Finite Group Databasebeta

← C2×#11722393 #13164497

Subgroups — C2×#11722393 #13164497

Order 1728.

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

normal subgroups only

This group has 764 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
1C2×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722393C214yesmaximal, direct product factor
3C2×#88611C211yesmaximal, characteristic, complemented
4C2×#88610C211yesmaximal, characteristic, complemented
5C2×C3×#985—33nomaximal
6C2×C3×#950—31nomaximal
7C2×C4×#272—41nomaximal
8#41⋊H(3,𝔽3)C2212yescomplemented
9C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
10#88611C2212yescomplemented
11#88610C2212yescomplemented
12C2×C32×#41D311yescharacteristic, complemented
13C2×C3×#235—31no—
14C2×C3×#41⋊C3D313yescomplemented
15C3×#985—312no—
16C2×C3×GL(2,𝔽3)—33no—
17C2×C3×#226—33no—
18C3×#950—34no—
19C3×#950—34no—
20C2×C3×#216—31no—
21C2×C3×#215—31no—
22C2×C3×#214—31no—
23C2×C3×#213—31no—
24C22×C3×C3⋊C8—31no—
25C22×#272—41no—
26C2×C4×H(3,𝔽3)—41no—
27C2×#1084—41no—
28C4×#272—44no—
29Q8⋊H(3,𝔽3)C2311yescharacteristic
30C2×#985—33no—
31C2×C3×#135—91no—
32C2×#950—31no—
33C32×#41—32no—
34C32×#41C2×D312yescomplemented
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×D4—31no—
37C22×C32×C4—31no—
38C2×C3×#67—61no—
39C3×#235—34no—
40C3×#235—62no—
41C2×C3×D12—31no—
42C2×C3×C4×D3—61no—
43C2×C3×C4×D3—123no—
44C2×C3×Dic6—31no—
45C3×#41⋊C3C2×D316yescomplemented
46C2×C3×SL(2,𝔽3)C2×D313yes—
47C3×GL(2,𝔽3)—36no—
48C3×#226—36no—
49C3×#216—32no—
50C3×#216—32no—
51C3×#215—32no—
52C3×#215—32no—
53C3×#214—34no—
54C3×#213—34no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—34no—
58C22×H(3,𝔽3)—41no—
59C2×#272—42no—
60C2×#272—44no—
61C4×H(3,𝔽3)—42no—
62#1084—42no—
63C2×C3×#41—31no—
64C2×C3×#41—91no—
65C2×C3×#41C3⋊D311yescharacteristic
66C2×#235—31no—
67C2×#41⋊C3—33no—
68#985—312no—
69C2×GL(2,𝔽3)—33no—
70C2×#226—33no—
71C3×#135—94no—
72C3×#135—94no—
73C2×C3×Q16—91no—
74C2×C3×SD16—91no—
75C2×C3×SD16—91no—
76C2×C3×D8—91no—
77C22×C3×C8—91no—
78#950—34no—
79#950—34no—
80C2×#216—31no—
81C2×#215—31no—
82C2×#214—31no—
83C2×#213—31no—
84C22×C3⋊C8—31no—
85C23×C32—31no—
86C22×C3×D3—61no—
87C22×C3×D3—123no—
88C32×Q8—31no—
89C32×Q8C22×D311yescharacteristic
90C32×D4—32no—
91C32×D4—61no—
92C2×C32×C4—31no—
93C2×C32×C4—32no—
94C2×C32×C4—32no—
95C2×C32×C4—43no—
96C2×C32×C4S411yescharacteristic
97C3×#67—64no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—123no—
100C3×D12—32no—
101C3×D12—61no—
102C3×C4×D3—64no—
103C3×C4×D3—1212no—
104C3×Dic6—32no—
105C3×Dic6—61no—
106C3×SL(2,𝔽3)C22×D313yes—
107C3×C3⋊C8—32no—
108C3×C3⋊C8—32no—
109C2×#135—91noSylow
110C2×H(3,𝔽3)—42no—
111C2×H(3,𝔽3)—41no—
112#272—44no—
113C3×#41—32no—
114C3×#41—32no—
115C3×#41—32no—
116C3×#41—62no—
117C3×#41—94no—
118C3×#41—182no—
119C3×#41C2×C3⋊D312yes—
120C2×C3×Q8—31no—
121C2×C3×Q8—91no—
122C2×C3×Q8C2×C3⋊D311yescharacteristic
123C2×C3×D4—31no—
124C2×C3×D4—31no—
125C2×C3×D4—61no—
126C2×C3×D4—91no—
127C2×C3×D4—181no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—61no—
131C22×C3×C4—181no—
132C2×#67—61no—
133#235—34no—
134#235—62no—
135C2×D12—31no—
136C2×C4×D3—61no—
137C2×C4×D3—123no—
138C2×Dic6—31no—
139#41⋊C3—36no—
140C2×SL(2,𝔽3)—33no—
141GL(2,𝔽3)—36no—
142#226—36no—
143C3×Q16—92no—
144C3×Q16—92no—
145C3×SD16—94no—
146C3×SD16—92no—
147C3×SD16—92no—
148C3×D8—94no—
149C2×C3×C8—91no—
150C2×C3×C8—91no—
151C2×C3×C8—94no—
152#216—32no—
153#216—32no—
154#215—32no—
155#215—32no—
156#214—34no—
157#213—34no—
158C2×C3⋊C8—31no—
159C2×C3⋊C8—31no—
160C2×C3⋊C8—34no—
161C22×C32—32no—
162C22×C32—43no—
163C22×C32—62no—
164C22×C32C2×S411yescharacteristic
165C2×C3×D3—62no—
166C2×C3×D3—126no—
167C2×C3×D3—1212no—
168C2×C3×D3—122no—
169C32×C4—31no—
170C32×C4—31no—
171C32×C4—46no—
172C32×C4C2×S412yes—
173C3×Dic3—62no—
174C3×Dic3—126no—
175C2×#41—91no—
176C2×#41#27211yescharacteristic, complemented
177#135—94no—
178#135—94no—
179C2×Q16—91no—
180C2×SD16—91no—
181C2×SD16—91no—
182C2×D8—91no—
183C22×C8—91no—
184H(3,𝔽3)—41noSylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3—181no—
189C22×D3—61no—
190C22×D3—123no—
191C3×Q8—31no—
192C3×Q8—31no—
193C3×Q8—31no—
194C3×Q8—61no—
195C3×Q8—92no—
196C3×Q8—181no—
197C3×Q8C22×C3⋊D311yescharacteristic
198C3×D4—32no—
199C3×D4—32no—
200C3×D4—62no—
201C3×D4—61no—
202C3×D4—61no—
203C3×D4—62no—
204C3×D4—92no—
205C3×D4—181no—
206C3×D4—184no—
207C2×C3×C4—31no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—31no—
211C2×C3×C4—32no—
212C2×C3×C4—32no—
213C2×C3×C4—31no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—61no—
217C2×C3×C4—123no—
218C2×C3×C4—181no—
219C2×C3×C4—184no—
220C2×C3×C4C3⋊S411yescharacteristic
221#67—64no—
222C2×Dic3—61no—
223C2×Dic3—123no—
224D12—32no—
225D12—61no—
226C4×D3—64no—
227C4×D3—1212no—
228Dic6—32no—
229Dic6—61no—
230SL(2,𝔽3)—33no—
231C3×C8—92no—
232C3×C8—92no—
233C3⋊C8—32no—
234C3⋊C8—32no—
235C2×C32—46no—
236C2×C32—43no—
237C2×C32—62no—
238C2×C32#98512yescomplemented
239C2×C32C22×S411yescharacteristic
240C3×D3—1212no—
241C3×D3—122no—
242#41—32no—
243#41—94no—
244#41—182no—
245#41C2×#27212yescomplemented
246C2×Q8—91no—
247C2×Q8C2×#27211yescharacteristic
248C2×D4—31no—
249C2×D4—91no—
250C2×D4—181no—
251C22×C4—31no—
252C22×C4—181no—
253Q16—92no—
254Q16—92no—
255SD16—94no—
256SD16—92no—
257SD16—92no—
258D8—94no—
259C2×C8—91no—
260C2×C8—91no—
261C2×C8—94no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—31no—
265C22×C3—62no—
266C22×C3—62no—
267C22×C3—62no—
268C22×C3—122no—
269C22×C3—123no—
270C22×C3—182no—
271C22×C3—362no—
272C22×C3C2×C3⋊S411yescharacteristic
273C2×D3—62no—
274C2×D3—126no—
275C2×D3—122no—
276C2×D3—1212no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—32no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—126no—
285C3×C4—182no—
286C3×C4C2×C3⋊S412yes—
287Dic3—62no—
288Dic3—126no—
289C32—43no—
290C32C2×#98511yescharacteristic, complemented
291C23—31no—
292C23—181no—
293Q8—31no—
294Q8—92no—
295Q8—181no—
296Q8C22×#27211yescharacteristic
297D4—32no—
298D4—61no—
299D4—92no—
300D4—181no—
301D4—184no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—181no—
306C2×C4—184no—
307C2×C4#625811yescharacteristic
308C8—92no—
309C8—92no—
310C2×C3—31no—
311C2×C3—32no—
312C2×C3—62no—
313C2×C3—62no—
314C2×C3—122no—
315C2×C3—123no—
316C2×C3—126no—
317C2×C3—362no—
318C2×C3#6426612yes—
319C2×C3C22×C3⋊S411yescharacteristic
320D3—122no—
321D3—1212no—
322C22—32no—
323C22—62no—
324C22—182no—
325C22—362no—
326C22C2×#625811yescharacteristic
327C4—31no—
328C4—31no—
329C4—182no—
330C4C2×#625812yes—
331C3—31no—
332C3—123no—
333C3C2×#6426611yescharacteristic
334C2—62no—
335C2—362no—
336C2#1172239312yesdirect product factor
337C2C22×#625811yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC2×C32×#41D3
socleC22×C3C2×C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. #11722393C2direct (N × H)
3. C2×#88611C2semidirect
4. C2×#88610C2semidirect
8. #41⋊H(3,𝔽3)C22semidirect
10. #88611C22semidirect
11. #88610C22semidirect
12. C2×C32×#41D3semidirect
14. C2×C3×#41⋊C3D3semidirect
34. C32×#41C2×D3semidirect
45. C3×#41⋊C3C2×D3semidirect
176. C2×#41#272semidirect
238. C2×C32#985semidirect
245. #41C2×#272semidirect
290. C32C2×#985semidirect
336. C2#11722393direct (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
#117223932
C2×#886112
C2×#886102
C2×C3×#9853
C2×C3×#9503
C2×C4×#2724