Finite Group Databasebeta

← #82556⋊C3 #11921278

Subgroups — #82556⋊C3 #11921278

Order 1152.

407 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 53 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×#1007⋊C3C211yesmaximal, characteristic
2C3×#519⋊C3C211yesmaximal, characteristic, complemented
3#5812⋊C3C211yesmaximal, characteristic, complemented
4#82556C311yesmaximal, characteristic, complemented
5#2884⋊C3—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7#1007⋊C3—21no—
8#1007⋊C3—21no—
9C2×D3×SL(2,𝔽3)C411yescharacteristic, complemented
10C2×C3⋊(#41⋊C3)C411yescharacteristic, complemented
11C3×C4×SL(2,𝔽3)—21no—
12Dic3×SL(2,𝔽3)—21no—
13C3×#806—41nomaximal
14C2×#1007C2×C311yescharacteristic
15C2×2−1+4⋊C3—31no—
16C3×#519C2×C311yescharacteristic, complemented
17#5812C2×C311yescharacteristic, complemented
18#519⋊C3—21no—
19#519⋊C3—31no—
20#519⋊C3D311yescharacteristic, complemented
21#5006—31no—
22#5006—31no—
23#5006—31no—
24#5006—31no—
25C2×C3×#67—41no—
26C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29D3×SL(2,𝔽3)—21no—
30C3⋊(#41⋊C3)—21no—
31C32×C22⋊C4—41no—
32C3×#217—41no—
33#2884—31noSylow
34C22×C3×Q8C22×C311yescharacteristic
35#1007—21no—
36#1007—21no—
37#1007—61no—
38#1007—61no—
39C2×#239—31no—
40C2×#239C3×C411yescharacteristic, complemented
41C2×D3×Q8—31no—
42C2×D3×Q8C3×C411yescharacteristic, complemented
43C2×#235—31no—
44C2×#235—31no—
452−1+4⋊C3—61no—
462−1+4⋊C3—61no—
47C2×#41⋊C3—31no—
48C22×SL(2,𝔽3)—21no—
49C22×SL(2,𝔽3)—31no—
50C22×SL(2,𝔽3)C2×D311yescharacteristic
51C3×C4×Q8—21no—
52C3×C4×Q8—61no—
53C3×#117—31no—
54C3×#117—31no—
55Q8×Dic3—21no—
56Q8×Dic3—61no—
57#926—31no—
58#926—31no—
59C4×SL(2,𝔽3)—22no—
60C4×SL(2,𝔽3)—21no—
61C4×SL(2,𝔽3)—61no—
62C3×#99—121no—
63#806—32no—
64#806—31no—
65#806—31no—
66#806—31no—
67#806A411yescharacteristic
68C23×C32—41no—
69C22×C3×D3—41no—
70C2×C32×C4—81no—
71C3×#67—81no—
72C3×#67—81no—
73C2×C3×Dic3—41no—
74C2×C3×Dic3—81no—
75C3×SL(2,𝔽3)C22⋊C411yescharacteristic
76C2×2−1+4—31no—
77#519—31no—
78#519C3×D311yescharacteristic, complemented
79#410—91no—
80#410—91no—
81#410—91no—
82#410—91no—
83C2×C3×Q8—31no—
84C2×C3×Q8—31no—
85C2×C3×Q8—31no—
86C2×C3×Q8C2×C3×C411yescharacteristic
87C2×C3×Q8C3×D411yescharacteristic
88C2×C3×Q8C3×D411yescharacteristic
89C2×C3×D4—121no—
90C22×C3×C4—31no—
91C2×#67—31no—
92C2×#67C2×A411yescharacteristic
93#239—21no—
94#239—62no—
95#239—61no—
96#239—61no—
97#239—61no—
98D3×Q8—21no—
99D3×Q8—62no—
100D3×Q8—61no—
101D3×Q8—61no—
102D3×Q8—61no—
103#235—61no—
104#235—61no—
105#235—61no—
106#235—61no—
107#235—61no—
108#235—61no—
109#235—61no—
110#235—61no—
111C2×D12—31no—
112C2×D12—31no—
113C2×C4×D3—32no—
114C2×C4×D3—31no—
115C2×C4×D3—31no—
116C2×Dic6—31no—
117C2×Dic6—31no—
118#41⋊C3—61no—
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)—21no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—61no—
123C2×SL(2,𝔽3)C4×D311yescharacteristic
124C2×SL(2,𝔽3)D1211yescharacteristic
125C2×SL(2,𝔽3)#6711yescharacteristic
126C3×C4⋊C4—61no—
127C3×C4⋊C4—61no—
128C3×C22⋊C4—31no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—81no—
131C3×C22⋊C4—121no—
132C3×C22⋊C4C2×A411yescharacteristic
133C3×C42—61no—
134C3×C42—61no—
135#217—31no—
136#217C2×A411yescharacteristic
137C4⋊Dic3—61no—
138C4⋊Dic3—61no—
139C4×Dic3—61no—
140C4×Dic3—61no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32—161no—
145C2×C3×D3—81no—
146C2×C3×D3—161no—
147C32×C4—161no—
148C3×Dic3—81no—
149C3×Dic3—161no—
1502−1+4—61no—
1512−1+4—61no—
1522−1+4—181no—
1532−1+4—181no—
154C2×#41—31no—
155C2×#41—91no—
156C2×#41—91no—
157C2×#41—91no—
158C22×Q8—31no—
159C22×Q8—91no—
160C22×Q8C2×C3×D311yescharacteristic
161C4×Q8—21no—
162C4×Q8—61no—
163C4×Q8—61no—
164C4×Q8—181no—
165#117—31no—
166#117—31no—
167#117—91no—
168#117—91no—
169#99—31no—
170#99—91no—
171#99—91no—
172#99—92no—
173#99—91no—
174C23×C3—41no—
175C23×C3—81no—
176C23×C3—121no—
177C23×C3C22×A411yescharacteristic
178C22×D3—31no—
179C22×D3C4×A411yescharacteristic
180C3×Q8—31no—
181C3×Q8—31no—
182C3×Q8—31no—
183C3×Q8—32no—
184C3×Q8C3×C22⋊C411yescharacteristic
185C3×D4—241no—
186C3×D4—241no—
187C2×C3×C4—21no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—61no—
195C2×C3×C4—81no—
196C2×C3×C4—82no—
197C2×C3×C4—121no—
198C2×C3×C4—241no—
199#67—21no—
200#67—21no—
201#67—61no—
202#67—61no—
203C2×Dic3—21no—
204C2×Dic3—31no—
205C2×Dic3—61no—
206C2×Dic3C4×A411yescharacteristic
207D12—61no—
208D12—61no—
209D12—61no—
210D12—61no—
211C4×D3—61no—
212C4×D3—61no—
213C4×D3—61no—
214C4×D3—61no—
215C4×D3—62no—
216C4×D3—62no—
217Dic6—61no—
218Dic6—61no—
219Dic6—61no—
220Dic6—61no—
221SL(2,𝔽3)—21no—
222SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
223C2×C32—41no—
224C2×C32—81no—
225C2×C32—81no—
226C2×C32—81no—
227C3×D3—161no—
228#41—61no—
229#41—181no—
230#41—181no—
231#41—181no—
232#41—181no—
233#41—181no—
234#41—181no—
235#41—181no—
236#41—181no—
237#41—182no—
238#41—181no—
239#41—181no—
240#41—181no—
241C2×Q8—31no—
242C2×Q8—31no—
243C2×Q8—31no—
244C2×Q8—61no—
245C2×Q8—91no—
246C2×Q8—91no—
247C2×Q8—181no—
248C2×Q8—181no—
249C2×Q8—182no—
250C2×Q8—181no—
251C2×Q8C3×C4×D311yescharacteristic
252C2×Q8C3×D1211yescharacteristic
253C2×Q8C3×#6711yescharacteristic
254C2×D4—31no—
255C2×D4—91no—
256C2×D4—91no—
257C2×D4—91no—
258C22×C4—31no—
259C22×C4—91no—
260C22×C4—91no—
261C22×C4—92no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—181no—
265C4⋊C4—181no—
266C22⋊C4—31no—
267C22⋊C4—31no—
268C22⋊C4—91no—
269C22⋊C4D3×A411yescharacteristic
270C42—61no—
271C42—61no—
272C42—181no—
273C42—181no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—161no—
282C22×C3—162no—
283C22×C3—241no—
284C22×C3—481no—
285C22×C3C2×C4×A411yescharacteristic
286C22×C3D4×A411yescharacteristic
287C22×C3D4×A411yescharacteristic
288C2×D3—21no—
289C2×D3—41no—
290C2×D3—61no—
291C2×D3—121no—
292C3×C4—31no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—41no—
297C3×C4—121no—
298C3×C4—161no—
299C3×C4—162no—
300C3×C4—241no—
301C3×C4—481no—
302Dic3—21no—
303Dic3—41no—
304Dic3—61no—
305Dic3—121no—
306C32—41noSylow
307C23—31no—
308C23—91no—
309C23C2×D3×A411yescharacteristic
310Q8—31no—
311Q8—31no—
312Q8—31no—
313Q8—32no—
314Q8—181no—
315Q8—181no—
316Q8—181no—
317Q8—181no—
318Q8C3×(C2×D3)⋊C411yescharacteristic
319D4—61no—
320D4—61no—
321D4—181no—
322D4—181no—
323D4—181no—
324D4—181no—
325D4—181no—
326D4—181no—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4—181no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3—81no—
351C2×C3—81no—
352C2×C3—82no—
353C2×C3—81no—
354C2×C3—161no—
355C2×C3—161no—
356C2×C3—481no—
357C2×C3A4×C22⋊C411yescharacteristic
358D3—41no—
359D3—121no—
360C22—41no—
361C22—61no—
362C22—121no—
363C22—181no—
364C22—361no—
365C22C4×D3×A411yescharacteristic
366C22A4×D1211yescharacteristic
367C22A4×#6711yescharacteristic
368C4—31no—
369C4—31no—
370C4—31no—
371C4—31no—
372C4—41no—
373C4—61no—
374C4—121no—
375C4—121no—
376C4—181no—
377C4—361no—
378C3—41no—
379C3—81no—
380C3#2884⋊C311yescharacteristic, complemented
381C2—21no—
382C2—21no—
383C2—21no—
384C2—121no—
385C2—361no—
386C2A4×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC2A4×(C2×D3)⋊C4
commutatorC2×C3×Q8C2×C3×C4
FrattiniC23C2×D3×A4
FittingC3×#519C2×C3
socleC2×C3A4×C22⋊C4
prime radicalC2×#1007⋊C3C2
cosocleC22×C3×Q8C22×C3
hypercenterC22⋊C4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#519⋊C3C2semidirect
3. #5812⋊C3C2semidirect
4. #82556C3semidirect
9. C2×D3×SL(2,𝔽3)C4semidirect
10. C2×C3⋊(#41⋊C3)C4semidirect
16. C3×#519C2×C3semidirect
17. #5812C2×C3semidirect
20. #519⋊C3D3semidirect
40. C2×#239C3×C4semidirect
42. C2×D3×Q8C3×C4semidirect
78. #519C3×D3semidirect
380. C3#2884⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1007⋊C32
C3×#519⋊C32
#5812⋊C32
#825563
#2884⋊C33
C3×#8064