Finite Group Databasebeta

← C22×#64230 #11900306

Subgroups — C22×#64230 #11900306

Order 1152.

3890 conjugacy classes of subgroups; only the 543 normal subgroups are shown.

normal subgroups only

This group has 3890 conjugacy classes of subgroups — too many to draw. Only its 543 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
1C23×GO+(4,𝔽2)C212yesmaximal, complemented
2C22×#3802C211yesmaximal, characteristic, complemented
3C22×C4×D32C211yesmaximal, characteristic, complemented
4C22×C4⋊#157C211yesmaximal, characteristic, complemented
5C22×D32⋊C4C212yesmaximal, complemented
6C2×#64230C2124yesmaximal, direct product factor
7C23×D32C2212yes—
8C23×D32C2211yescharacteristic
9C23×#157C2212yes—
10C22×GO+(4,𝔽2)—216no—
11C22×GO+(4,𝔽2)C22124yescomplemented
12C22×C4×C3⋊D3C2211yescharacteristic
13C22×C3×D12—21no—
14C22×C3×C4×D3—21no—
15C22×#625—21no—
16C22×#624C2211yescharacteristic
17C22×D3×Dic3—21no—
18C2×#3802—26no—
19C2×#3802C22112yescomplemented
20C2×C4×D32—26no—
21C2×C4×D32C22112yescomplemented
22C2×C4⋊#157C22112yescomplemented
23C2×D32⋊C4C22124yescomplemented
24#64230C22164yesdirect product factor
25C22×D3×D4—61no—
26C23×C4×D3—61no—
27C23×C3⋊D3C2311yescharacteristic
28C23×C3×D3—21no—
29C23×C3×D3—22no—
30C22×D32—23no—
31C22×D32—24no—
32C22×D32—28no—
33C22×D32—28no—
34C22×D32—26no—
35C22×D32D418yescomplemented
36C22×D32C23112yes—
37C22×D32C2316yes—
38C22×#157—28no—
39C22×#157C23112yes—
40C2×GO+(4,𝔽2)—296no—
41C2×GO+(4,𝔽2)C23132yescomplemented
42C22×C32×C4D411yescharacteristic, complemented
43C22×C3⋊Dic3D411yescharacteristic, complemented
44C2×C4×C3⋊D3D416yescomplemented
45C2×C4×C3⋊D3C2316yes—
46C22×C3×Dic3—21no—
47C2×C3×D12—212no—
48C2×C3×C4×D3—212no—
49C2×#625—212no—
50C2×#624—23no—
51C2×#624C2316yes—
52C2×D3×Dic3—212no—
53#3802—224no—
54#3802C23116yescomplemented
55C4×D32—224no—
56C4×D32C23116yescomplemented
57C4⋊#157C23116yescomplemented
58D32⋊C4C23132yescomplemented
59C22×#121—91nomaximal, Sylow
60C24×D3—61no—
61C24×D3—62no—
62C22×C3×D4—61no—
63C23×C3×C4—61no—
64C22×#67—62no—
65C23×Dic3—61no—
66C2×D3×D4—612no—
67C2×D3×D4—612no—
68C22×D12—21no—
69C22×C4×D3—21no—
70C22×C4×D3—612no—
71C22×C4×D3—61no—
72C22×C4×D3—61no—
73C22×C4×D3—61no—
74C22×C4×D3—612no—
75C22×C4×D3—61no—
76C23×C32C2×D411yescharacteristic
77C22×C3⋊D3C2×D413yes—
78C22×C3⋊D3C2×D414yes—
79C22×C3⋊D3#4114yes—
80C22×C3⋊D3C2413yes—
81C22×C3×D3—28no—
82C22×C3×D3—26no—
83C22×C3×D3—212no—
84C22×C3×D3—48no—
85C2×D32—224no—
86C2×D32—26no—
87C2×D32—212no—
88C2×D32—212no—
89C2×D32—224no—
90C2×D32—224no—
91C2×D32—424no—
92C2×D32C2×D4124yescomplemented
93C2×D32C2418yes—
94C2×D32C2414yes—
95C2×#157—224no—
96C2×#157C2418yes—
97GO+(4,𝔽2)—264no—
98C2×C32×C4C2×D416yescomplemented
99C2×C3⋊Dic3C2×D416yescomplemented
100C4×C3⋊D3C2×D4112yescomplemented
101C4×C3⋊D3C2414yes—
102C2×C3×Dic3—26no—
103C3×D12—216no—
104C3×C4×D3—216no—
105#625—216no—
106#624—26no—
107#624C2414yes—
108D3×Dic3—216no—
109C23×D4—91no—
110C23×D4—92no—
111C24×C4—91no—
112C2×#121—924no—
113C22×C4⋊C4—91no—
114C22×C22⋊C4—92no—
115C24×C3—61no—
116C24×C3—62no—
117C23×D3—21no—
118C23×D3—22no—
119C23×D3—68no—
120C23×D3—66no—
121C23×D3—612no—
122C23×D3—61no—
123C23×D3—61no—
124C23×D3—66no—
125C23×D3—68no—
126C23×D3—612no—
127C23×D3—128no—
128C23×D3—128no—
129C2×C3×D4—612no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—612no—
133C22×C3×C4—61no—
134C2×#67—624no—
135C22×Dic3—21no—
136C22×Dic3—61no—
137C22×Dic3—61no—
138C22×Dic3—612no—
139D3×D4—616no—
140D3×D4—648no—
141C2×D12—212no—
142C2×C4×D3—212no—
143C2×C4×D3—616no—
144C2×C4×D3—66no—
145C2×C4×D3—66no—
146C2×C4×D3—66no—
147C2×C4×D3—66no—
148C2×C4×D3—66no—
149C2×C4×D3—648no—
150C2×C4×D3—66no—
151C2×C4×D3—612no—
152C22×C32#12114yescomplemented
153C22×C32C22×D413yes—
154C2×C3⋊D3#121112yescomplemented
155C2×C3⋊D3C22×D413yes—
156C2×C3⋊D3C22×D416yes—
157C2×C3⋊D3C2×#4116yes—
158C2×C3⋊D3C2511yescharacteristic
159C2×C3×D3—224no—
160C2×C3×D3—24no—
161C2×C3×D3—28no—
162C2×C3×D3—424no—
163D32—224no—
164D32—24no—
165D32—28no—
166D32—28no—
167D32—424no—
168D32C22×D418yescomplemented
169#157—28no—
170C32×C4C22×D414yescomplemented
171C3⋊Dic3C22×D414yescomplemented
172C3×Dic3—24no—
173C25—91no—
174C25—92no—
175C22×D4—61no—
176C22×D4—912no—
177C22×D4—924no—
178C22×D4—186no—
179C22×D4—181no—
180C22×D4—1816no—
181C23×C4—61no—
182C23×C4—91no—
183C23×C4—912no—
184C23×C4—91no—
185C23×C4—92no—
186C23×C4—186no—
187C23×C4—181no—
188#121—964no—
189C2×C4⋊C4—912no—
190C2×C22⋊C4—924no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—66no—
194C23×C3—68no—
195C23×C3—612no—
196C23×C3—128no—
197C22×D3—26no—
198C22×D3—28no—
199C22×D3—212no—
200C22×D3—48no—
201C22×D3—624no—
202C22×D3—64no—
203C22×D3—68no—
204C22×D3—63no—
205C22×D3—63no—
206C22×D3—64no—
207C22×D3—64no—
208C22×D3—64no—
209C22×D3—64no—
210C22×D3—63no—
211C22×D3—63no—
212C22×D3—612no—
213C22×D3—648no—
214C22×D3—624no—
215C22×D3—624no—
216C22×D3—1224no—
217C22×D3—1224no—
218C22×D3—1248no—
219C3×D4—616no—
220C2×C3×C4—26no—
221C2×C3×C4—26no—
222C2×C3×C4—616no—
223C2×C3×C4—66no—
224#67—632no—
225C2×Dic3—26no—
226C2×Dic3—66no—
227C2×Dic3—66no—
228C2×Dic3—616no—
229D12—216no—
230C4×D3—216no—
231C4×D3—64no—
232C4×D3—612no—
233C4×D3—612no—
234C4×D3—64no—
235C4×D3—64no—
236C4×D3—612no—
237C4×D3—616no—
238C2×C32C2×#12116yescomplemented
239C2×C32C23×D411yescharacteristic
240C3⋊D3C2×#12116yescomplemented
241C3⋊D3C23×D411yescharacteristic
242C3⋊D3C22×#4111yescharacteristic
243C3×D3—28no—
244C3×D3—48no—
245C24—61no—
246C24—62no—
247C24—96no—
248C24—98no—
249C24—912no—
250C24—91no—
251C24—183no—
252C24—184no—
253C24—186no—
254C24—188no—
255C24—188no—
256C2×D4—612no—
257C2×D4—916no—
258C2×D4—932no—
259C2×D4—1824no—
260C2×D4—1812no—
261C2×D4—1896no—
262C22×C4—61no—
263C22×C4—612no—
264C22×C4—96no—
265C22×C4—916no—
266C22×C4—91no—
267C22×C4—96no—
268C22×C4—96no—
269C22×C4—912no—
270C22×C4—1824no—
271C22×C4—183no—
272C22×C4—1812no—
273C22×C4—188no—
274C22×C4GO+(4,𝔽2)11yescharacteristic, complemented
275C4⋊C4—916no—
276C22⋊C4—932no—
277C22×C3—23no—
278C22×C3—23no—
279C22×C3—24no—
280C22×C3—24no—
281C22×C3—64no—
282C22×C3—624no—
283C22×C3—68no—
284C22×C3—1224no—
285C2×D3—24no—
286C2×D3—224no—
287C2×D3—28no—
288C2×D3—424no—
289C2×D3—68no—
290C2×D3—63no—
291C2×D3—61no—
292C2×D3—63no—
293C2×D3—61no—
294C2×D3—612no—
295C2×D3—66no—
296C2×D3—66no—
297C2×D3—612no—
298C2×D3—66no—
299C2×D3—66no—
300C2×D3—648no—
301C2×D3—68no—
302C2×D3—128no—
303C2×D3—128no—
304C2×D3—1248no—
305C3×C4—24no—
306C3×C4—24no—
307C3×C4—64no—
308Dic3—24no—
309Dic3—64no—
310Dic3—64no—
311C32C22×#12111yescharacteristic, complemented, Sylow
312C23—66no—
313C23—68no—
314C23—612no—
315C23—94no—
316C23—924no—
317C23—98no—
318C23—93no—
319C23—93no—
320C23—94no—
321C23—94no—
322C23—128no—
323C23—186no—
324C23—1812no—
325C23—1824no—
326C23—1812no—
327C23—1824no—
328C23—1824no—
329C23—3624no—
330C23C2×GO+(4,𝔽2)11yescharacteristic
331D4—616no—
332D4—1816no—
333D4—1864no—
334C2×C4—66no—
335C2×C4—616no—
336C2×C4—94no—
337C2×C4—96no—
338C2×C4—94no—
339C2×C4—912no—
340C2×C4—98no—
341C2×C4—186no—
342C2×C4—1816no—
343C2×C4—1824no—
344C2×C4C2×GO+(4,𝔽2)16yescomplemented
345C2×C3—26no—
346C2×C3—26no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—68no—
350C2×C3—128no—
351D3—28no—
352D3—48no—
353D3—66no—
354D3—61no—
355D3—61no—
356D3—66no—
357D3—61no—
358D3—61no—
359C22—624no—
360C22—64no—
361C22—68no—
362C22—98no—
363C22—96no—
364C22—96no—
365C22—912no—
366C22—91no—
367C22—93no—
368C22—1224no—
369C22—184no—
370C22—1824no—
371C22—188no—
372C22—188no—
373C22—3624no—
374C22#6423014yesdirect product factor
375C22C22×GO+(4,𝔽2)13yes—
376C4—64no—
377C4—94no—
378C4—188no—
379C4C22×GO+(4,𝔽2)14yescomplemented
380C3—21no—
381C3—21no—
382C2—68no—
383C2—91no—
384C2—91no—
385C2—96no—
386C2—128no—
387C2C2×#6423016yesdirect product factor
388C2C23×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC23C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C25
FrattiniC2C23×GO+(4,𝔽2)
FittingC22×C32×C4D4
socleC23×C32C2×D4
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32C22×#121
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C23×GO+(4,𝔽2)C2semidirect
2. C22×#3802C2semidirect
3. C22×C4×D32C2semidirect
4. C22×C4⋊#157C2semidirect
5. C22×D32⋊C4C2semidirect
6. C2×#64230C2direct (N × H)
11. C22×GO+(4,𝔽2)C22semidirect
19. C2×#3802C22semidirect
21. C2×C4×D32C22semidirect
22. C2×C4⋊#157C22semidirect
23. C2×D32⋊C4C22semidirect
24. #64230C22direct (N × H)
35. C22×D32D4semidirect
41. C2×GO+(4,𝔽2)C23semidirect
42. C22×C32×C4D4semidirect
43. C22×C3⋊Dic3D4semidirect
44. C2×C4×C3⋊D3D4semidirect
54. #3802C23semidirect
56. C4×D32C23semidirect
57. C4⋊#157C23semidirect
58. D32⋊C4C23semidirect
92. C2×D32C2×D4semidirect
98. C2×C32×C4C2×D4semidirect
99. C2×C3⋊Dic3C2×D4semidirect
100. C4×C3⋊D3C2×D4semidirect
152. C22×C32#121semidirect
154. C2×C3⋊D3#121semidirect
168. D32C22×D4semidirect
170. C32×C4C22×D4semidirect
171. C3⋊Dic3C22×D4semidirect
238. C2×C32C2×#121semidirect
240. C3⋊D3C2×#121semidirect
274. C22×C4GO+(4,𝔽2)semidirect
311. C32C22×#121semidirect
344. C2×C4C2×GO+(4,𝔽2)semidirect
374. C22#64230direct (N × H)
379. C4C22×GO+(4,𝔽2)semidirect
387. C2C2×#64230direct (N × H)

Maximal subgroups

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

subgroupindex
C23×GO+(4,𝔽2)2
C22×#38022
C22×C4×D322
C22×C4⋊#1572
C22×D32⋊C42
C2×#642302
C22×#1219