Finite Group Databasebeta

← #11777736

Subgroups — #11777736

Order 1152.

365 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C4×#3715C211yesmaximal, characteristic
2#10590339C211yesmaximal, characteristic
3#10590250C211yesmaximal, characteristic, complemented
4#10589321C211yesmaximal, characteristic, complemented
5C3×#4459C211yesmaximal, characteristic
6C3×#4456C211yesmaximal, characteristic, complemented
7#10589100C211yesmaximal, characteristic, complemented
8#68223—31nomaximal
9#68159—31nomaximal
10C3×C4×D12C2211yescharacteristic
11C3×C4×Dic6C2211yescharacteristic
12C2×#3715C411yescharacteristic
13C2×#3711—21no—
14C32×C4×C8C2211yescharacteristic
15C4×C3⋊(C3⋊C8)C2211yescharacteristic
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×C4⋊(C3⋊C8)C2211yescharacteristic
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19#63574C411yescharacteristic
20Dic3⋊(C3⋊C8)—21no—
21#63565C411yescharacteristic
22#63562C411yescharacteristic
23(C2×D3)⋊(C3⋊C8)—21no—
24Dic3×C3⋊C8—21no—
25C4×#215—31no—
26Q8×C3⋊C8—31no—
27C4×#214—31no—
28#5012—31no—
29#4838—31no—
30#4830—31no—
31C3×Q8⋊C8—31no—
32C3×D4⋊C8—31no—
33#4479—31no—
34#4479—31no—
35#4459D311yescharacteristic
36#4456D311yescharacteristic, complemented
37#4455—31no—
38#4455—31no—
39#4455—61no—
40C2×C3×D12C2×C411yescharacteristic
41C2×C3×C4×D3—21no—
42C2×C3×Dic6C2×C411yescharacteristic
43C2×C32×C8—21no—
44C32×C42C2311yescharacteristic
45C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
46C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
47C2×C3⋊(C3⋊C8)D411yescharacteristic
48C2×C3⋊(C3⋊C8)D411yescharacteristic
49C3×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3C2×C411yescharacteristic
51C3×C4⋊Dic3C2×C411yescharacteristic
52C3×Dic3⋊C4—21no—
53C3×C4×Dic3—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56#3715—22no—
57#3711—42no—
58#1588—91noSylow
59C3×C4×Q8—31no—
60C3×C4×D4—31no—
61C2×#215—31no—
62C2×#214—31no—
63C2×#208—61no—
64C2×ΓL(1,𝔽25)—61no—
65C4×D12C2×D311yescharacteristic
66C4×Dic6C2×D311yescharacteristic
67C3×C4⋊C8—31no—
68C3×C4⋊C8—31no—
69C3×C4×C8—21no—
70C3×C4×C8C2×D311yescharacteristic
71C3×C4×C8C2×D311yescharacteristic
72#835—31no—
73#832—31no—
74#830—61no—
75(C2×D3)⋊C8—61no—
76Dic3⋊C8—61no—
77C8×Dic3—61no—
78#810—31no—
79#809—31no—
80#808—31no—
81#808—31no—
82C4⋊(C3⋊C8)—61no—
83C4⋊(C3⋊C8)C2×D311yescharacteristic
84C4⋊(C3⋊C8)C2×D311yescharacteristic
85C4×C3⋊C8—31no—
86C4×C3⋊C8—31no—
87C4×C3⋊C8—31no—
88C4×C3⋊C8—31no—
89C4×C3⋊C8—61no—
90C4×C3⋊C8—61no—
91C4×C3⋊C8—61no—
92C22×C3×D3—21no—
93C2×C32×C4C22×C411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C3×D12—21no—
100C3×D12—21no—
101C3×C4×D3—41no—
102C3×C4×D3—41no—
103C3×Dic6—21no—
104C3×Dic6M1612yescomplemented
105C32×C8—22no—
106C3⋊(C3⋊C8)—21no—
107C3⋊(C3⋊C8)—21no—
108C3⋊(C3⋊C8)—21no—
109C3⋊(C3⋊C8)—21no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8—41no—
112C8×Q8—91no—
113C4×SD16—91no—
114#436—91no—
115#335—91no—
116Q8⋊C8—31no—
117D4⋊C8—31no—
118#322—91no—
119C2×C3×Q8—31no—
120C2×C3×D4—31no—
121C22×C3×C4—61no—
122C2×D12C4×D311yescharacteristic
123C2×C4×D3—21no—
124C2×Dic6C4×D311yescharacteristic
125C2×C3×C8—22no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—61no—
129C2×C3×C8—61no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4—31no—
132C3×C4⋊C4—61no—
133C3×C22⋊C4—61no—
134C3×C42—21no—
135C3×C42—61no—
136C3×C42C22×D311yescharacteristic
137C3×C42C22×D311yescharacteristic
138#215—62no—
139#214—62no—
140(C2×D3)⋊C4—21no—
141C4⋊Dic3C4×D311yescharacteristic
142C4⋊Dic3C4×D311yescharacteristic
143Dic3⋊C4—21no—
144C4×Dic3—21no—
145#208—122no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—31no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—31no—
152C2×C3⋊C8—31no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C2×C3⋊C8—61no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—61no—
160ΓL(1,𝔽25)—122no—
161C22×C32C4×D411yescharacteristic
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C2×C3×D3—41no—
165C32×C4—21no—
166C32×C4C2×M1611yescharacteristic
167C32×C4#13111yescharacteristic
168C32×C4AGL(1,ℤ/8)11yescharacteristic
169C32×C4#13711yescharacteristic
170C3×Dic3—22no—
171C3×Dic3—41no—
172C3×Dic3—41no—
173C2×SD16—91no—
174C2×M16—181no—
175C4×Q8—31no—
176C4×D4—31no—
177#106—91no—
178C4⋊C8—31no—
179C4⋊C8—31no—
180C4⋊C8—181no—
181#103—91no—
182#102—91no—
183C22⋊C8—181no—
184C4×C8—91no—
185C4×C8—91no—
186C4×C8—181no—
187C4×C8D3211yescharacteristic
188C23×C3—61no—
189C22×D3—21no—
190C3×Q8—32no—
191C3×Q8—61no—
192C3×D4—61no—
193C3×D4—61no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—121no—
201C2×C3×C4—121no—
202C2×C3×C4C2×C4×D311yescharacteristic
203C2×C3×C4C2×C4×D311yescharacteristic
204C2×C3×C4D3×D411yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4#23711yescharacteristic
207C2×C3×C4#23711yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211D12—21no—
212D12—21no—
213C4×D3—41no—
214C4×D3—41no—
215Dic6—21no—
216Dic6ΓL(1,𝔽25)12yescomplemented
217C3×C8—24no—
218C3×C8—22no—
219C3×C8—22no—
220C3×C8—121no—
221C3×C8—121no—
222C3⋊C8—41no—
223C3⋊C8—41no—
224C3⋊C8—61no—
225C3⋊C8—61no—
226C3⋊C8—61no—
227C3⋊C8—61no—
228C3⋊C8—61no—
229C3⋊C8—61no—
230C3⋊C8—61no—
231C3⋊C8—61no—
232C3⋊C8—121no—
233C3⋊C8—121no—
234C3⋊C8—121no—
235C3⋊C8—121no—
236C2×C32#43511yescharacteristic
237C2×C32#44011yescharacteristic
238C2×C32#44311yescharacteristic
239C3×D3—81no—
240C2×Q8—31no—
241C2×D4—31no—
242C22×C4—61no—
243SD16—182no—
244M16—362no—
245C2×C8—21no—
246C2×C8—61no—
247C2×C8—61no—
248C2×C8—91no—
249C2×C8—91no—
250C2×C8—91no—
251C2×C8—91no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—61no—
255C22⋊C4—61no—
256C42—61no—
257C42C2×D3211yescharacteristic
258C22×C3—21no—
259C22×C3—121no—
260C22×C3—121no—
261C22×C3—121no—
262C22×C3#67⋊C411yescharacteristic
263C22×C3#67⋊C411yescharacteristic
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—62no—
275C3×C4—121no—
276C3×C4—121no—
277C3×C4C2×ΓL(1,𝔽25)11yescharacteristic
278C3×C4#90111yescharacteristic
279C3×C4D3×M1611yescharacteristic
280C3×C4#90711yescharacteristic
281C3×C4#91111yescharacteristic
282C3×C4#91411yescharacteristic
283C3×C4#91511yescharacteristic
284C3×C4#91811yescharacteristic
285Dic3—22no—
286Dic3—41no—
287Dic3—41no—
288C32#158811yescharacteristic, complemented, Sylow
289C23—61no—
290Q8—32no—
291Q8—61no—
292D4—61no—
293D4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4C4×D3211yescharacteristic
300C2×C4#380211yescharacteristic
301C2×C4#380511yescharacteristic
302C8—22no—
303C8—121no—
304C8—121no—
305C8—181no—
306C8—181no—
307C8—181no—
308C8—181no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—241no—
313C2×C3#472811yescharacteristic
314C2×C3#473011yescharacteristic
315C2×C3#475711yescharacteristic
316C2×C3#481711yescharacteristic
317C2×C3#482411yescharacteristic
318C2×C3#482411yescharacteristic
319D3—81no—
320C22—121no—
321C22—121no—
322C22—121no—
323C22#6390211yescharacteristic
324C4—21no—
325C4—62no—
326C4—121no—
327C4—121no—
328C4D3×ΓL(1,𝔽25)11yescharacteristic
329C4#6379811yescharacteristic
330C4#6380411yescharacteristic
331C4#6381111yescharacteristic
332C3—21no—
333C3#6815911yescharacteristic, complemented
334C3#6822311yescharacteristic, complemented
335C2—241no—
336C2#1059038911yescharacteristic
337C2#1059072511yescharacteristic
338C2#1059074311yescharacteristic
subgroupG/H
centerC2×C4#3802
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4×C8C22
socleC22×C32C4×D4
prime radicalC2×C3×D12C2×C4
cosocleC32×C42C23
hypercenterC4×C8D32
hypocenterC32#1588
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
3. #10590250C2semidirect
4. #10589321C2semidirect
6. C3×#4456C2semidirect
7. #10589100C2semidirect
36. #4456D3semidirect
104. C3×Dic6M16semidirect
216. Dic6ΓL(1,𝔽25)semidirect
288. C32#1588semidirect
333. C3#68159semidirect
334. C3#68223semidirect

Maximal subgroups

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

subgroupindex
C4×#37152
#105903392
#105902502
#105893212
C3×#44592
C3×#44562
#105891002
#682233
#681593