Finite Group Databasebeta

← Dic6×#41⋊C3 #11923192

Subgroups — Dic6×#41⋊C3 #11923192

Order 1152.

559 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×C4×#41⋊C3C211yesmaximal, characteristic
2Dic3×#41⋊C3C212yesmaximal
3#5570⋊C3C211yesmaximal, characteristic, complemented
4#5569⋊C3C212yesmaximal, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#41×Dic6C311yesmaximal, characteristic, complemented
7Q8×#41⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×C4×Dic6—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2212yes—
14C3×C4×#41C2×C311yescharacteristic
15Dic3×#41C2×C312yes—
16#5570C2×C311yescharacteristic
17#5569C2×C312yes—
18Q8×Dic6C2×C311yescharacteristic
19#5543—31no—
20#5539—32no—
21D4×Dic6—31no—
22#5517—32no—
23#5470—31no—
24C2×C4×Dic6—31no—
25Q8×SL(2,𝔽3)—31no—
26#557⋊C3—32no—
27#557⋊C3—31no—
28C4×#41⋊C3—21no—
29C4×#41⋊C3—32no—
30C4×#41⋊C3D311yescharacteristic
31C2×C3×Dic6—41no—
32C3×#41⋊C3Q812yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C32×C42—41no—
35C3×C4⋊Dic3—41no—
36C3×Dic3⋊C4—42no—
37C3×C4×Dic3—42no—
38Q8×#41—31noSylow
39C2×C3×#41C22×C311yescharacteristic
40C22×Dic6—31no—
41C2×#41⋊C3—21no—
42C2×#41⋊C3C2×D311yescharacteristic
43C3×C4×Q8—121no—
44C3×C4×Q8C22×C311yescharacteristic
45C3×C4×Q8C22×C311yescharacteristic
46C3×C4×D4—31no—
47C3×C4×D4—31no—
48C3×#117—31no—
49C2×C3×C42—31no—
50Q8×Dic3C22×C312yes—
51Q8×Dic3C22×C312yes—
52D4×Dic3—32no—
53D4×Dic3—32no—
54#926—32no—
55C2×C4⋊Dic3—31no—
56#924—32no—
57#924—32no—
58C2×Dic3⋊C4—32no—
59C2×C4×Dic3—32no—
60#890—32no—
61#890—32no—
62#888—32no—
63#888—32no—
64#878—34no—
65#878—34no—
66#870—32no—
67#869—31no—
68#869—31no—
69C4×Dic6—32no—
70C4×Dic6—31no—
71C4×Dic6—31no—
72C4×Dic6—31no—
73C4×Dic6—32no—
74C4×Dic6—32no—
75C4×Dic6A411yescharacteristic
76C4×SL(2,𝔽3)—21no—
77C4×SL(2,𝔽3)—21no—
78C4×SL(2,𝔽3)—32no—
79C4×SL(2,𝔽3)—32no—
80C4×SL(2,𝔽3)C2×D311yescharacteristic
81C4×SL(2,𝔽3)C2×D311yescharacteristic
82C2×C32×C4—41no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—42no—
87C3×Dic6—44no—
88C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
89Q82—31no—
90#557—32no—
91#557—31no—
92#554—92no—
93#554—91no—
94D4×Q8—91no—
95#533—91no—
96#533—92no—
97C4×#41—32no—
98C4×#41C3×D311yescharacteristic
99C2×C4×Q8—91no—
100C3×#41—32no—
101C3×#41C3×Q812yescomplemented
102C2×C3×Q8—121no—
103C2×C3×Q8C23×C311yescharacteristic
104C2×C3×D4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×Dic3—32no—
109C22×Dic3—32no—
110C2×Dic6—31no—
111C2×Dic6—32no—
112C2×Dic6—32no—
113C2×Dic6—64no—
114C2×Dic6C2×A411yescharacteristic
115#41⋊C3—22no—
116#41⋊C3Dic612yesdirect product factor
117C2×SL(2,𝔽3)—21no—
118C2×SL(2,𝔽3)C22×D311yescharacteristic
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—122no—
122C3×C4⋊C4—121no—
123C3×C22⋊C4—32no—
124C3×C42—31no—
125C3×C42—31no—
126C3×C42—31no—
127C3×C42—41no—
128C3×C42—81no—
129C3×C42—122no—
130C3×C42C2×A411yescharacteristic
131#217—34no—
132C4⋊Dic3—32no—
133C4⋊Dic3—31no—
134C4⋊Dic3—32no—
135C4⋊Dic3—32no—
136C4⋊Dic3—32no—
137C4⋊Dic3C2×A411yescharacteristic
138Dic3⋊C4—34no—
139Dic3⋊C4—32no—
140Dic3⋊C4—32no—
141Dic3⋊C4—32no—
142Dic3⋊C4C2×A412yes—
143C4×Dic3—32no—
144C4×Dic3—32no—
145C4×Dic3—32no—
146C4×Dic3C2×A412yes—
147C22×C32—41no—
148C32×C4—42no—
149C32×C4—42no—
150C32×C4—81no—
151C3×Dic3—44no—
152C3×Dic3—82no—
153C2×#41C2×C3×D311yescharacteristic
154C22×Q8—91no—
155C4⋊Q8—91no—
156C4⋊Q8—91no—
157C4⋊Q8—92no—
158C4⋊Q8—92no—
159#125—92no—
160#125—92no—
161#125—92no—
162C22⋊Q8—94no—
163C22⋊Q8—94no—
164C22⋊Q8—92no—
165C22⋊Q8—92no—
166C4×Q8—31no—
167C4×Q8—32no—
168C4×Q8—32no—
169C4×Q8—92no—
170C4×Q8—92no—
171C4×Q8—91no—
172C4×Q8—91no—
173C4×Q8—92no—
174C4×Q8—91no—
175C4×Q8C2×C3×D311yescharacteristic
176C4×Q8C2×C3×D311yescharacteristic
177C4×D4—31no—
178C4×D4—31no—
179C4×D4—92no—
180C4×D4—92no—
181#117—31no—
182#117—92no—
183C2×C4⋊C4—92no—
184C2×C4⋊C4—91no—
185C2×C42—31no—
186C2×C42—92no—
187C23×C3—31no—
188C3×Q8—31no—
189C3×Q8—124no—
190C3×Q8C2×C3×Q811yescharacteristic
191C3×D4—32no—
192C3×D4—32no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—62no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—122no—
206C2×C3×C4—122no—
207C2×C3×C4—121no—
208C2×C3×C4C22×A411yescharacteristic
209C2×C3×C4C22×A411yescharacteristic
210C2×C3×C4C22×A411yescharacteristic
211C2×Dic3—32no—
212C2×Dic3—32no—
213C2×Dic3—64no—
214C2×Dic3—122no—
215C2×Dic3C22×A412yes—
216C2×Dic3C22×A412yes—
217Dic6—62no—
218Dic6—64no—
219Dic6—122no—
220Dic6#41⋊C314yesdirect product factor
221SL(2,𝔽3)—21no—
222SL(2,𝔽3)C2×Dic611yescharacteristic, complemented
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226#41—32no—
227#41C3×Dic612yescomplemented
228C2×Q8—31no—
229C2×Q8—92no—
230C2×Q8—92no—
231C2×Q8—91no—
232C2×Q8—184no—
233C2×Q8C22×C3×D311yescharacteristic
234C2×D4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—92no—
239C22×C4—92no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—32no—
243C4⋊C4—31no—
244C4⋊C4—92no—
245C4⋊C4—92no—
246C4⋊C4—92no—
247C4⋊C4—92no—
248C4⋊C4—92no—
249C4⋊C4—94no—
250C4⋊C4—92no—
251C4⋊C4—92no—
252C4⋊C4—91no—
253C22⋊C4—32no—
254C22⋊C4—94no—
255C42—31no—
256C42—31no—
257C42—31no—
258C42—32no—
259C42—92no—
260C42—92no—
261C42—92no—
262C42D3×A411yescharacteristic
263C22×C3—32no—
264C22×C3—41no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—81no—
268C22×C3C23×A411yescharacteristic
269C3×C4—21no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—42no—
273C3×C4—42no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—81no—
277C3×C4—82no—
278C3×C4—82no—
279C3×C4—82no—
280C3×C4—124no—
281C3×C4—242no—
282C3×C4Q8×A412yes—
283C3×C4C2×#41⋊C312yes—
284Dic3—22no—
285Dic3—62no—
286Dic3—62no—
287Dic3C2×#41⋊C314yes—
288C32—41noSylow
289C23—31no—
290Q8—31no—
291Q8—34no—
292Q8—184no—
293Q8—182no—
294Q8—362no—
295Q8C2×C3×Dic611yescharacteristic
296D4—32no—
297D4—32no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—32no—
305C2×C4—62no—
306C2×C4—92no—
307C2×C4—92no—
308C2×C4—121no—
309C2×C4—184no—
310C2×C4—362no—
311C2×C4C2×D3×A411yescharacteristic
312C2×C4C2×D3×A411yescharacteristic
313C2×C4C2×D3×A411yescharacteristic
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—62no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3—81no—
321C2×C3C2×Q8×A411yescharacteristic
322C2×C3C22×#41⋊C311yescharacteristic
323C2×C3#594511yescharacteristic
324C22—32no—
325C22—61no—
326C22—61no—
327C22C22×D3×A411yescharacteristic
328C4—21no—
329C4—34no—
330C4—31no—
331C4—31no—
332C4—61no—
333C4—61no—
334C4—62no—
335C4—182no—
336C4—182no—
337C4A4×Dic612yes—
338C4D3×#41⋊C312yes—
339C3—41no—
340C3—81no—
341C3Q8×#41⋊C311yescharacteristic, complemented
342C2—62no—
343C2C2×A4×Dic611yescharacteristic
344C2C2×D3×#41⋊C311yescharacteristic
345C2#5964⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
prime radicalC2×C3×#41⋊C3C22
hypercenterC42D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #5570⋊C3C2semidirect
4. #5569⋊C3C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #41×Dic6C3semidirect
32. C3×#41⋊C3Q8semidirect
88. C3×SL(2,𝔽3)C2×Q8semidirect
101. C3×#41C3×Q8semidirect
116. #41⋊C3Dic6direct (N × H)
220. Dic6#41⋊C3direct (N × H)
222. SL(2,𝔽3)C2×Dic6semidirect
227. #41C3×Dic6semidirect
341. C3Q8×#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
Dic3×#41⋊C32
#5570⋊C32
#5569⋊C32
SL(2,𝔽3)×Dic62
#41×Dic63
Q8×#41⋊C33
C3×C4×Dic64