Finite Group Databasebeta

← #13138750

Subgroups — #13138750

Order 1728.

353 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721095C211yesmaximal, characteristic
2#11721084C211yesmaximal, characteristic
3#11720547C211yesmaximal, characteristic
4#63644⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic
6#11720315C211yesmaximal, characteristic
7#11720313C211yesmaximal, characteristic
8#10590815—31nomaximal
9#10590809—31nomaximal
10C2×#634⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#88448—21no—
18#88448—21no—
19#88448—21no—
20#88447—21no—
21#63869—31no—
22#63869—31no—
23#63844—31no—
24Dic3×Dic6—31no—
25#63644D311yescharacteristic
26#63643D311yescharacteristic
27C3×#817—31no—
28C3×#816—31no—
29C3×Dic3⋊C8—31no—
30C3×Dic3⋊C8—31no—
31C3×Dic3⋊C8—61no—
32#63575—31no—
33#63575—31no—
34#63567—31no—
35#63566—31no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084D411yescharacteristic
38C2×#1084D411yescharacteristic
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43#634⋊C3—21no—
44#634⋊C3Q812yes—
45C8×H(3,𝔽3)—21no—
46#6180—21no—
47#4857—91no—
48#4829—91no—
49#4799—91no—
50C2×#634C2×D311yescharacteristic
51C2×C3×Dic6—31no—
52C2×C32×C8—21no—
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4×Dic3—31no—
62C3×C4×Dic3—31no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—61no—
67#3724—61no—
68#3724—61no—
69#3724—61no—
70Dic3⋊Dic3—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic32—61no—
74Dic32—61no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)C2×Q811yescharacteristic
77C4×H(3,𝔽3)#4111yescharacteristic
78#1084—21no—
79#1084—21no—
80(C3⋊Dic3)⋊C3—22no—
81(C3⋊Dic3)⋊C3—41no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84Q8×Dic3—91no—
85#890—91no—
86#890—91no—
87#887—91no—
88#870—91no—
89C4×Dic6—91no—
90C3×#106—91no—
91C3×C4⋊C8—91no—
92C3×#103—91no—
93#835—91no—
94#818—91no—
95#817—31no—
96#817—91no—
97#816—31no—
98#816—91no—
99#816—91no—
100Dic3⋊C8—31no—
101Dic3⋊C8—31no—
102Dic3⋊C8—61no—
103#810—91no—
104#807—91no—
105#807—91no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—21no—
113#634—21no—
114#634Dic612yes—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3×Dic6—32no—
126C3×Dic6—61no—
127C32×C8—21no—
128C32×C8—21no—
129C32×C8—41no—
130C3×C3⋊C8—61no—
131C3×C3⋊C8—61no—
132C3×C3⋊C8—121no—
133#479—271noSylow
134C2×H(3,𝔽3)C22⋊Q811yescharacteristic
135C2×H(3,𝔽3)#13511yescharacteristic
136C2×H(3,𝔽3)#13711yescharacteristic
137C2×C3×Q8—91no—
138C2×Dic6—31no—
139C2×Dic6—91no—
140C2×C3×C8—31no—
141C2×C3×C8—31no—
142C2×C3×C8—61no—
143C2×C3×C8—91no—
144C2×C3×C8D3211yescharacteristic
145C3×C4⋊C4—91no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—91no—
148C3×C42—91no—
149C4⋊Dic3—31no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157C4⋊Dic3—181no—
158Dic3⋊C4—181no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164C4×Dic3—31no—
165C4×Dic3—31no—
166C4×Dic3—61no—
167C4×Dic3—181no—
168C4×Dic3—181no—
169C4×Dic3—181no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—61no—
173C22×C32—21no—
174C22×C32D3×D411yescharacteristic
175C22×C32D3×D411yescharacteristic
176C32×C4—21no—
177C32×C4—21no—
178C32×C4C2×Dic611yescharacteristic
179C32×C4#23511yescharacteristic
180C32×C4#23711yescharacteristic
181C32×C4D3×Q811yescharacteristic
182C3⋊Dic3—22no—
183C3⋊Dic3—41no—
184C3⋊Dic3—41no—
185C3⋊Dic3—41no—
186C3×Dic3—62no—
187C3×Dic3—62no—
188C3×Dic3—61no—
189C3×Dic3—61no—
190C3×Dic3—61no—
191C3×Dic3—61no—
192C3×Dic3—121no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196#125—271no—
197C4×Q8—271no—
198#107—271no—
199#106—91no—
200C4⋊C8—91no—
201#103—91no—
202#103—271no—
203H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
204C3×Q8—92no—
205C3×Q8—181no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—61no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4C2×D3211yescharacteristic
216C2×Dic3—31no—
217C2×Dic3—31no—
218C2×Dic3—31no—
219C2×Dic3—31no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C2×Dic3—181no—
230Dic6—32no—
231Dic6—61no—
232Dic6—92no—
233Dic6—181no—
234C3×C8—21no—
235C3×C8—61no—
236C3×C8—61no—
237C3×C8—121no—
238C3×C8—181no—
239C3⋊C8—61no—
240C3⋊C8—61no—
241C3⋊C8—121no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32#87811yescharacteristic
246C2×C32#89611yescharacteristic
247C2×C32#91511yescharacteristic
248C2×C32#91611yescharacteristic
249C2×C32#91611yescharacteristic
250C2×C32#91811yescharacteristic
251C2×Q8—91no—
252C2×C8—91no—
253C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C4⋊C4—91no—
255C4⋊C4—91no—
256C4⋊C4—91no—
257C4⋊C4—541no—
258C4⋊C4—541no—
259C4⋊C4—541no—
260C42—91no—
261C42—541no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—61no—
265C22×C3#380211yescharacteristic
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—182no—
273C3×C4—181no—
274C3×C4—181no—
275C3×C4—361no—
276C3×C4—361no—
277C3×C4—361no—
278C3×C4D3×Dic611yescharacteristic
279C3×C4#380411yescharacteristic
280Dic3—61no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—62no—
285Dic3—62no—
286Dic3—121no—
287Dic3—121no—
288Dic3—121no—
289Dic3—121no—
290Dic3—182no—
291Dic3—361no—
292Dic3—361no—
293Dic3—361no—
294C32—21no—
295C32#479911yescharacteristic, complemented
296C32#485711yescharacteristic, complemented
297Q8—92no—
298Q8—181no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4—181no—
304C2×C4—181no—
305C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C8—21no—
307C8—181no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—61no—
315C2×C3—61no—
316C2×C3—61no—
317C2×C3#6380911yescharacteristic
318C2×C3#6381111yescharacteristic
319C2×C3#6389911yescharacteristic
320C22#8865511yescharacteristic
321C4—181no—
322C4—181no—
323C4—182no—
324C4—361no—
325C4—361no—
326C4—361no—
327C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
328C4#8867111yescharacteristic
329C3—31no—
330C3—31no—
331C3—61no—
332C3#1059082011yescharacteristic
333C2#1172084311yescharacteristic
334C2#1172084511yescharacteristic
335C2#1172113411yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#479
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
203. H(3,𝔽3)#479semidirect
295. C32#4799semidirect
296. C32#4857semidirect

Maximal subgroups

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

subgroupindex
#117210952
#117210842
#117205472
#63644⋊C32
#63643⋊C32
#117203152
#117203132
#105908153
#105908093