Finite Group Databasebeta

← #11920706

Subgroups — #11920706

Order 1152.

375 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 45 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×#64266C211yesmaximal, characteristic
2C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
3C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73127—31nomaximal
5#70377—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#64266—21no—
8#64266—21no—
9C2×C3⋊GL(2,𝔽3)C411yescharacteristic, complemented
10C2×C3⋊#226C411yescharacteristic, complemented
11C3⋊(#131⋊C3)—21no—
12C3×GU(2,𝔽3)—21no—
13#63557—41nomaximal
14C2×#985—31no—
15C2×#985—61no—
16C2×#950—31no—
17C2×#948D311yescharacteristic, complemented
18C2×#131⋊C3—31no—
19C2×GU(2,𝔽3)D311yescharacteristic, complemented
20C2×C3×#104—31no—
21#5129—31no—
22C2×#805—31no—
23#4558—31no—
24#4524—31no—
25C2×C4×C3⋊D3—41no—
26C3×#41⋊C3D411yescharacteristic
27C3×#41⋊C3D411yescharacteristic
28C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
29C3⋊GL(2,𝔽3)—21no—
30C3⋊#226—21no—
31C2×C3×C3⋊C8—41no—
32C2×C3×C3⋊C8—41no—
33#1876—91noSylow
34C2×C3×#41C2×D311yescharacteristic
35C2×#235—31no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3C2×D311yescharacteristic
38#985—62no—
39#985—62no—
40#985—61no—
41#985—61no—
42C2×GL(2,𝔽3)—31no—
43C2×GL(2,𝔽3)—61no—
44C2×#226—31no—
45C2×#226—61no—
46C2×C3×M16—31no—
47C2×C3×C42—31no—
48#950—61no—
49#950—61no—
50#950—61no—
51#950—61no—
52#948—21no—
53#948—61no—
54C2×#216—31no—
55C2×#215—31no—
56C2×#214—31no—
57C2×#213—31no—
58C2×#208—31no—
59C22×C3⋊C8—31no—
60C22×C3⋊C8—31no—
61#131⋊C3—61no—
62GU(2,𝔽3)—21no—
63C3×#104—61no—
64C3×#104—61no—
65(C2×D3)⋊C8—61no—
66(C2×D3)⋊C8—61no—
67(C2×D3)⋊C8—121no—
68#805—62no—
69C22×C3⋊D3—41no—
70C2×C32×C4—41no—
71C2×C3⋊Dic3—41no—
72C4×C3⋊D3—81no—
73C4×C3⋊D3—81no—
74C3×SL(2,𝔽3)C22⋊C411yescharacteristic
75C3×C3⋊C8—81no—
76C3×C3⋊C8—81no—
77C2×#135—91no—
78C2×#131—31no—
79C2×#104—31no—
80C2×#104—91no—
81#408—91no—
82#341—91no—
83#336—91no—
84C3×#41—31no—
85C3×#41—31no—
86C3×#41D1211yescharacteristic
87C3×#41#6711yescharacteristic
88C2×C3×Q8C4×D311yescharacteristic
89C2×C3×D4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—61no—
92C2×#67—61no—
93#235—61no—
94#235—61no—
95#235—61no—
96#235—61no—
97C2×D12—31no—
98C2×C4×D3—61no—
99C2×C4×D3—121no—
100C2×C4×D3—241no—
101C2×Dic6—31no—
102#41⋊C3—21no—
103#41⋊C3—21no—
104#41⋊C3D1211yescharacteristic
105#41⋊C3#6711yescharacteristic
106C2×SL(2,𝔽3)—21no—
107C2×SL(2,𝔽3)C4×D311yescharacteristic
108GL(2,𝔽3)—62no—
109GL(2,𝔽3)—61no—
110#226—62no—
111#226—61no—
112C3×M16—61no—
113C3×M16—61no—
114C2×C3×C8—61no—
115C2×C3×C8—121no—
116C3×C42—61no—
117C3×C42—61no—
118#216—61no—
119#216—61no—
120#215—61no—
121#215—61no—
122#214—61no—
123#214—61no—
124#213—61no—
125#213—61no—
126#208—32no—
127#208—61no—
128C2×C3⋊C8—31no—
129C2×C3⋊C8—31no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—61no—
133C2×C3⋊C8—61no—
134C2×C3⋊C8—61no—
135C2×C3⋊C8—61no—
136C2×C3⋊C8S411yescharacteristic
137C22×C32—41no—
138C2×C3⋊D3—42no—
139C2×C3⋊D3—81no—
140C2×C3⋊D3—81no—
141C32×C4—41no—
142C32×C4—41no—
143C3⋊Dic3—81no—
144C2×#41—91no—
145C2×#41D3211yescharacteristic
146#135—181no—
147#135—181no—
148#135—181no—
149#135—181no—
150C2×Q16—91no—
151C2×SD16—91no—
152C2×SD16—91no—
153C2×D8—91no—
154#131—61no—
155#131—181no—
156C2×M16—31no—
157C2×M16—91no—
158C22×C8—91no—
159C22×C8—91no—
160C2×C42—31no—
161#104—61no—
162#104—61no—
163#104—182no—
164C22⋊C8—181no—
165C22⋊C8—181no—
166C23×C3—31no—
167C22×D3—61no—
168C22×D3—121no—
169C22×D3—241no—
170C3×Q8—31no—
171C3×Q8(C2×D3)⋊C411yescharacteristic
172C3×D4—31no—
173C3×D4—31no—
174C3×D4—61no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—41no—
181C2×C3×C4—62no—
182C2×C3×C4—81no—
183C2×C3×C4—121no—
184C2×C3×C4—121no—
185C2×C3×C4C2×S411yescharacteristic
186#67—121no—
187#67—121no—
188C2×Dic3—61no—
189C2×Dic3—121no—
190C2×Dic3—241no—
191D12—61no—
192D12—61no—
193C4×D3—121no—
194C4×D3—121no—
195C4×D3—241no—
196C4×D3—241no—
197C4×D3—242no—
198C4×D3—242no—
199Dic6—61no—
200Dic6—61no—
201SL(2,𝔽3)—21no—
202SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
203C3×C8—121no—
204C3×C8—241no—
205C3⋊C8—21no—
206C3⋊C8—61no—
207C3⋊C8—61no—
208C3⋊C8—61no—
209C3⋊C8—81no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C3⋊D3—82no—
214#41—31no—
215#41—31no—
216#41—181no—
217#41—181no—
218#41—181no—
219#41—181no—
220#41#62611yescharacteristic
221#41#62611yescharacteristic
222C2×Q8—91no—
223C2×Q8#62411yescharacteristic
224C2×D4—31no—
225C2×D4—91no—
226C2×D4—181no—
227C22×C4—31no—
228C22×C4—61no—
229C22×C4—181no—
230Q16—181no—
231Q16—181no—
232SD16—181no—
233SD16—181no—
234SD16—181no—
235SD16—181no—
236D8—181no—
237D8—181no—
238M16—61no—
239M16—61no—
240M16—92no—
241M16—181no—
242C2×C8—31no—
243C2×C8—61no—
244C2×C8—91no—
245C2×C8—91no—
246C2×C8—91no—
247C2×C8—181no—
248C2×C8—181no—
249C2×C8—181no—
250C2×C8—181no—
251C42—61no—
252C42—61no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—41no—
256C22×C3—61no—
257C22×C3—61no—
258C22×C3—81no—
259C22×C3C4×S411yescharacteristic
260C2×D3—122no—
261C2×D3—121no—
262C2×D3—121no—
263C2×D3—121no—
264C2×D3—241no—
265C2×D3—241no—
266C2×D3—242no—
267C2×D3—242no—
268C2×D3—242no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—81no—
274C3×C4—81no—
275C3×C4—122no—
276C3×C4C4⋊S411yescharacteristic
277C3×C4GL(2,ℤ/4)11yescharacteristic
278Dic3—121no—
279Dic3—241no—
280Dic3—242no—
281C32—41noSylow
282C23—31no—
283C23—181no—
284Q8—31no—
285Q8—181no—
286Q8—181no—
287Q8#372211yescharacteristic
288D4—31no—
289D4—31no—
290D4—61no—
291D4—181no—
292D4—181no—
293D4—361no—
294D4—361no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—62no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—181no—
304C2×C4—361no—
305C2×C4—361no—
306C2×C4D3×S411yescharacteristic
307C8—61no—
308C8—121no—
309C8—181no—
310C8—181no—
311C8—181no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—61no—
316C2×C3—61no—
317C2×C3—81no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3#540411yescharacteristic
321C2×C3#540511yescharacteristic
322C2×C3(C2×S4)⋊C411yescharacteristic
323D3—242no—
324D3—244no—
325D3—241no—
326C22—31no—
327C22—31no—
328C22—61no—
329C22—61no—
330C22—361no—
331C22—361no—
332C22—361no—
333C22(C3⋊S4)⋊C411yescharacteristic
334C4—31no—
335C4—31no—
336C4—122no—
337C4—361no—
338C4#6420711yescharacteristic
339C4#6421011yescharacteristic
340C3—41no—
341C3—81no—
342C3#7312711yescharacteristic, complemented
343C2—61no—
344C2—61no—
345C2—721no—
346C2#1059322011yescharacteristic
347C2#1059322211yescharacteristic
348C2#1059324211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D3×S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC2×C3×#41C2×D3
socleC22×C3C4×S4
prime radicalC2×#64266C2
cosocleC2×C3×#41⋊C3C22
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(#131⋊C3)C2semidirect
3. C2×C3×GU(2,𝔽3)C2semidirect
9. C2×C3⋊GL(2,𝔽3)C4semidirect
10. C2×C3⋊#226C4semidirect
17. C2×#948D3semidirect
19. C2×GU(2,𝔽3)D3semidirect
342. C3#73127semidirect

Maximal subgroups

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

subgroupindex
C2×#642662
C2×C3⋊(#131⋊C3)2
C2×C3×GU(2,𝔽3)2
#731273
#703773
#635574