Finite Group Databasebeta

← #13137878

Subgroups — #13137878

Order 1728.

357 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11721758C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic
3#63635⋊C3C211yesmaximal, characteristic
4#63634⋊C3C211yesmaximal, characteristic
5#11720307C211yesmaximal, characteristic
6#11720306C211yesmaximal, characteristic
7#11720302C211yesmaximal, characteristic
8#10590414—31nomaximal
9#10590413—31nomaximal
10C4⋊C4×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084—21no—
12C4⋊#1084C2211yescharacteristic
13#88540—21no—
14C4×#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21C3×#890—31no—
22C3×#890—31no—
23C3×#890—61no—
24#63869—31no—
25#63869—31no—
26#63635D311yescharacteristic
27#63634D311yescharacteristic
28C3×#808—31no—
29C3×#807—31no—
30#63576—31no—
31#63575—31no—
32#63574—31no—
33#63573—31no—
34Dic3×C3⋊C8—31no—
35Dic3×C3⋊C8—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084D411yescharacteristic
40C2×#1084D411yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43(C3⋊(C3⋊C8))⋊C3Q812yes—
44(C3⋊(C3⋊C8))⋊C3Q812yes—
45#5065—91no—
46#4878—91no—
47#4858—91no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4C2×D311yescharacteristic
50C32×C4⋊C4C2×D311yescharacteristic
51C4⋊(C3⋊Dic3)C2×D311yescharacteristic
52C4⋊(C3⋊Dic3)C2×D311yescharacteristic
53C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3—61no—
61C3×C4⋊Dic3—61no—
62C3×C4⋊Dic3—62no—
63C3×Dic3⋊C4—61no—
64C3×Dic3⋊C4—61no—
65C3×Dic3⋊C4—62no—
66C3×C4×Dic3—31no—
67C3×C4×Dic3—31no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71#3724—61no—
72#3724—61no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—61no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)—41no—
77C4×H(3,𝔽3)C2×Q811yescharacteristic
78C4×H(3,𝔽3)C2×Q811yescharacteristic
79#1084—21no—
80#1084—21no—
81#1084—41no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84C3×#125—91no—
85#890—31no—
86#890—31no—
87#890—61no—
88#890—91no—
89#890—91no—
90#870—91no—
91C3×#107—91no—
92C3×#106—91no—
93#818—91no—
94#817—91no—
95C8×Dic3—91no—
96C8×Dic3—91no—
97#808—31no—
98#808—91no—
99#808—91no—
100#808—91no—
101#807—31no—
102#807—91no—
103#807—91no—
104#807—91no—
105C4×C3⋊C8—91no—
106C2×C32×C4—22no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3⋊(C3⋊C8)Dic612yes—
126C3⋊(C3⋊C8)Dic612yes—
127C3×C3⋊C8—32no—
128C3×C3⋊C8—32no—
129#499—271noSylow
130C2×H(3,𝔽3)C4⋊Q811yescharacteristic
131C2×H(3,𝔽3)#13511yescharacteristic
132C2×H(3,𝔽3)#13511yescharacteristic
133C2×C3×C8—91no—
134C2×C3×C8—91no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—91no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4—181no—
143C3×C4⋊C4D3211yescharacteristic
144C3×C42—91no—
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—61no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—62no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157Dic3⋊C4—61no—
158Dic3⋊C4—61no—
159Dic3⋊C4—62no—
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164C4×Dic3—31no—
165C4×Dic3—31no—
166C4×Dic3—61no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—91no—
171C22×C32—21no—
172C22×C32D3×D411yescharacteristic
173C22×C32D3×D411yescharacteristic
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—42no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4C2×Dic611yescharacteristic
180C32×C4C2×Dic611yescharacteristic
181C32×C4D3×Q811yescharacteristic
182C32×C4D3×Q811yescharacteristic
183C3⋊Dic3—41no—
184C3⋊Dic3—41no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—61no—
188C3×Dic3—61no—
189C3×Dic3—121no—
190C3×Dic3—121no—
191C3×Dic3—122no—
192C3×Dic3—121no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196#125—91no—
197#125—271no—
198#107—91no—
199#107—271no—
200#106—91no—
201#106—271no—
202C4×C8—271no—
203H(3,𝔽3)#49911yescharacteristic, complemented, Sylow
204C2×C3×C4—21no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—62no—
210C2×C3×C4—61no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—31no—
218C2×Dic3—31no—
219C2×Dic3—31no—
220C2×Dic3—31no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—62no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—181no—
229C2×Dic3—181no—
230C3×C8—92no—
231C3×C8—92no—
232C3⋊C8—32no—
233C3⋊C8—32no—
234C3⋊C8—92no—
235C3⋊C8—92no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32#88811yescharacteristic
240C2×C32#88811yescharacteristic
241C2×C32#91211yescharacteristic
242C2×C32#91611yescharacteristic
243C2×C32#91611yescharacteristic
244C2×C32#91911yescharacteristic
245C2×C8—91no—
246C2×C8—91no—
247C4⋊C4—91no—
248C4⋊C4—91no—
249C4⋊C4—91no—
250C4⋊C4—181no—
251C4⋊C4—181no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C42—91no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—61no—
259C22×C3#381111yescharacteristic
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—41no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—121no—
268C3×C4—121no—
269C3×C4—122no—
270C3×C4—181no—
271C3×C4—181no—
272C3×C4—361no—
273C3×C4—361no—
274C3×C4—361no—
275C3×C4D3×Dic611yescharacteristic
276C3×C4D3×Dic611yescharacteristic
277Dic3—61no—
278Dic3—61no—
279Dic3—61no—
280Dic3—61no—
281Dic3—121no—
282Dic3—121no—
283Dic3—121no—
284Dic3—121no—
285Dic3—122no—
286Dic3—121no—
287Dic3—121no—
288Dic3—361no—
289Dic3—361no—
290C32—21no—
291C32#485811yescharacteristic, complemented
292C32#487811yescharacteristic, complemented
293C2×C4—21no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4—181no—
299C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C8—92no—
301C8—92no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—61no—
309C2×C3—61no—
310C2×C3—61no—
311C2×C3#6384011yescharacteristic
312C2×C3#6393611yescharacteristic
313C2×C3#6395011yescharacteristic
314C22#8867711yescharacteristic
315C4—41no—
316C4—181no—
317C4—181no—
318C4—361no—
319C4—361no—
320C4—361no—
321C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
322C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
323C3—31no—
324C3—31no—
325C3—61no—
326C3#1059041711yescharacteristic
327C2#1172108011yescharacteristic
328C2#1172119211yescharacteristic
329C2#1172123011yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#499
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
203. H(3,𝔽3)#499semidirect
291. C32#4858semidirect
292. C32#4878semidirect

Maximal subgroups

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

subgroupindex
#117217582
#117210952
#63635⋊C32
#63634⋊C32
#117203072
#117203062
#117203022
#105904143
#105904133