322C23C23×#692311yescharacteristic
323D4—2443no—
324D4—2441no—
325D4—2443no—
326D4—2446no—
327D4—2443no—
328C2×C4—24no—
329C2×C4—412no—
330C2×C4—46no—
331C2×C4—46no—
332C2×C4—44no—
333C2×C4—613no—
334C2×C4—611no—
335C2×C4—1223no—
336C2×C4—1223no—
337C2×C4—1223no—
338C2×C4—1223no—
339C2×C4—1224no—
340C2×C4—48812no—
341C2×C4—2444no—
342C2×C4—24412no—
343C8—2444no—
344C22—26no—
345C22—26no—
346C22—23no—
347C22—21no—
348C22—43no—
349C22—41no—
350C22—46no—
351C22—46no—
352C22—46no—
353C22—46no—
354C22—1221no—
355C22—1223no—
356C22—4883no—
357C22—4883no—
358C22—2441no—
359C22—2443no—
360C22—2446no—
361C22—2446no—
362C22#1684066414yesdirect product factor
363C22C2×#1451983513yes—
364C22C2×#1451983513yes—
365C22C22×C22⋊#692311yescharacteristic
366C4—44no—
367C4—1221no—
368C4—1223no—
369C4—4884no—
370C4—2444no—
371C2—21no—
372C2—23no—
373C2—43no—
374C2—41no—
375C2—2441no—
376C2—2443no—
377C2C2×#1684066416yesdirect product factor
378C2C22×#1451983511yescharacteristic
subgroupG/H
centerC23#14519835
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC24×C4×C61C4
socleC23×C61#99
prime radicalC23×C22⋊D61C22
cosocleC2×C22⋊D61C24
hypercenterC24×C4#6923
hypocenterC61C22×#352
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24 ◁ C25 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
3. C2×#16840664C2direct (N × H)
11. #16840664C22direct (N × H)
15. C2×#14519760C4semidirect
18. C22×C22⋊Dic61C4semidirect
20. C24×C4×C61C4semidirect
43. #14519760C2×C4semidirect
48. C2×C22⋊Dic61C2×C4semidirect
56. C23×C4×C61C2×C4semidirect
78. C22×C4×C61C22×C4semidirect
104. C22⋊Dic61C22×C4semidirect
165. C22×C61#352semidirect
190. C2×C61C2×#352semidirect
195. C24×C4#6923semidirect
210. C61C22×#352semidirect
226. C23×C4C2×#6923semidirect
287. C22×C4C22×#6923semidirect
362. C22#16840664direct (N × H)
377. C2C2×#16840664direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198372
C22×#145198352
C2×#168406642
C22×#145197602
C22×#35261