Finite Group Databasebeta

← #13144488

Subgroups — #13144488

Order 1728.

836 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 113 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×#88740C211yesmaximal, characteristic
2H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#88550C211yesmaximal, characteristic, complemented
4C2×#88549C211yesmaximal, characteristic, complemented
5#11720573C211yesmaximal, characteristic, complemented
6#11720532C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#5047—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12#88740—22no—
13#88740—22no—
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272—21no—
16C2×#6229C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)—21no—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C22⋊C4×H(3,𝔽3)—21no—
20#88550—22no—
21#88549—22no—
22C2×#6180C2211yescharacteristic
23C2×#6180C2211yescharacteristic
24C2×C3×#235—34no—
25C32×C22⋊Q8D314yescomplemented
26C2×C3×#216—34no—
27C2×C3×#215—34no—
28C3×#830—34no—
29C3×#810—34no—
30C3×#809—34no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×#272—21no—
33Q8×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#6235—24no—
42C2×#1084—21no—
43C4⋊#272—21no—
44C4⋊#272D412yes—
45C4×#272—24no—
46#6229—21no—
47#6229D412yes—
48#6180—21no—
49#6180—21no—
50C3×#452—91no—
51#5047—34no—
52C2×C32×Q8C2×D314yes—
53C22×C32×C4C2×D314yes—
54C2×C3×#67—64no—
55C3×#235—68no—
56C3×#235—68no—
57C2×C3×D12—34no—
58C2×C3×C4×D3—64no—
59C2×C3×Dic6—34no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4C2×D314yes—
62C32×C22⋊C4—24no—
63C3×#216—68no—
64C3×#215—68no—
65C2×C3×C3⋊C8—34no—
66C2×C3×C3⋊C8—34no—
67C22×H(3,𝔽3)—21no—
68C22×H(3,𝔽3)—21no—
69C22×H(3,𝔽3)—21no—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C2×#272—22no—
72C2×#272—41no—
73C2×#272—41no—
74C4×H(3,𝔽3)—21no—
75C4×H(3,𝔽3)—41no—
76C4×H(3,𝔽3)—41no—
77C4×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)C2×D411yescharacteristic
79#1084—22no—
80C2×C3×#41—91no—
81C2×#235—34no—
82C2×C3×Q16—91no—
83C2×C3×SD16—91no—
84C3×C22⋊Q8—34no—
85C3×C22⋊Q8C3⋊D311yescharacteristic
86C2×#216—34no—
87C2×#215—34no—
88C3×#103—91no—
89C3×#102—91no—
90C3×C22⋊C8—91no—
91#830—34no—
92#810—34no—
93#809—34no—
94C23×C32#6714yes—
95C22×C3×D3—64no—
96C32×Q8—24no—
97C32×Q8—24no—
98C2×C32×C4—24no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4—24no—
102C2×C32×C4#6714yes—
103C2×C32×C4C22×D314yes—
104C3×#67—616no—
105C2×C3×Dic3—64no—
106C3×D12—38no—
107C3×D12—64no—
108C3×C4×D3—616no—
109C3×Dic6—38no—
110C3×Dic6—64no—
111C3×C3⋊C8—64no—
112C3×C3⋊C8—64no—
113#452—91noSylow
114C2×H(3,𝔽3)—41no—
115C2×H(3,𝔽3)#12011yescharacteristic
116C2×H(3,𝔽3)#13511yescharacteristic
117C2×H(3,𝔽3)#13711yescharacteristic
118#272—42no—
119C3×#41—182no—
120C3×#41—182no—
121C2×C3×Q8—34no—
122C2×C3×Q8—91no—
123C2×C3×Q8C2×C3⋊D311yescharacteristic
124C2×C3×D4—91no—
125C2×C3×D4—181no—
126C22×C3×C4—34no—
127C22×C3×C4—181no—
128C22×C3×C4C2×C3⋊D311yescharacteristic
129C2×#67—64no—
130#235—68no—
131#235—68no—
132C2×D12—34no—
133C2×C4×D3—64no—
134C2×Dic6—34no—
135C3×Q16—182no—
136C3×SD16—182no—
137C2×C3×C8—91no—
138C2×C3×C8—91no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—34no—
141C3×C4⋊C4—64no—
142C3×C4⋊C4C2×C3⋊D311yescharacteristic
143C3×C22⋊C4—21no—
144C3×C22⋊C4—64no—
145#216—68no—
146#215—68no—
147C2×C3⋊C8—34no—
148C2×C3⋊C8—34no—
149C22×C32—24no—
150C22×C32—24no—
151C22×C32—24no—
152C22×C32C2×#6714yes—
153C2×C3×D3—68no—
154C2×C3×D3—124no—
155C2×C3×D3—124no—
156C32×C4—24no—
157C32×C4—44no—
158C32×C4—44no—
159C32×C4D3×D414yes—
160C32×C4D3×D414yes—
161C3×Dic3—68no—
162C2×#41—91no—
163C2×Q16—91no—
164C2×SD16—91no—
165C22⋊Q8#27211yescharacteristic, complemented
166#103—91no—
167#102—91no—
168C22⋊C8—91no—
169H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
170C23×C3—34no—
171C23×C3—181no—
172C23×C3#63811yescharacteristic
173C22×D3—64no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—64no—
177C3×Q8—64no—
178C3×Q8—92no—
179C3×Q8—181no—
180C3×D4—92no—
181C3×D4—184no—
182C3×D4—181no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—34no—
188C2×C3×C4—34no—
189C2×C3×C4—64no—
190C2×C3×C4—64no—
191C2×C3×C4—64no—
192C2×C3×C4—64no—
193C2×C3×C4—184no—
194C2×C3×C4—181no—
195C2×C3×C4#63811yescharacteristic
196C2×C3×C4C22×C3⋊D311yescharacteristic
197#67—616no—
198C2×Dic3—64no—
199D12—38no—
200D12—64no—
201C4×D3—616no—
202Dic6—38no—
203Dic6—64no—
204C3×C8—181no—
205C3×C8—181no—
206C3⋊C8—64no—
207C3⋊C8—64no—
208C2×C32—44no—
209C2×C32#93714yes—
210C2×C32#94214yes—
211C2×C32#95014yes—
212C3×D3—128no—
213#41—182no—
214#41—182no—
215C2×Q8—91no—
216C2×Q8C2×#27211yescharacteristic
217C2×D4—91no—
218C2×D4—181no—
219C22×C4—181no—
220C22×C4C2×#27211yescharacteristic
221Q16—182no—
222SD16—182no—
223C2×C8—91no—
224C2×C8—91no—
225C4⋊C4—21no—
226C4⋊C4C2×#27211yescharacteristic
227C22⋊C4—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—34no—
232C22×C3—64no—
233C22×C3—64no—
234C22×C3—64no—
235C22×C3—182no—
236C22×C3—361no—
237C22×C3—361no—
238C22×C3C2×#63811yescharacteristic
239C2×D3—68no—
240C2×D3—124no—
241C2×D3—124no—
242C3×C4—21no—
243C3×C4—34no—
244C3×C4—34no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—64no—
248C3×C4—124no—
249C3×C4—124no—
250C3×C4—182no—
251C3×C4D4×C3⋊D311yescharacteristic
252C3×C4D4×C3⋊D311yescharacteristic
253Dic3—68no—
254C32#504714yescomplemented
255C23—181no—
256C23#623511yescharacteristic
257Q8—21no—
258Q8—21no—
259Q8—92no—
260Q8—181no—
261D4—92no—
262D4—184no—
263D4—181no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—184no—
269C2×C4—181no—
270C2×C4#623511yescharacteristic
271C2×C4C22×#27211yescharacteristic
272C8—181no—
273C8—181no—
274C2×C3—34no—
275C2×C3—34no—
276C2×C3—34no—
277C2×C3—41no—
278C2×C3—124no—
279C2×C3—362no—
280C2×C3#6414611yescharacteristic
281C2×C3#6415111yescharacteristic
282C2×C3#6415911yescharacteristic
283D3—128no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—182no—
288C22—361no—
289C22—361no—
290C22C2×#623511yescharacteristic
291C4—21no—
292C4—41no—
293C4—41no—
294C4—182no—
295C4D4×#27211yescharacteristic
296C4D4×#27211yescharacteristic
297C3—34no—
298C3#1059274611yescharacteristic
299C2—41no—
300C2—362no—
301C2#1172184711yescharacteristic
302C2#1172185911yescharacteristic
303C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC2×#88740C2
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#452
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×C22⋊Q8C2semidirect
3. C2×#88550C2semidirect
4. C2×#88549C2semidirect
5. #11720573C2semidirect
6. #11720532C2semidirect
7. #11720531C2semidirect
25. C32×C22⋊Q8D3semidirect
165. C22⋊Q8#272semidirect
169. H(3,𝔽3)#452semidirect
254. C32#5047semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
H(3,𝔽3)×C22⋊Q82
C2×#885502
C2×#885492
#117205732
#117205322
#117205312
C3×#50473