Finite Group Databasebeta

← #11773985

Subgroups — #11773985

Order 1152.

840 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 131 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×#64103C211yesmaximal, characteristic
2#10592425C211yesmaximal, characteristic, complemented
3C3⋊D3×C22⋊C8C211yesmaximal, characteristic, complemented
4C32×#324C211yesmaximal, characteristic, complemented
5#10589394C211yesmaximal, characteristic, complemented
6#10589334C211yesmaximal, characteristic, complemented
7#10589334C211yesmaximal, characteristic, complemented
8#68146—34nomaximal
9C22×C4⋊(C3⋊D3)C411yescharacteristic
10C22×C4×C3⋊D3C2211yescharacteristic
11C2×C32×C4⋊C4C2211yescharacteristic
12C2×#3752C411yescharacteristic
13C2×#3752C411yescharacteristic
14C2×#3752C2211yescharacteristic
15C2×#3750C411yescharacteristic
16C2×#3743—21no—
17C2×C8×C3⋊D3—21no—
18#64103—21no—
19#64103—21no—
20#64103—21no—
21#64103—21no—
22C32×C22⋊C8C2211yescharacteristic
23C32×C22⋊C8C2211yescharacteristic
24#63657C2211yescharacteristic
25#63657C2211yescharacteristic
26#63647—21no—
27#63647—21no—
28C2×#894—34no—
29#4726—34no—
30D3×C22⋊C8—34no—
31C3×#324D314yescomplemented
32#4528—34no—
33#4468—34no—
34#4468—34no—
35C23×C3⋊D3C2×C411yescharacteristic
36C23×C3⋊D3C2×C411yescharacteristic
37C22×C32×C4C2×C411yescharacteristic
38C22×C32×C4C2×C411yescharacteristic
39C22×C32×C4C2311yescharacteristic
40C22×C3⋊Dic3C2×C411yescharacteristic
41C22×C3⋊Dic3C2×C411yescharacteristic
42C2×C4⋊(C3⋊D3)—21no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C4⋊(C3⋊D3)—42no—
45C2×C4×C3⋊D3—42no—
46C2×C4×C3⋊D3D411yescharacteristic
47C2×C4×C3⋊D3D411yescharacteristic
48C2×C4×C3⋊D3D411yescharacteristic
49C2×C4×C3⋊D3D411yescharacteristic
50C2×C32×C8—21no—
51C2×C32×C8—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54#3752—21no—
55#3752—21no—
56#3752—21no—
57#3752—21no—
58#3752—21no—
59#3752—21no—
60#3750—21no—
61#3750—21no—
62C2×C3⋊(C3⋊C8)—21no—
63C2×C3⋊(C3⋊C8)—21no—
64#3743—42no—
65C8×C3⋊D3—42no—
66#1468—91noSylow
67C22×D12—34no—
68C22×C4×D3—34no—
69C2×C3×C4⋊C4C2×D314yes—
70C2×(C2×D3)⋊C4—34no—
71C2×(C2×D3)⋊C4—34no—
72C2×(C2×D3)⋊C4—34no—
73C2×Dic3⋊C4—34no—
74C2×ΓL(1,𝔽25)—64no—
75C2×D3×C8—64no—
76#894—64no—
77#894—64no—
78#894—64no—
79#894—64no—
80C3×C22⋊C8C2×D314yes—
81C3×C22⋊C8C2×D314yes—
82#830—34no—
83#830—34no—
84(C2×D3)⋊C8—64no—
85(C2×D3)⋊C8—64no—
86C23×C32C22×C411yescharacteristic
87C22×C3⋊D3—21no—
88C22×C3⋊D3—21no—
89C22×C3⋊D3—22no—
90C22×C3⋊D3—21no—
91C22×C3⋊D3—21no—
92C22×C3⋊D3—21no—
93C22×C3⋊D3—41no—
94C22×C3⋊D3—41no—
95C22×C3⋊D3—41no—
96C22×C3⋊D3C22⋊C411yescharacteristic
97C22×C3⋊D3C22⋊C411yescharacteristic
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—41no—
108C2×C3⋊Dic3—41no—
109C2×C3⋊Dic3C22⋊C411yescharacteristic
110C2×C3⋊Dic3C22⋊C411yescharacteristic
111C4⋊(C3⋊D3)—42no—
112C4⋊(C3⋊D3)—42no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3—22no—
115C4×C3⋊D3—41no—
116C4×C3⋊D3—41no—
117C32×C8—41no—
118C32×C8—41no—
119C3⋊(C3⋊C8)—41no—
120C3⋊(C3⋊C8)—41no—
121C2×#123—91no—
122#407—91no—
123C2×C22⋊C8—91no—
124#324—91no—
125#324C3⋊D311yescharacteristic, complemented
126#323—91no—
127#323—91no—
128C23×D3—34no—
129C23×D3—34no—
130C22×C3×C4C4×D314yes—
131C22×C3×C4C4×D314yes—
132C22×C3×C4C22×D314yes—
133C22×Dic3—34no—
134C22×Dic3—34no—
135C2×D12—64no—
136C2×D12—64no—
137C2×D12—128no—
138C2×C4×D3—34no—
139C2×C4×D3—34no—
140C2×C4×D3—34no—
141C2×C4×D3—34no—
142C2×C4×D3—128no—
143C2×C3×C8—24no—
144C2×C3×C8—24no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—24no—
147(C2×D3)⋊C4—64no—
148(C2×D3)⋊C4—64no—
149(C2×D3)⋊C4—64no—
150(C2×D3)⋊C4—64no—
151(C2×D3)⋊C4—64no—
152(C2×D3)⋊C4—64no—
153Dic3⋊C4—64no—
154Dic3⋊C4—64no—
155C2×C3⋊C8—64no—
156C2×C3⋊C8—64no—
157ΓL(1,𝔽25)—128no—
158D3×C8—128no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32C2×C22⋊C411yescharacteristic
162C22×C32#13111yescharacteristic
163C22×C32#13111yescharacteristic
164C2×C3⋊D3—21no—
165C2×C3⋊D3—21no—
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3⋊D3—42no—
170C2×C3⋊D3—42no—
171C2×C3⋊D3—41no—
172C2×C3⋊D3—41no—
173C2×C3⋊D3—41no—
174C2×C3⋊D3—81no—
175C2×C3⋊D3—81no—
176C2×C3⋊D3#9912yes—
177C32×C4—21no—
178C32×C4—21no—
179C32×C4—41no—
180C32×C4—41no—
181C3⋊Dic3—21no—
182C3⋊Dic3—21no—
183C3⋊Dic3—41no—
184C22×D4—91no—
185C23×C4—91no—
186C2×M16—181no—
187C22×C8—181no—
188#123—181no—
189#123—181no—
190#123—181no—
191#123—181no—
192C2×C4⋊C4—91no—
193C2×C4⋊C4C2×C3⋊D311yescharacteristic
194C2×C22⋊C4—91no—
195C2×C22⋊C4—91no—
196C2×C22⋊C4—91no—
197C22⋊C8—91no—
198C22⋊C8—91no—
199C22⋊C8—181no—
200C22⋊C8—181no—
201C22⋊C8C2×C3⋊D311yescharacteristic
202C22⋊C8C2×C3⋊D311yescharacteristic
203C23×C3C2×C4×D314yes—
204C22×D3—34no—
205C22×D3—34no—
206C22×D3—68no—
207C22×D3—64no—
208C22×D3—64no—
209C22×D3—64no—
210C22×D3—64no—
211C22×D3—64no—
212C22×D3—124no—
213C22×D3—124no—
214C22×D3—124no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—44no—
220C2×C3×C4—44no—
221C2×C3×C4D3×D414yes—
222C2×C3×C4D3×D414yes—
223C2×Dic3—34no—
224C2×Dic3—34no—
225C2×Dic3—64no—
226C2×Dic3—124no—
227C2×Dic3—124no—
228D12—128no—
229D12—128no—
230C4×D3—68no—
231C4×D3—68no—
232C4×D3—124no—
233C4×D3—124no—
234C3×C8—44no—
235C3×C8—44no—
236C3⋊C8—124no—
237C3⋊C8—124no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32#40811yescharacteristic
241C2×C32C2×#9911yescharacteristic
242C2×C32#41311yescharacteristic
243C3⋊D3—22no—
244C3⋊D3—41no—
245C3⋊D3—81no—
246C24—91no—
247C24—91no—
248C2×D4—181no—
249C2×D4—181no—
250C2×D4—362no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4—91no—
254C22×C4—91no—
255C22×C4—91no—
256C22×C4—91no—
257C22×C4—362no—
258C22×C4C4×C3⋊D311yescharacteristic
259C22×C4C4×C3⋊D311yescharacteristic
260C22×C4C22×C3⋊D311yescharacteristic
261M16—362no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—181no—
265C2×C8—181no—
266C2×C8—362no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—181no—
270C4⋊C4—181no—
271C22⋊C4—181no—
272C22⋊C4—181no—
273C22⋊C4—181no—
274C22⋊C4—181no—
275C22⋊C4—181no—
276C22⋊C4—181no—
277C22×C3—24no—
278C22×C3—24no—
279C22×C3D3×C22⋊C414yes—
280C22×C3#90114yes—
281C22×C3#90714yes—
282C2×D3—38no—
283C2×D3—64no—
284C2×D3—64no—
285C2×D3—64no—
286C2×D3—64no—
287C2×D3—64no—
288C2×D3—128no—
289C2×D3—128no—
290C2×D3—124no—
291C2×D3—124no—
292C2×D3—124no—
293C2×D3—244no—
294C2×D3—244no—
295C3×C4—24no—
296C3×C4—24no—
297C3×C4—44no—
298C3×C4—44no—
299Dic3—64no—
300Dic3—64no—
301Dic3—124no—
302C32#146811yescharacteristic, complemented, Sylow
303C23—91no—
304C23—91no—
305C23—182no—
306C23—181no—
307C23—181no—
308C23—181no—
309C23—181no—
310C23—181no—
311C23—361no—
312C23—361no—
313C23—361no—
314C23C2×C4×C3⋊D311yescharacteristic
315D4—362no—
316D4—362no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—91no—
324C2×C4—91no—
325C2×C4—182no—
326C2×C4—182no—
327C2×C4—181no—
328C2×C4—361no—
329C2×C4—361no—
330C2×C4—361no—
331C2×C4—361no—
332C2×C4D4×C3⋊D311yescharacteristic
333C2×C4D4×C3⋊D311yescharacteristic
334C8—41no—
335C8—41no—
336C8—361no—
337C8—361no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3#472714yes—
341C2×C3D3×#9914yes—
342C2×C3#475114yes—
343D3—68no—
344D3—124no—
345D3—244no—
346C22—21no—
347C22—21no—
348C22—92no—
349C22—181no—
350C22—181no—
351C22—181no—
352C22—181no—
353C22—181no—
354C22—362no—
355C22—362no—
356C22—361no—
357C22—361no—
358C22—361no—
359C22—721no—
360C22—721no—
361C22C22⋊C4×C3⋊D311yescharacteristic
362C22#6411011yescharacteristic
363C22#6411611yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—181no—
369C4—181no—
370C4—361no—
371C3#6814614yescomplemented
372C2—21no—
373C2—21no—
374C2—182no—
375C2—361no—
376C2—721no—
377C2#1059242611yescharacteristic
378C2C3⋊D3×#9911yescharacteristic
379C2#1059245011yescharacteristic
subgroupG/H
centerC22C22⋊C4×C3⋊D3
commutatorC23×C32C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×#324C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C4⋊(C3⋊D3)C4
cosocleC22×C32×C4C23
hypercenter#324C3⋊D3
hypocenterC32#1468
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #10592425C2semidirect
3. C3⋊D3×C22⋊C8C2semidirect
4. C32×#324C2semidirect
5. #10589394C2semidirect
6. #10589334C2semidirect
7. #10589334C2semidirect
31. C3×#324D3semidirect
125. #324C3⋊D3semidirect
302. C32#1468semidirect
371. C3#68146semidirect

Maximal subgroups

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

subgroupindex
C2×#641032
#105924252
C3⋊D3×C22⋊C82
C32×#3242
#105893942
#105893342
#681463