Finite Group Databasebeta

← C2×#11722388 #13164489

Subgroups — C2×#11722388 #13164489

Order 1728.

760 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722388C214yesmaximal, direct product factor
3C2×#88611C211yesmaximal, characteristic, complemented
4C2×#88610C211yesmaximal, characteristic, complemented
5C2×C3×#983—33nomaximal
6C2×C3×#942—31nomaximal
7C2×#6235—41nomaximal
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9C2×Q8⋊H(3,𝔽3)C2212yescomplemented
10#88611C2212yescomplemented
11#88610C2212yescomplemented
12C22×C32×Q8D311yescharacteristic, complemented
13C2×C3×#235—31no—
14C22×C3×SL(2,𝔽3)D313yescomplemented
15C3×#983—312no—
16C2×C3×GL(2,𝔽3)—33no—
17C2×C3×#226—33no—
18C2×C3×#216—31no—
19C2×C3×#216—31no—
20C3×#942—34no—
21C3×#942—34no—
22C2×C3×#215—31no—
23C2×C3×#215—31no—
24C2×C3×#208—31no—
25C23×H(3,𝔽3)—41no—
26C22×#272—41no—
27#6235—44no—
28C2×#1084—41no—
29Q8⋊H(3,𝔽3)C2311yescharacteristic
30C2×#983—33no—
31C2×C3×#137—91no—
32C2×#942—31no—
33C2×C32×Q8—31no—
34C2×C32×Q8—32no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×Q8C2×D312yescomplemented
37C22×C32×C4—31no—
38C2×C3×#67—61no—
39C2×C3×#67—123no—
40C3×#235—34no—
41C3×#235—62no—
42C2×C3×D12—31no—
43C2×C3×C4×D3—61no—
44C2×C3×Dic6—31no—
45C2×C3×SL(2,𝔽3)C2×D313yes—
46C2×C3×SL(2,𝔽3)C2×D316yescomplemented
47C3×GL(2,𝔽3)—36no—
48C3×#226—36no—
49C3×#216—32no—
50C3×#216—34no—
51C3×#216—32no—
52C3×#215—32no—
53C3×#215—34no—
54C3×#215—32no—
55C3×#208—34no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C22×H(3,𝔽3)—41no—
59C22×H(3,𝔽3)—42no—
60C22×H(3,𝔽3)—82no—
61C2×#272—42no—
62C2×#272—82no—
63#1084—42no—
64C2×C3×#41—91no—
65C22×C3×Q8—31no—
66C22×C3×Q8C3⋊D311yescharacteristic
67C2×#235—31no—
68C22×SL(2,𝔽3)—33no—
69#983—312no—
70C2×GL(2,𝔽3)—33no—
71C2×#226—33no—
72C3×#137—94no—
73C3×#137—94no—
74C2×C3×Q16—91no—
75C2×C3×Q16—91no—
76C2×C3×SD16—91no—
77C2×C3×SD16—91no—
78C2×C3×M16—91no—
79C2×#216—31no—
80C2×#216—31no—
81#942—34no—
82#942—34no—
83C2×#215—31no—
84C2×#215—31no—
85C2×#208—31no—
86C23×C32—43no—
87C23×C32S411yescharacteristic
88C22×C3×D3—61no—
89C22×C3×D3—123no—
90C32×Q8—31no—
91C32×Q8—32no—
92C32×Q8—61no—
93C32×Q8C22×D311yescharacteristic
94C2×C32×C4—31no—
95C2×C32×C4—32no—
96C2×C32×C4—31no—
97C2×C32×C4—32no—
98C3×#67—64no—
99C3×#67—1212no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—123no—
102C3×D12—32no—
103C3×D12—61no—
104C3×C4×D3—64no—
105C3×Dic6—32no—
106C3×Dic6—61no—
107C3×SL(2,𝔽3)C22×D313yes—
108C3×C3⋊C8—32no—
109C3×C3⋊C8—32no—
110C2×#137—91noSylow
111C2×H(3,𝔽3)—42no—
112C2×H(3,𝔽3)—41no—
113C2×H(3,𝔽3)—82no—
114#272—82no—
115C3×#41—94no—
116C3×#41—182no—
117C2×C3×Q8—31no—
118C2×C3×Q8—32no—
119C2×C3×Q8—31no—
120C2×C3×Q8—32no—
121C2×C3×Q8—31no—
122C2×C3×Q8—32no—
123C2×C3×Q8—61no—
124C2×C3×Q8—62no—
125C2×C3×Q8—91no—
126C2×C3×Q8C2×C3⋊D311yescharacteristic
127C2×C3×Q8C2×C3⋊D312yes—
128C2×C3×D4—91no—
129C2×C3×D4—181no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—61no—
133C22×C3×C4—181no—
134C2×#67—61no—
135C2×#67—123no—
136#235—34no—
137#235—62no—
138C2×D12—31no—
139C2×C4×D3—61no—
140C2×Dic6—31no—
141C2×SL(2,𝔽3)—33no—
142C2×SL(2,𝔽3)—36no—
143GL(2,𝔽3)—36no—
144#226—36no—
145C3×Q16—94no—
146C3×Q16—92no—
147C3×Q16—92no—
148C3×SD16—94no—
149C3×SD16—92no—
150C3×SD16—92no—
151C3×M16—94no—
152C2×C3×C8—91no—
153C2×C3×C8—91no—
154#216—32no—
155#216—34no—
156#216—32no—
157#215—32no—
158#215—34no—
159#215—32no—
160#208—34no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163C22×C32—22no—
164C22×C32—46no—
165C22×C32—43no—
166C22×C32—86no—
167C22×C32C2×S412yes—
168C22×C32C2×S411yescharacteristic
169C2×C3×D3—62no—
170C2×C3×D3—126no—
171C2×C3×D3—122no—
172C2×C3×D3—246no—
173C32×C4—31no—
174C32×C4—31no—
175C32×C4—32no—
176C3×Dic3—62no—
177C3×Dic3—126no—
178C2×#41—91no—
179C22×Q8#27211yescharacteristic, complemented
180#137—94no—
181#137—94no—
182C2×Q16—91no—
183C2×Q16—91no—
184C2×SD16—91no—
185C2×SD16—91no—
186C2×M16—91no—
187H(3,𝔽3)—41noSylow
188C23×C3—31no—
189C23×C3—123no—
190C23×C3—181no—
191C23×C3C3⋊S411yescharacteristic
192C22×D3—61no—
193C22×D3—123no—
194C3×Q8—31no—
195C3×Q8—32no—
196C3×Q8—31no—
197C3×Q8—32no—
198C3×Q8—31no—
199C3×Q8—61no—
200C3×Q8—62no—
201C3×Q8—61no—
202C3×Q8—61no—
203C3×Q8—62no—
204C3×Q8—92no—
205C3×Q8—181no—
206C3×Q8C22×C3⋊D311yescharacteristic
207C3×D4—92no—
208C3×D4—181no—
209C3×D4—184no—
210C2×C3×C4—31no—
211C2×C3×C4—32no—
212C2×C3×C4—31no—
213C2×C3×C4—32no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—61no—
219C2×C3×C4—62no—
220C2×C3×C4—61no—
221C2×C3×C4—62no—
222C2×C3×C4—184no—
223C2×C3×C4—181no—
224#67—64no—
225#67—1212no—
226C2×Dic3—61no—
227C2×Dic3—123no—
228D12—32no—
229D12—61no—
230C4×D3—64no—
231Dic6—32no—
232Dic6—61no—
233SL(2,𝔽3)—33no—
234C3×C8—92no—
235C3×C8—92no—
236C3⋊C8—32no—
237C3⋊C8—32no—
238C2×C32—22no—
239C2×C32—46no—
240C2×C32—43no—
241C2×C32—86no—
242C2×C32#98312yescomplemented
243C2×C32C22×S411yescharacteristic
244C3×D3—122no—
245C3×D3—246no—
246#41—94no—
247#41—182no—
248C2×Q8—31no—
249C2×Q8—32no—
250C2×Q8—91no—
251C2×Q8C2×#27211yescharacteristic
252C2×Q8C2×#27212yescomplemented
253C2×D4—91no—
254C2×D4—181no—
255C22×C4—31no—
256C22×C4—181no—
257Q16—92no—
258Q16—94no—
259Q16—92no—
260SD16—92no—
261SD16—94no—
262SD16—92no—
263M16—94no—
264C2×C8—91no—
265C2×C8—91no—
266C22×C3—22no—
267C22×C3—31no—
268C22×C3—32no—
269C22×C3—62no—
270C22×C3—123no—
271C22×C3—126no—
272C22×C3—182no—
273C22×C3—246no—
274C22×C3—362no—
275C22×C3C2×C3⋊S412yes—
276C22×C3C2×C3⋊S411yescharacteristic
277C2×D3—62no—
278C2×D3—126no—
279C2×D3—122no—
280C2×D3—246no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—32no—
284C3×C4—31no—
285C3×C4—32no—
286C3×C4—31no—
287C3×C4—61no—
288C3×C4—62no—
289C3×C4—61no—
290C3×C4—182no—
291Dic3—62no—
292Dic3—126no—
293C32—43no—
294C32C2×#98311yescharacteristic, complemented
295C23—181no—
296C23#625811yescharacteristic
297Q8—31no—
298Q8—32no—
299Q8—61no—
300Q8—92no—
301Q8—181no—
302Q8C22×#27211yescharacteristic
303D4—92no—
304D4—181no—
305D4—184no—
306C2×C4—31no—
307C2×C4—32no—
308C2×C4—31no—
309C2×C4—32no—
310C2×C4—184no—
311C2×C4—181no—
312C8—92no—
313C8—92no—
314C2×C3—22no—
315C2×C3—31no—
316C2×C3—32no—
317C2×C3—62no—
318C2×C3—123no—
319C2×C3—126no—
320C2×C3—246no—
321C2×C3—362no—
322C2×C3#6426412yes—
323C2×C3C22×C3⋊S411yescharacteristic
324D3—122no—
325D3—246no—
326C22—22no—
327C22—182no—
328C22—362no—
329C22C2×#625812yes—
330C22C2×#625811yescharacteristic
331C4—31no—
332C4—32no—
333C4—31no—
334C4—182no—
335C3—31no—
336C3—123no—
337C3C2×#6426411yescharacteristic
338C2—22no—
339C2—362no—
340C2#1172238812yesdirect product factor
341C2C22×#625811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC22×C32×Q8D3
hypercenterC23×C3C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. #11722388C2direct (N × H)
3. C2×#88611C2semidirect
4. C2×#88610C2semidirect
9. C2×Q8⋊H(3,𝔽3)C22semidirect
10. #88611C22semidirect
11. #88610C22semidirect
12. C22×C32×Q8D3semidirect
14. C22×C3×SL(2,𝔽3)D3semidirect
36. C2×C32×Q8C2×D3semidirect
46. C2×C3×SL(2,𝔽3)C2×D3semidirect
179. C22×Q8#272semidirect
242. C2×C32#983semidirect
252. C2×Q8C2×#272semidirect
294. C32C2×#983semidirect
340. C2#11722388direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
#117223882
C2×#886112
C2×#886102
C2×C3×#9833
C2×C3×#9423
C2×#62354