Finite Group Databasebeta

← #11920705

Subgroups — #11920705

Order 1152.

351 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 47 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×C3×#985C211yesmaximal, characteristic
2C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73127—31nomaximal
5#72087—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#985—21no—
8C3×#985—21no—
9C2×C3×GL(2,𝔽3)C411yescharacteristic, complemented
10C2×C3×#226C411yescharacteristic, complemented
11C3⋊(#131⋊C3)—21no—
12C3⋊GU(2,𝔽3)—21no—
13(C2×D3)⋊(C3⋊C8)—41nomaximal
14C2×#985D311yescharacteristic
15C2×C3×#135—31no—
16C2×#948D311yescharacteristic, complemented
17C2×#131⋊C3—31no—
18C2×GU(2,𝔽3)—31no—
19C2×GU(2,𝔽3)—61no—
20C2×#837—31no—
21C2×#837—31no—
22#5234—31no—
23#4552—31no—
24#4550—31no—
25C2×C3×C4×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×#41—31no—
35C2×C3×#41C2×D311yescharacteristic
36C2×#41⋊C3—21no—
37C2×#41⋊C3C2×D311yescharacteristic
38#985—21no—
39#985—21no—
40C2×GL(2,𝔽3)Dic311yescharacteristic, complemented
41C2×#226Dic311yescharacteristic, complemented
42C3×#135—61no—
43C3×#135—61no—
44C3×#135—61no—
45C3×#135—61no—
46C2×C3×Q16—31no—
47C2×C3×SD16—31no—
48C2×C3×SD16—31no—
49C2×C3×D8—31no—
50C22×C3×C8—31no—
51#948—21no—
52#948—61no—
53C2×C4×Dic3—31no—
54C2×#208—31no—
55C2×#208—31no—
56C22×C3⋊C8—31no—
57#131⋊C3—61no—
58GU(2,𝔽3)—62no—
59GU(2,𝔽3)—61no—
60#837—62no—
61#837—61no—
62#837—61no—
63#830—61no—
64#830—61no—
65(C2×D3)⋊C8—121no—
66C22×C3×D3—41no—
67C2×C32×C4—41no—
68C2×C3×Dic3—41no—
69C3×C4×D3—81no—
70C3×C4×D3—81no—
71C3×SL(2,𝔽3)C22⋊C411yescharacteristic
72C3⋊(C3⋊C8)—81no—
73C3×C3⋊C8—81no—
74C2×#135—31no—
75C2×#131—31no—
76C2×#104—91no—
77C2×#104—91no—
78#408—91no—
79#341—91no—
80#336—91no—
81C3×#41—31no—
82C3×#41—31no—
83C3×#41—61no—
84C3×#41—61no—
85C3×#41—61no—
86C3×#41—61no—
87C3×#41D1211yescharacteristic
88C3×#41#6711yescharacteristic
89C2×C3×Q8—31no—
90C2×C3×Q8C4×D311yescharacteristic
91C2×C3×D4—31no—
92C2×C3×D4—31no—
93C2×C3×D4—61no—
94C22×C3×C4—31no—
95C22×C3×C4—61no—
96C22×Dic3—61no—
97C2×C4×D3—41no—
98#41⋊C3—21no—
99#41⋊C3—21no—
100#41⋊C3#6711yescharacteristic
101#41⋊C3#6711yescharacteristic
102C2×SL(2,𝔽3)—21no—
103C2×SL(2,𝔽3)C2×Dic311yescharacteristic
104GL(2,𝔽3)—21no—
105#226—21no—
106C3×Q16—61no—
107C3×Q16—61no—
108C3×SD16—61no—
109C3×SD16—61no—
110C3×SD16—61no—
111C3×SD16—61no—
112C3×D8—61no—
113C3×D8—61no—
114C2×C3×C8—31no—
115C2×C3×C8—31no—
116C2×C3×C8—61no—
117C2×C3×C8—61no—
118C2×C3×C8—121no—
119C4×Dic3—61no—
120C4×Dic3—61no—
121#208—32no—
122#208—61no—
123#208—61no—
124#208—61no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—61no—
127C2×C3⋊C8—61no—
128C2×C3⋊C8—61no—
129C2×C3⋊C8—121no—
130C2×C3⋊C8—241no—
131C2×C3⋊C8S411yescharacteristic
132C22×C32—41no—
133C2×C3×D3—42no—
134C2×C3×D3—81no—
135C2×C3×D3—81no—
136C32×C4—41no—
137C32×C4—41no—
138C3×Dic3—81no—
139C2×#41—31no—
140C2×#41D3211yescharacteristic
141#135—61no—
142#135—61no—
143#135—61no—
144#135—61no—
145C2×Q16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×D8—31no—
149#131—61no—
150#131—181no—
151C2×M16—91no—
152C2×M16—91no—
153C22×C8—31no—
154C22×C8—91no—
155C2×C42—91no—
156#104—182no—
157#104—181no—
158#104—181no—
159C22⋊C8—181no—
160C22⋊C8—181no—
161C23×C3—31no—
162C23×C3—61no—
163C22×D3—41no—
164C3×Q8—31no—
165C3×Q8—61no—
166C3×Q8—61no—
167C3×Q8(C2×D3)⋊C411yescharacteristic
168C3×D4—31no—
169C3×D4—31no—
170C3×D4—61no—
171C3×D4—61no—
172C3×D4—61no—
173C3×D4—121no—
174C3×D4—121no—
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—61no—
182C2×C3×C4—81no—
183C2×C3×C4—121no—
184C2×C3×C4—121no—
185C2×C3×C4C2×S411yescharacteristic
186C2×Dic3—41no—
187C2×Dic3—62no—
188C2×Dic3—121no—
189C2×Dic3—121no—
190C4×D3—81no—
191C4×D3—81no—
192SL(2,𝔽3)—21no—
193SL(2,𝔽3)#21711yescharacteristic
194C3×C8—61no—
195C3×C8—61no—
196C3×C8—241no—
197C3⋊C8—21no—
198C3⋊C8—61no—
199C3⋊C8—121no—
200C3⋊C8—241no—
201C3⋊C8—242no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C3×D3—82no—
206#41—31no—
207#41—31no—
208#41—61no—
209#41—61no—
210#41—61no—
211#41—61no—
212#41#62511yescharacteristic
213#41#62611yescharacteristic
214C2×Q8—31no—
215C2×Q8D3×Dic311yescharacteristic
216C2×D4—31no—
217C2×D4—31no—
218C2×D4—61no—
219C22×C4—31no—
220C22×C4—61no—
221C22×C4—181no—
222Q16—61no—
223Q16—61no—
224SD16—61no—
225SD16—61no—
226SD16—61no—
227SD16—61no—
228D8—61no—
229D8—61no—
230M16—92no—
231M16—181no—
232M16—181no—
233M16—181no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—61no—
238C2×C8—61no—
239C2×C8—91no—
240C2×C8—181no—
241C2×C8—181no—
242C2×C8—181no—
243C42—181no—
244C42—181no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—41no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—81no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3—121no—
254C22×C3C4×S411yescharacteristic
255C2×D3—42no—
256C2×D3—81no—
257C2×D3—81no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4—121no—
265C3×C4C4⋊S411yescharacteristic
266C3×C4GL(2,ℤ/4)11yescharacteristic
267Dic3—81no—
268Dic3—122no—
269C32—41noSylow
270C23—31no—
271C23—61no—
272Q8—31no—
273Q8—61no—
274Q8—61no—
275Q8(C2×D3)⋊Dic311yescharacteristic
276D4—31no—
277D4—31no—
278D4—61no—
279D4—61no—
280D4—61no—
281D4—121no—
282D4—121no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—61no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—182no—
292C2×C4—361no—
293C2×C4—361no—
294C2×C4D3×S411yescharacteristic
295C8—61no—
296C8—61no—
297C8—61no—
298C8—181no—
299C8—361no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—61no—
304C2×C3—61no—
305C2×C3—81no—
306C2×C3—81no—
307C2×C3—81no—
308C2×C3—241no—
309C2×C3#540411yescharacteristic
310C2×C3#540511yescharacteristic
311C2×C3(C2×S4)⋊C411yescharacteristic
312D3—82no—
313C22—31no—
314C22—31no—
315C22—61no—
316C22—61no—
317C22—121no—
318C22—121no—
319C22—121no—
320C22Dic3×S411yescharacteristic
321C4—31no—
322C4—31no—
323C4—121no—
324C4—362no—
325C4#6420711yescharacteristic
326C4#6420911yescharacteristic
327C3—41no—
328C3—81no—
329C3#7312711yescharacteristic, complemented
330C2—61no—
331C2—61no—
332C2—241no—
333C2#1059321911yescharacteristic
334C2#1059322111yescharacteristic
335C2(C2×S4)⋊Dic311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D3×S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC2×C3×#41C2×D3
socleC22×C3C4×S4
prime radicalC2×C3×#985C2
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
16. C2×#948D3semidirect
40. C2×GL(2,𝔽3)Dic3semidirect
41. C2×#226Dic3semidirect
329. C3#73127semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9852
C2×C3⋊(#131⋊C3)2
C2×C3⋊GU(2,𝔽3)2
#731273
#720873
(C2×D3)⋊(C3⋊C8)4