Finite Group Databasebeta

← #11786398

Subgroups — #11786398

Order 1152.

749 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3712C211yesmaximal, characteristic
2C2×#63647C211yesmaximal, characteristic
3C2×C3×Dic3⋊C8C212yesmaximal, complemented
4C4×#3722C211yesmaximal, characteristic
5#10589155C212yesmaximal, complemented
6#70478—32nomaximal
7C22×C4×C3⋊D3C2211yescharacteristic
8C22×C32×C8C2211yescharacteristic
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C2×C3×C4×Dic3C2212yes—
11C22×C3×C3⋊C8C2212yes—
12C2×#3722C412yes—
13C2×#3712—22no—
14C2×#3712—24no—
15#63647—22no—
16C3×Dic3⋊C8—24no—
17#63579C411yescharacteristic
18#63579C411yescharacteristic
19C22×ΓL(1,𝔽25)—32no—
20C2×C3×C4⋊C8—32no—
21C2×(C2×D3)⋊C8—32no—
22C2×(C2×D3)⋊C8—61no—
23C2×Dic3⋊C8D312yescomplemented
24C4×(C2×D3)⋊C4—32no—
25#4550—32no—
26#4524—32no—
27C23×C3⋊D3C2×C411yescharacteristic
28C22×C32×C4C2311yescharacteristic
29C22×C3⋊Dic3C2×C411yescharacteristic
30C2×C4×C3⋊D3—22no—
31C2×C4×C3⋊D3—22no—
32C2×C4×C3⋊D3D412yes—
33C2×C4×C3⋊D3Q812yes—
34C22×C3×Dic3C2×C412yes—
35C22×C3×Dic3C2×C412yes—
36C2×C32×C8—21no—
37C2×C32×C8—22no—
38C2×C3⋊(C3⋊C8)—21no—
39C2×C3⋊(C3⋊C8)—22no—
40C3×C4×Dic3—22no—
41C3×C4×Dic3—22no—
42C2×C3×C3⋊C8—22no—
43C2×C3×C3⋊C8—22no—
44C2×C3×C3⋊C8D414yes—
45#3722—24no—
46#3712—28no—
47#1851—91noSylow
48C22×C4×D3—32no—
49C22×C4×D3—61no—
50C22×C3×C8—21no—
51C22×C3×C8—32no—
52C22×C3×C8C2×D312yes—
53C2×C3×C42—32no—
54C2×(C2×D3)⋊C4—32no—
55C2×(C2×D3)⋊C4—32no—
56C2×C4×Dic3C2×D312yes—
57C22×C3⋊C8—32no—
58C22×C3⋊C8—61no—
59C22×C3⋊C8C2×D312yes—
60C2×ΓL(1,𝔽25)—64no—
61C2×ΓL(1,𝔽25)—64no—
62C2×ΓL(1,𝔽25)—64no—
63C3×C4⋊C8—64no—
64C3⋊#95—32no—
65C3⋊#95—32no—
66(C2×D3)⋊C8—64no—
67(C2×D3)⋊C8—64no—
68Dic3⋊C8—24no—
69C23×C32C22×C411yescharacteristic
70C22×C3⋊D3—21no—
71C22×C3⋊D3—21no—
72C22×C3⋊D3—22no—
73C22×C3⋊D3—22no—
74C22×C3⋊D3C4⋊C412yes—
75C2×C32×C4C2×D411yescharacteristic
76C2×C32×C4C2×D412yes—
77C2×C32×C4C2×Q811yescharacteristic
78C2×C32×C4#4112yes—
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3C4⋊C412yes—
81C4×C3⋊D3—22no—
82C4×C3⋊D3—24no—
83C4×C3⋊D3—22no—
84C2×C3×Dic3—22no—
85C2×C3×Dic3—22no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3M1614yescomplemented
90C32×C8—41no—
91C3⋊(C3⋊C8)—41no—
92C3×C3⋊C8—24no—
93C22×M16—91no—
94C2×C4⋊C8—32no—
95C2×C22⋊C8—91no—
96C4×C22⋊C4—91no—
97#336—92no—
98C23×D3—32no—
99C23×D3—61no—
100C22×C3×C4—21no—
101C22×C3×C4—32no—
102C22×C3×C4—32no—
103C22×C3×C4C22×D312yes—
104C22×Dic3—32no—
105C22×Dic3—61no—
106C22×Dic3C4×D312yes—
107C22×Dic3C4×D312yes—
108C2×C4×D3—34no—
109C2×C4×D3—34no—
110C2×C4×D3—62no—
111C2×C4×D3—62no—
112C2×C4×D3—62no—
113C2×C4×D3—62no—
114C2×C4×D3—62no—
115C2×C4×D3—62no—
116C2×C4×D3—122no—
117C2×C4×D3—122no—
118C2×C3×C8—22no—
119C2×C3×C8—22no—
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C2×C3×C8—34no—
123C2×C3×C8—42no—
124C2×C3×C8—62no—
125C2×C3×C8—62no—
126C3×C42—62no—
127C3×C42—62no—
128(C2×D3)⋊C4—64no—
129(C2×D3)⋊C4—64no—
130C4×Dic3—22no—
131C4×Dic3—22no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—62no—
135C2×C3⋊C8—62no—
136C2×C3⋊C8—62no—
137C2×C3⋊C8—62no—
138C2×C3⋊C8—122no—
139C2×C3⋊C8#6714yes—
140ΓL(1,𝔽25)—616no—
141C22×C32C2×C4⋊C411yescharacteristic
142C22×C32C4×D412yes—
143C22×C32C2×M1612yes—
144C22×C32#13111yescharacteristic
145C22×C32#13111yescharacteristic
146C2×C3⋊D3—22no—
147C2×C3⋊D3—22no—
148C2×C3⋊D3—22no—
149C2×C3⋊D3—42no—
150C2×C3⋊D3—42no—
151C32×C4#12011yescharacteristic
152C32×C4C22⋊Q812yes—
153C32×C4#12311yescharacteristic
154C3⋊Dic3—22no—
155C3×Dic3—24no—
156C3×Dic3—42no—
157C23×C4—91no—
158C2×M16—184no—
159C2×M16—182no—
160C22×C8—32no—
161C22×C8—91no—
162C22×C8D3211yescharacteristic
163C2×C22⋊C4—92no—
164C2×C42—32no—
165C4⋊C8—64no—
166C22⋊C8—182no—
167#95—91no—
168#95—91no—
169C23×C3—21no—
170C23×C3C2×C4×D312yes—
171C22×D3—34no—
172C22×D3—62no—
173C22×D3—62no—
174C22×D3—62no—
175C22×D3—62no—
176C22×D3—62no—
177C22×D3—62no—
178C22×D3—62no—
179C22×D3—62no—
180C22×D3—62no—
181C22×D3—122no—
182C22×D3—122no—
183C2×C3×C4—21no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—21no—
187C2×C3×C4—34no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4#23512yes—
194C2×C3×C4D3×D412yes—
195C2×C3×C4D3×D412yes—
196C2×C3×C4D3×Q812yes—
197C2×C3×C4#23912yes—
198C2×C3×C4C2×#6712yes—
199C2×Dic3—22no—
200C2×Dic3—22no—
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—22no—
204C2×Dic3—34no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—122no—
209C2×Dic3ΓL(1,𝔽25)14yescomplemented
210C4×D3—64no—
211C4×D3—64no—
212C4×D3—64no—
213C4×D3—64no—
214C4×D3—124no—
215C4×D3—122no—
216C4×D3—122no—
217C3×C8—42no—
218C3×C8—42no—
219C3×C8—64no—
220C3⋊C8—24no—
221C3⋊C8—122no—
222C3⋊C8—122no—
223C2×C32#38511yescharacteristic
224C2×C32#42311yescharacteristic
225C2×C32#42411yescharacteristic
226C2×C32#43512yes—
227C2×C32#43612yes—
228C3⋊D3—42no—
229C24—91no—
230C22×C4—32no—
231C22×C4—32no—
232C22×C4—91no—
233C22×C4—92no—
234C22×C4—92no—
235C22×C4—182no—
236C22×C4—182no—
237C22×C4C2×D3211yescharacteristic
238M16—188no—
239C2×C8—22no—
240C2×C8—21no—
241C2×C8—34no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—182no—
245C2×C8—181no—
246C22⋊C4—184no—
247C42—62no—
248C42—62no—
249C22×C3—22no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—22no—
254C22×C3D3×C4⋊C412yes—
255C22×C3#89312yes—
256C22×C3C2×ΓL(1,𝔽25)12yes—
257C22×C3#90112yes—
258C22×C3D3×M1612yes—
259C22×C3#90712yes—
260C22×C3C4×#6712yes—
261C2×D3—64no—
262C2×D3—64no—
263C2×D3—64no—
264C2×D3—64no—
265C2×D3—122no—
266C2×D3—122no—
267C2×D3—124no—
268C2×D3—124no—
269C2×D3—122no—
270C2×D3—122no—
271C2×D3—122no—
272C2×D3—122no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—64no—
277C3×C4—122no—
278C3×C4#89412yes—
279C3×C4#89612yes—
280C3×C4#93712yes—
281C3×C4#94612yes—
282Dic3—24no—
283Dic3—42no—
284Dic3—64no—
285Dic3—122no—
286C32#185111yescharacteristic, complemented, Sylow
287C23—92no—
288C23—182no—
289C23—181no—
290C23—181no—
291C23—182no—
292C23C4×D3211yescharacteristic
293C2×C4—34no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—92no—
300C2×C4—182no—
301C2×C4—184no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4#379711yescharacteristic
305C2×C4#379912yes—
306C2×C4D3×#6712yes—
307C2×C4#381111yescharacteristic
308C8—41no—
309C8—64no—
310C8—361no—
311C2×C3—22no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3#483612yes—
317C2×C3#483712yes—
318C2×C3#483812yes—
319C2×C3#483912yes—
320C2×C3#498912yes—
321C2×C3#511112yes—
322C2×C3#512812yes—
323D3—124no—
324D3—124no—
325C22—182no—
326C22—182no—
327C22—182no—
328C22—362no—
329C22—362no—
330C22D3×ΓL(1,𝔽25)12yes—
331C22#6380311yescharacteristic
332C22#6380511yescharacteristic
333C22#6388311yescharacteristic
334C22#6390412yes—
335C4—64no—
336C4—122no—
337C4—182no—
338C4#6388812yes—
339C4#6389711yescharacteristic
340C4#6398311yescharacteristic
341C3—21no—
342C3#7047812yescomplemented
343C2—362no—
344C2#1059066611yescharacteristic
345C2#1059070111yescharacteristic
346C2#1059072712yes—
347C2#1059073112yes—
348C2#1059137411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C2×D32
commutator, socleC23×C32C22×C4
FittingC22×C32×C8C22
prime radicalC23×C3⋊D3C2×C4
cosocleC22×C32×C4C23
hypercenterC22×C8D32
hypocenterC32#1851
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. C2×C3×Dic3⋊C8C2semidirect
5. #10589155C2semidirect
23. C2×Dic3⋊C8D3semidirect
89. C2×C3×Dic3M16semidirect
209. C2×Dic3ΓL(1,𝔽25)semidirect
286. C32#1851semidirect
342. C3#70478semidirect

Maximal subgroups

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

subgroupindex
C22×#37122
C2×#636472
C2×C3×Dic3⋊C82
C4×#37222
#105891552
#704783