Finite Group Databasebeta

← #13147060

Subgroups — #13147060

Order 1728.

414 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 48 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
2C2×#88610C211yesmaximal, characteristic, complemented
3#11720738C211yesmaximal, characteristic, complemented
4C3×#5419—33nomaximal
5C3×#5228—31nomaximal
6C2×Q8⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8#88610—21no—
9#88552—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C22×C3×SL(2,𝔽3)D313yes—
12C2×C3×#226—33no—
13C2×C3×#216—31no—
14C2×C3×#216—31no—
15C3×#924—31no—
16C3×#859—33no—
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20C23×H(3,𝔽3)—41no—
21C2×#1084—41no—
22C2×#1084—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic
24#5419—33no—
25C3×#451—91no—
26#5228—31no—
27C2×C32×Q8—21no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C2×C32×Q8C2×D311yescharacteristic
31C22×C32×C4—31no—
32C2×C3×Dic6—31no—
33C2×C3×SL(2,𝔽3)—23no—
34C2×C3×SL(2,𝔽3)C2×D313yes—
35C3×#226—63no—
36C3×#217—61no—
37C3×#217—123no—
38C3×#216—61no—
39C3×#216—62no—
40C3×#216—61no—
41C3×C4⋊Dic3—31no—
42C3×Dic3⋊C4—61no—
43C2×C3×C3⋊C8—31no—
44C2×C3×C3⋊C8—31no—
45C22×H(3,𝔽3)—41no—
46C22×H(3,𝔽3)—42no—
47C22×H(3,𝔽3)—81no—
48C22×H(3,𝔽3)—81no—
49#1084—81no—
50#1084—81no—
51C22×C3×Q8—31no—
52C22×C3×Q8C3⋊D311yescharacteristic
53C22×SL(2,𝔽3)—33no—
54C2×#226—33no—
55C2×C3×Q16—91no—
56C2×C3×Q16—91no—
57C3×C22⋊Q8—91no—
58C2×#216—31no—
59C2×#216—31no—
60#924—31no—
61#859—33no—
62C3×#103—91no—
63C3×#103—91no—
64C3×C22⋊C8—91no—
65#835—31no—
66#835—31no—
67#830—31no—
68C23×C32—43no—
69C23×C32S411yescharacteristic
70C32×Q8—31no—
71C32×Q8—32no—
72C32×Q8—61no—
73C32×Q8#6711yescharacteristic
74C2×C32×C4—31no—
75C2×C32×C4—31no—
76C2×C32×C4—61no—
77C2×C32×C4—61no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—123no—
81C2×C3×Dic3—123no—
82C3×Dic6—61no—
83C3×Dic6—61no—
84C3×SL(2,𝔽3)#6713yes—
85C3×C3⋊C8—61no—
86C3×C3⋊C8—61no—
87#451—91noSylow
88C2×H(3,𝔽3)—41no—
89C2×H(3,𝔽3)—41no—
90C2×H(3,𝔽3)—41no—
91C2×H(3,𝔽3)—82no—
92C2×C3×Q8—21no—
93C2×C3×Q8—31no—
94C2×C3×Q8—31no—
95C2×C3×Q8—31no—
96C2×C3×Q8—61no—
97C2×C3×Q8—61no—
98C2×C3×Q8—62no—
99C2×C3×Q8—61no—
100C2×C3×Q8—61no—
101C2×C3×Q8—91no—
102C2×C3×Q8C2×C3⋊D311yescharacteristic
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—61no—
106C2×Dic6—31no—
107C2×SL(2,𝔽3)—33no—
108C2×SL(2,𝔽3)—63no—
109#226—63no—
110C3×Q16—181no—
111C3×Q16—182no—
112C3×Q16—181no—
113C2×C3×C8—91no—
114C2×C3×C8—91no—
115C3×C4⋊C4—91no—
116C3×C4⋊C4—181no—
117C3×C22⋊C4—181no—
118#217—61no—
119#217—123no—
120#216—61no—
121#216—62no—
122#216—61no—
123C4⋊Dic3—31no—
124Dic3⋊C4—61no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—31no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—43no—
130C22×C32—46no—
131C22×C32—83no—
132C22×C32—83no—
133C22×C32#22612yescomplemented
134C22×C32C2×S411yescharacteristic
135C32×C4—31no—
136C32×C4—31no—
137C32×C4—61no—
138C3×Dic3—121no—
139C3×Dic3—121no—
140C3×Dic3—243no—
141C3×Dic3—243no—
142C22×Q8#27211yescharacteristic
143C2×Q16—91no—
144C2×Q16—91no—
145C22⋊Q8—91no—
146#103—91no—
147#103—91no—
148C22⋊C8—91no—
149H(3,𝔽3)—41noSylow
150C23×C3—31no—
151C23×C3—123no—
152C23×C3C3⋊S411yescharacteristic
153C3×Q8—31no—
154C3×Q8—31no—
155C3×Q8—32no—
156C3×Q8—32no—
157C3×Q8—31no—
158C3×Q8—61no—
159C3×Q8—62no—
160C3×Q8—61no—
161C3×Q8—61no—
162C3×Q8—62no—
163C3×Q8—181no—
164C3×Q8—181no—
165C3×Q8#63811yescharacteristic
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—123no—
183C2×Dic3—123no—
184Dic6—61no—
185Dic6—61no—
186SL(2,𝔽3)—33no—
187C3×C8—181no—
188C3×C8—181no—
189C3⋊C8—61no—
190C3⋊C8—61no—
191C2×C32—22no—
192C2×C32—43no—
193C2×C32—43no—
194C2×C32—43no—
195C2×C32—86no—
196C2×C32C2×#22611yescharacteristic
197C2×C32#98311yescharacteristic
198C2×C32GL(2,ℤ/4)11yescharacteristic
199C2×Q8—21no—
200C2×Q8—31no—
201C2×Q8—61no—
202C2×Q8—91no—
203C2×Q8C2×#27211yescharacteristic
204C22×C4—31no—
205Q16—181no—
206Q16—182no—
207Q16—181no—
208C2×C8—91no—
209C2×C8—91no—
210C4⋊C4—91no—
211C4⋊C4—181no—
212C22⋊C4—181no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—32no—
216C22×C3—31no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3—126no—
220C22×C3—123no—
221C22×C3—243no—
222C22×C3—243no—
223C22×C3C3⋊#22612yes—
224C22×C3C2×C3⋊S411yescharacteristic
225C3×C4—31no—
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—61no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4—62no—
233C3×C4—61no—
234C3×C4—361no—
235C3×C4—361no—
236Dic3—121no—
237Dic3—121no—
238Dic3—243no—
239Dic3—243no—
240C32—43no—
241C32#541911yescharacteristic, complemented
242C23#625811yescharacteristic
243Q8—31no—
244Q8—32no—
245Q8—61no—
246Q8—181no—
247Q8—181no—
248Q8#623511yescharacteristic
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—181no—
254C2×C4—181no—
255C8—181no—
256C8—181no—
257C2×C3—22no—
258C2×C3—31no—
259C2×C3—31no—
260C2×C3—31no—
261C2×C3—62no—
262C2×C3—123no—
263C2×C3—123no—
264C2×C3—123no—
265C2×C3—246no—
266C2×C3C2×C3⋊#22611yescharacteristic
267C2×C3#6426411yescharacteristic
268C2×C3#6426911yescharacteristic
269C22—21no—
270C22—21no—
271C22#8861012yescomplemented
272C22C2×#625811yescharacteristic
273C4—31no—
274C4—31no—
275C4—61no—
276C4—361no—
277C4—361no—
278C3—31no—
279C3—123no—
280C3#1059365311yescharacteristic
281C2—22no—
282C2C2×#8861011yescharacteristic
283C2#1172238811yescharacteristic
284C2#1172239811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, cosocleC2×Q8⋊H(3,𝔽3)C22
FittingC22×C32×Q8D3
prime radicalC22×Q8⋊H(3,𝔽3)C2
hypercenterC23×C3C3⋊S4
hypocenterQ8⋊H(3,𝔽3)D4
LCSG ▷ C2×Q8⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×#88610C2semidirect
3. #11720738C2semidirect
133. C22×C32#226semidirect
241. C32#5419semidirect
271. C22#88610semidirect

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
C2×#886102
#117207382
C3×#54193
C3×#52283
#885524