Finite Group Databasebeta

← C2×#64266⋊C3 #13164428

Subgroups — C2×#64266⋊C3 #13164428

Order 1728.

618 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 71 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64266⋊C3C214yesmaximal, direct product factor
3C2×(C3⋊GL(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
4C2×(C3⋊#226)⋊C3C211yesmaximal, characteristic, complemented
5C2×#64266C311yesmaximal, characteristic, complemented
6C2×C3×#950—31nomaximal
7C2×C4×(C3⋊D3)⋊C3—41nomaximal
8#41⋊H(3,𝔽3)C2212yescomplemented
9C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
10(C3⋊GL(2,𝔽3))⋊C3C2212yescomplemented
11(C3⋊#226)⋊C3C2212yescomplemented
12C2×C32×#41D311yescharacteristic, complemented
13C2×C3×#235—31no—
14C2×C3×#41⋊C3—21no—
15C2×C3×#41⋊C3C2×C311yescharacteristic, complemented
16#64266C2×C314yescomplemented
17C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
18C2×C3⋊#226C2×C311yescharacteristic, complemented
19C3×#950—34no—
20C3×#950—34no—
21C2×C3×#216—31no—
22C2×C3×#215—31no—
23C2×C3×#214—31no—
24C2×C3×#213—31no—
25C22×C3×C3⋊C8—31no—
26C22×(C3⋊D3)⋊C3—41no—
27C2×C4×H(3,𝔽3)—41no—
28C2×(C3⋊Dic3)⋊C3—41no—
29C4×(C3⋊D3)⋊C3—44no—
30Q8⋊H(3,𝔽3)C2311yescharacteristic
31C2×#985—91no—
32C2×C3×#135—91no—
33C2×#950—31no—
34C32×#41—32no—
35C32×#41C2×D312yescomplemented
36C2×C32×Q8C2×D311yescharacteristic
37C2×C32×D4—31no—
38C22×C32×C4—31no—
39C2×C4×C3⋊D3—41no—
40C2×C3×#67—61no—
41C3×#235—34no—
42C3×#235—62no—
43C2×C3×D12—31no—
44C2×C3×C4×D3—61no—
45C2×C3×Dic6—31no—
46C3×#41⋊C3—22no—
47C3×#41⋊C3C22×C312yescomplemented
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
50C3⋊GL(2,𝔽3)C22×C312yescomplemented
51C3⋊#226C22×C312yescomplemented
52C3×#216—32no—
53C3×#216—32no—
54C3×#215—32no—
55C3×#215—32no—
56C3×#214—34no—
57C3×#213—34no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8—34no—
61C22×H(3,𝔽3)—41no—
62C2×(C3⋊D3)⋊C3—42no—
63C2×(C3⋊D3)⋊C3—44no—
64C4×H(3,𝔽3)—42no—
65(C3⋊Dic3)⋊C3—42no—
66C2×C3×#41—31no—
67C2×C3×#41—91no—
68C2×C3×#41C3×D311yescharacteristic
69C2×#235—31no—
70C2×#41⋊C3—31no—
71C2×#41⋊C3—61no—
72#985—94no—
73C2×GL(2,𝔽3)—91no—
74C2×#226—91no—
75C3×#135—94no—
76C3×#135—94no—
77C2×C3×Q16—91no—
78C2×C3×SD16—91no—
79C2×C3×SD16—91no—
80C2×C3×D8—91no—
81C22×C3×C8—91no—
82#950—34no—
83#950—34no—
84C2×#216—31no—
85C2×#215—31no—
86C2×#214—31no—
87C2×#213—31no—
88C22×C3⋊C8—31no—
89C23×C32—31no—
90C22×C3⋊D3—41no—
91C22×C3×D3—61no—
92C32×Q8—31no—
93C32×Q8C22×D311yescharacteristic
94C32×D4—32no—
95C32×D4—61no—
96C2×C32×C4—31no—
97C2×C32×C4—32no—
98C2×C32×C4—32no—
99C2×C32×C4—41no—
100C2×C32×C4—81no—
101C2×C32×C4S411yescharacteristic
102C2×C3⋊Dic3—41no—
103C4×C3⋊D3—44no—
104C3×#67—64no—
105C2×C3×Dic3—61no—
106C3×D12—32no—
107C3×D12—61no—
108C3×C4×D3—64no—
109C3×Dic6—32no—
110C3×Dic6—61no—
111C3×SL(2,𝔽3)—21no—
112C3×SL(2,𝔽3)C23×C311yescharacteristic
113C3×C3⋊C8—32no—
114C3×C3⋊C8—32no—
115C2×#135—91noSylow
116C2×H(3,𝔽3)—42no—
117C2×H(3,𝔽3)—41no—
118(C3⋊D3)⋊C3—44no—
119C3×#41—32no—
120C3×#41—32no—
121C3×#41—32no—
122C3×#41—62no—
123C3×#41—94no—
124C3×#41—182no—
125C3×#41C2×C3×D312yes—
126C2×C3×Q8—31no—
127C2×C3×Q8—91no—
128C2×C3×Q8C2×C3×D311yescharacteristic
129C2×C3×D4—31no—
130C2×C3×D4—31no—
131C2×C3×D4—61no—
132C2×C3×D4—91no—
133C2×C3×D4—181no—
134C22×C3×C4—31no—
135C22×C3×C4—31no—
136C22×C3×C4—61no—
137C22×C3×C4—181no—
138C2×#67—61no—
139#235—34no—
140#235—62no—
141C2×D12—31no—
142C2×C4×D3—61no—
143C2×C4×D3—361no—
144C2×Dic6—31no—
145#41⋊C3—32no—
146#41⋊C3—62no—
147C2×SL(2,𝔽3)—31no—
148C2×SL(2,𝔽3)—61no—
149GL(2,𝔽3)—92no—
150#226—92no—
151C3×Q16—92no—
152C3×Q16—92no—
153C3×SD16—94no—
154C3×SD16—92no—
155C3×SD16—92no—
156C3×D8—94no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C2×C3×C8—94no—
160#216—32no—
161#216—32no—
162#215—32no—
163#215—32no—
164#214—34no—
165#213—34no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—34no—
169C22×C32—32no—
170C22×C32—41no—
171C22×C32—62no—
172C22×C32—81no—
173C22×C32C2×S411yescharacteristic
174C2×C3⋊D3—42no—
175C2×C3⋊D3—44no—
176C2×C3×D3—62no—
177C2×C3×D3—122no—
178C32×C4—31no—
179C32×C4—31no—
180C32×C4—42no—
181C32×C4—82no—
182C32×C4C2×S412yes—
183C3⋊Dic3—42no—
184C3×Dic3—62no—
185C2×#41—91no—
186C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
187#135—94no—
188#135—94no—
189C2×Q16—91no—
190C2×SD16—91no—
191C2×SD16—91no—
192C2×D8—91no—
193C22×C8—91no—
194H(3,𝔽3)—41noSylow
195C23×C3—31no—
196C23×C3—31no—
197C23×C3—61no—
198C23×C3—181no—
199C22×D3—61no—
200C22×D3—361no—
201C3×Q8—31no—
202C3×Q8—31no—
203C3×Q8—31no—
204C3×Q8—61no—
205C3×Q8—92no—
206C3×Q8—181no—
207C3×Q8C22×C3×D311yescharacteristic
208C3×D4—32no—
209C3×D4—32no—
210C3×D4—62no—
211C3×D4—61no—
212C3×D4—61no—
213C3×D4—62no—
214C3×D4—92no—
215C3×D4—181no—
216C3×D4—184no—
217C2×C3×C4—31no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—31no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—31no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—121no—
228C2×C3×C4—184no—
229C2×C3×C4—181no—
230C2×C3×C4—241no—
231C2×C3×C4C3×S411yescharacteristic
232#67—64no—
233C2×Dic3—61no—
234C2×Dic3—361no—
235D12—32no—
236D12—61no—
237C4×D3—64no—
238C4×D3—364no—
239Dic6—32no—
240Dic6—61no—
241SL(2,𝔽3)—31no—
242SL(2,𝔽3)—61no—
243C3×C8—92no—
244C3×C8—92no—
245C3⋊C8—32no—
246C3⋊C8—32no—
247C2×C32—41no—
248C2×C32—42no—
249C2×C32—62no—
250C2×C32—81no—
251C2×C32—82no—
252C2×C32#98512yescomplemented
253C2×C32C22×S411yescharacteristic
254C3⋊D3—44no—
255C3×D3—122no—
256#41—32no—
257#41—94no—
258#41—182no—
259#41C2×(C3⋊D3)⋊C312yescomplemented
260C2×Q8—91no—
261C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
262C2×D4—31no—
263C2×D4—91no—
264C2×D4—181no—
265C22×C4—31no—
266C22×C4—181no—
267Q16—92no—
268Q16—92no—
269SD16—94no—
270SD16—92no—
271SD16—92no—
272D8—94no—
273C2×C8—91no—
274C2×C8—91no—
275C2×C8—94no—
276C22×C3—32no—
277C22×C3—32no—
278C22×C3—31no—
279C22×C3—62no—
280C22×C3—62no—
281C22×C3—62no—
282C22×C3—122no—
283C22×C3—121no—
284C22×C3—182no—
285C22×C3—241no—
286C22×C3—362no—
287C22×C3C2×C3×S411yescharacteristic
288C2×D3—62no—
289C2×D3—122no—
290C2×D3—362no—
291C2×D3—364no—
292C3×C4—31no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—32no—
297C3×C4—61no—
298C3×C4—61no—
299C3×C4—122no—
300C3×C4—182no—
301C3×C4—242no—
302C3×C4C2×C3×S412yes—
303Dic3—62no—
304Dic3—362no—
305C32—41no—
306C32—81no—
307C32C2×#98511yescharacteristic, complemented
308C23—31no—
309C23—181no—
310Q8—31no—
311Q8—92no—
312Q8—181no—
313Q8C22×(C3⋊D3)⋊C311yescharacteristic
314D4—32no—
315D4—61no—
316D4—92no—
317D4—181no—
318D4—184no—
319C2×C4—31no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—181no—
323C2×C4—184no—
324C2×C4(C3⋊S4)⋊C311yescharacteristic
325C8—92no—
326C8—92no—
327C2×C3—31no—
328C2×C3—32no—
329C2×C3—62no—
330C2×C3—62no—
331C2×C3—122no—
332C2×C3—121no—
333C2×C3—122no—
334C2×C3—241no—
335C2×C3—242no—
336C2×C3—362no—
337C2×C3C3×#98512yes—
338C2×C3C22×C3×S411yescharacteristic
339D3—122no—
340D3—364no—
341C22—32no—
342C22—62no—
343C22—182no—
344C22—362no—
345C22C2×(C3⋊S4)⋊C311yescharacteristic
346C4—31no—
347C4—31no—
348C4—182no—
349C4C2×(C3⋊S4)⋊C312yes—
350C3—31no—
351C3—121no—
352C3—241no—
353C3C2×C3×#98511yescharacteristic
354C2—62no—
355C2—362no—
356C2#64266⋊C312yesdirect product factor
357C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2×C4(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2×C3C22×C3×S4
FittingC2×C32×#41D3
socleC22×C3C2×C3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. #64266⋊C3C2direct (N × H)
3. C2×(C3⋊GL(2,𝔽3))⋊C3C2semidirect
4. C2×(C3⋊#226)⋊C3C2semidirect
5. C2×#64266C3semidirect
8. #41⋊H(3,𝔽3)C22semidirect
10. (C3⋊GL(2,𝔽3))⋊C3C22semidirect
11. (C3⋊#226)⋊C3C22semidirect
12. C2×C32×#41D3semidirect
15. C2×C3×#41⋊C3C2×C3semidirect
16. #64266C2×C3semidirect
17. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
18. C2×C3⋊#226C2×C3semidirect
35. C32×#41C2×D3semidirect
47. C3×#41⋊C3C22×C3semidirect
50. C3⋊GL(2,𝔽3)C22×C3semidirect
51. C3⋊#226C22×C3semidirect
186. C2×#41(C3⋊D3)⋊C3semidirect
252. C2×C32#985semidirect
259. #41C2×(C3⋊D3)⋊C3semidirect
307. C32C2×#985semidirect
356. C2#64266⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#64266⋊C32
C2×(C3⋊GL(2,𝔽3))⋊C32
C2×(C3⋊#226)⋊C32
C2×#642663
C2×C3×#9503
C2×C4×(C3⋊D3)⋊C34