Finite Group Databasebeta

← #13144861

Subgroups — #13144861

Order 1728.

836 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×#88550C211yesmaximal, characteristic, complemented
3C2×#88546C211yesmaximal, characteristic, complemented
4#11721824C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720578C211yesmaximal, characteristic, complemented
7#11720573C211yesmaximal, characteristic, complemented
8C3×#5239—34nomaximal
9#41×H(3,𝔽3)—22no—
10#41×H(3,𝔽3)—22no—
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×D4×H(3,𝔽3)—21no—
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)—21no—
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C2×#6229C2211yescharacteristic
17#88552—21no—
18#88550—22no—
19#88546—22no—
20C4⋊#1084C2211yescharacteristic
21#88540—21no—
22C2×#6180C2211yescharacteristic
23C2×#6180C2211yescharacteristic
24C2×C32×#41D314yes—
25C2×C3×#216—34no—
26C2×C3×#214—34no—
27C3×#924—34no—
28C3×#835—34no—
29C3×#832—34no—
30C3×#830—34no—
31C23×H(3,𝔽3)—21no—
32C23×H(3,𝔽3)D411yescharacteristic
33Q8×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)D412yes—
35D4×H(3,𝔽3)—24no—
36D4×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)D412yes—
38C2×C4×H(3,𝔽3)—24no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45C2×#1084—21no—
46#6229—21no—
47#6229—21no—
48#6180—21no—
49#6180—21no—
50C3×#452—91no—
51#5239—34no—
52C32×#41—28no—
53C32×#41—28no—
54C2×C32×Q8C2×D314yes—
55C2×C32×D4—24no—
56C2×C32×D4C2×D314yes—
57C22×C32×C4—24no—
58C22×C32×C4C2×D314yes—
59C2×C3×Dic6—34no—
60C3×#217—64no—
61C3×#216—68no—
62C3×#214—68no—
63C3×C4⋊Dic3—34no—
64C3×Dic3⋊C4—64no—
65C2×C3×C3⋊C8—34no—
66C2×C3×C3⋊C8—34no—
67C22×H(3,𝔽3)—21no—
68C22×H(3,𝔽3)—22no—
69C22×H(3,𝔽3)—21no—
70C22×H(3,𝔽3)—21no—
71C22×H(3,𝔽3)—41no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)—21no—
75C4×H(3,𝔽3)—22no—
76C4×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)C2×D411yescharacteristic
78#1084—41no—
79#1084—41no—
80C2×C3×#41—34no—
81C2×C3×#41C3⋊D311yescharacteristic
82C2×C3×Q16—91no—
83C2×C3×SD16—91no—
84C3×C22⋊Q8—91no—
85C2×#216—34no—
86C2×#214—34no—
87#924—34no—
88C3×#103—91no—
89C3×#102—91no—
90C3×C22⋊C8—91no—
91#835—34no—
92#832—34no—
93#830—34no—
94C23×C32—24no—
95C23×C32#6714yes—
96C32×Q8—24no—
97C32×Q8#6718yes—
98C32×D4—216no—
99C32×D4—24no—
100C32×D4#6718yes—
101C2×C32×C4—216no—
102C2×C32×C4—24no—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4#6714yes—
106C2×C32×C4C22×D314yes—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—64no—
109C3×Dic6—64no—
110C3×Dic6—64no—
111C3×C3⋊C8—64no—
112C3×C3⋊C8—64no—
113#452—91noSylow
114C2×H(3,𝔽3)—41no—
115C2×H(3,𝔽3)—42no—
116C2×H(3,𝔽3)#12011yescharacteristic
117C2×H(3,𝔽3)#13511yescharacteristic
118C2×H(3,𝔽3)#13711yescharacteristic
119C3×#41—22no—
120C3×#41—22no—
121C3×#41—68no—
122C3×#41—68no—
123C2×C3×Q8—34no—
124C2×C3×Q8—91no—
125C2×C3×Q8C2×C3⋊D311yescharacteristic
126C2×C3×D4—21no—
127C2×C3×D4—34no—
128C2×C3×D4—64no—
129C2×C3×D4C2×C3⋊D311yescharacteristic
130C22×C3×C4—21no—
131C22×C3×C4—34no—
132C22×C3×C4—64no—
133C22×C3×C4C2×C3⋊D311yescharacteristic
134C2×Dic6—34no—
135C3×Q16—182no—
136C3×SD16—182no—
137C2×C3×C8—91no—
138C2×C3×C8—91no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—181no—
141C3×C22⋊C4—181no—
142#217—64no—
143#216—68no—
144#214—68no—
145C4⋊Dic3—34no—
146Dic3⋊C4—64no—
147C2×C3⋊C8—34no—
148C2×C3⋊C8—34no—
149C22×C32—24no—
150C22×C32—24no—
151C22×C32—24no—
152C22×C32—28no—
153C22×C32—44no—
154C22×C32—44no—
155C22×C32C2×#6714yes—
156C32×C4—24no—
157C32×C4—28no—
158C32×C4C2×#6714yes—
159C32×C4C2×#6714yes—
160C3×Dic3—124no—
161C3×Dic3—124no—
162C2×#41#27211yescharacteristic
163C2×Q16—91no—
164C2×SD16—91no—
165C22⋊Q8—91no—
166#103—91no—
167#102—91no—
168C22⋊C8—91no—
169H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
170C23×C3—21no—
171C23×C3—34no—
172C23×C3—64no—
173C23×C3#63811yescharacteristic
174C3×Q8—21no—
175C3×Q8—38no—
176C3×Q8—64no—
177C3×Q8—181no—
178C3×Q8—181no—
179C3×Q8#63812yes—
180C3×D4—24no—
181C3×D4—21no—
182C3×D4—38no—
183C3×D4—64no—
184C3×D4—616no—
185C3×D4#63812yes—
186C2×C3×C4—24no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—34no—
191C2×C3×C4—34no—
192C2×C3×C4—64no—
193C2×C3×C4—64no—
194C2×C3×C4—64no—
195C2×C3×C4—616no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198C2×C3×C4#63811yescharacteristic
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—64no—
201C2×Dic3—64no—
202Dic6—64no—
203Dic6—64no—
204C3×C8—181no—
205C3×C8—181no—
206C3⋊C8—64no—
207C3⋊C8—64no—
208C2×C32—44no—
209C2×C32—48no—
210C2×C32#95014yes—
211C2×C32#95114yes—
212C2×C32#95314yes—
213#41—22no—
214#41—22no—
215C2×Q8—91no—
216C2×Q8C2×#27211yescharacteristic
217C2×D4—21no—
218C2×D4C2×#27211yescharacteristic
219C22×C4—21no—
220C22×C4C2×#27211yescharacteristic
221Q16—182no—
222SD16—182no—
223C2×C8—91no—
224C2×C8—91no—
225C4⋊C4—91no—
226C4⋊C4—181no—
227C22⋊C4—181no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—22no—
232C22×C3—34no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—68no—
236C22×C3—64no—
237C22×C3—64no—
238C22×C3—64no—
239C22×C3—124no—
240C22×C3—124no—
241C22×C3C2×#63811yescharacteristic
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—34no—
245C3×C4—34no—
246C3×C4—64no—
247C3×C4—68no—
248C3×C4—361no—
249C3×C4—361no—
250C3×C4C2×#63811yescharacteristic
251C3×C4C2×#63811yescharacteristic
252Dic3—124no—
253Dic3—124no—
254C32#523914yescomplemented
255C23—21no—
256C23#623511yescharacteristic
257Q8—21no—
258Q8—181no—
259Q8—181no—
260Q8#623512yes—
261D4—24no—
262D4—21no—
263D4#623512yes—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—181no—
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—42no—
279C2×C3—124no—
280C2×C3—128no—
281C2×C3#6415911yescharacteristic
282C2×C3#6416011yescharacteristic
283C2×C3#6416211yescharacteristic
284C22—21no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—41no—
289C22—41no—
290C22C2×#623511yescharacteristic
291C4—21no—
292C4—22no—
293C4—361no—
294C4—361no—
295C4C2×#623511yescharacteristic
296C4C2×#623511yescharacteristic
297C3—34no—
298C3#1059293811yescharacteristic
299C2—41no—
300C2—42no—
301C2#1172187111yescharacteristic
302C2#1172187211yescharacteristic
303C2#1172187611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
Fitting, prime radicalC2×#41×H(3,𝔽3)C2
hypercenterC2×C3×#41C3⋊D3
hypocenterH(3,𝔽3)#452
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
2. C2×#88550C2semidirect
3. C2×#88546C2semidirect
4. #11721824C2semidirect
5. #11720583C2semidirect
6. #11720578C2semidirect
7. #11720573C2semidirect
169. H(3,𝔽3)#452semidirect
254. C32#5239semidirect

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
C2×#885502
C2×#885462
#117218242
#117205832
#117205782
#117205732
C3×#52393