Finite Group Databasebeta

← #13137898

Subgroups — #13137898

Order 1728.

379 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 83 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#64097⋊C3C211yesmaximal, characteristic
2#11721116C211yesmaximal, characteristic
3#11720529C211yesmaximal, characteristic
4#63634⋊C3C211yesmaximal, characteristic
5#11720315C211yesmaximal, characteristic
6#11720310C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic
8#10590415—31nomaximal
9#10590411—31nomaximal
10C2×#634⋊C3—21no—
11C2×#6196—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#64097D311yescharacteristic
22C3×#888—31no—
23#63919—31no—
24#63918—31no—
25#63634D311yescharacteristic
26C3×#807—31no—
27C3×#807—31no—
28C3×#807—31no—
29C3×#807—61no—
30#63576—31no—
31#63575—31no—
32#63575—31no—
33#63575—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43#634⋊C3—42no—
44#6196—42no—
45#6180Q812yescomplemented
46(C3⋊(C3⋊C8))⋊C3Q812yescomplemented
47#5078—91no—
48#4874—91no—
49#4874—91no—
50C2×#634—21no—
51C2×C3×Dic6—61no—
52C2×#627—61no—
53C2×#627—61no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59#3750—21no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×Dic3⋊C4—61no—
68C3×C4×Dic3—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—61no—
73#3724—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—41no—
80(C3⋊Dic3)⋊C3—41no—
81(C3⋊Dic3)⋊C3—41no—
82(C3⋊Dic3)⋊C3D812yes—
83(C3⋊Dic3)⋊C3Q1612yes—
84C3×C4⋊Q8—91no—
85#888—31no—
86#888—91no—
87#888—91no—
88#888—91no—
89#869—91no—
90C3×#107—91no—
91C3×#107—91no—
92#818—91no—
93#818—91no—
94C8×Dic3—91no—
95C8×Dic3—91no—
96#807—31no—
97#807—31no—
98#807—31no—
99#807—61no—
100#807—91no—
101#807—91no—
102#807—91no—
103#807—91no—
104#807—91no—
105C4×C3⋊C8—91no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3#6711yescharacteristic
115C2×C3⋊Dic3#6711yescharacteristic
116#634—42no—
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—121no—
124C3×Dic6—122no—
125#627—122no—
126#627—122no—
127C3⋊(C3⋊C8)Dic612yescomplemented
128C3×C3⋊C8—32no—
129C3×C3⋊C8—32no—
130C3×C3⋊C8—32no—
131C3×C3⋊C8—62no—
132#500—271noSylow
133C2×H(3,𝔽3)C4⋊Q811yescharacteristic
134C2×H(3,𝔽3)C2×D811yescharacteristic
135C2×H(3,𝔽3)C2×Q1611yescharacteristic
136C2×C3×Q8—181no—
137C2×Dic6—61no—
138C2×Dic6—181no—
139C2×Dic6—181no—
140C2×Dic6—181no—
141C2×Dic6—181no—
142C2×C3×C8—91no—
143C2×C3×C8—91no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—61no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C4⋊C4D3211yescharacteristic
152C3×C42—91no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—61no—
158C4⋊Dic3—91no—
159C4⋊Dic3—91no—
160C4⋊Dic3—181no—
161Dic3⋊C4—61no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165C4×Dic3—31no—
166C4×Dic3—91no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—61no—
171C2×C3⋊C8—91no—
172C22×C32—21no—
173C22×C32D3×D411yescharacteristic
174C22×C32C2×#6711yescharacteristic
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4—81no—
180C32×C4C2×Dic611yescharacteristic
181C32×C4D3×Q811yescharacteristic
182C32×C4D3×Q811yescharacteristic
183C32×C4D3×Q811yescharacteristic
184C3⋊Dic3—41no—
185C3⋊Dic3—41no—
186C3⋊Dic3#21312yes—
187C3⋊Dic3#21612yes—
188C3×Dic3—32no—
189C3×Dic3—32no—
190C3×Dic3—121no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C3×Dic3—121no—
194C3×Dic3—241no—
195C4⋊Q8—91no—
196C4⋊Q8—271no—
197#107—91no—
198#107—91no—
199#107—271no—
200#107—271no—
201C4×C8—271no—
202H(3,𝔽3)#50011yescharacteristic, complemented, Sylow
203C3×Q8—362no—
204C2×C3×C4—21no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—121no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—31no—
218C2×Dic3—31no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—121no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228Dic6—122no—
229Dic6—362no—
230Dic6—362no—
231Dic6—362no—
232Dic6—362no—
233C3×C8—92no—
234C3×C8—92no—
235C3⋊C8—32no—
236C3⋊C8—32no—
237C3⋊C8—32no—
238C3⋊C8—62no—
239C3⋊C8—92no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32#88811yescharacteristic
244C2×C32D3×D811yescharacteristic
245C2×C32D3×Q1611yescharacteristic
246C2×C32C2×#21311yescharacteristic
247C2×C32C2×#21611yescharacteristic
248C2×C32#94411yescharacteristic
249C2×Q8—181no—
250C2×Q8—541no—
251C2×C8—91no—
252C2×C8—91no—
253C4⋊C4—91no—
254C4⋊C4—91no—
255C4⋊C4—91no—
256C4⋊C4—181no—
257C4⋊C4—541no—
258C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C42—91no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—61no—
263C22×C3D3×#6711yescharacteristic
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—41no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—92no—
272C3×C4—92no—
273C3×C4—121no—
274C3×C4—121no—
275C3×C4—241no—
276C3×C4—361no—
277C3×C4—361no—
278C3×C4—361no—
279C3×C4D3×Dic611yescharacteristic
280C3×C4#379711yescharacteristic
281Dic3—32no—
282Dic3—32no—
283Dic3—92no—
284Dic3—92no—
285Dic3—121no—
286Dic3—121no—
287Dic3—121no—
288Dic3—121no—
289Dic3—241no—
290Dic3—361no—
291Dic3—361no—
292C32—21no—
293C32#487411yescharacteristic, complemented
294C32#507811yescharacteristic, complemented
295Q8—362no—
296Q8—1082no—
297C2×C4—21no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
304C8—92no—
305C8—92no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3#6383911yescharacteristic
316C2×C3D3×#21311yescharacteristic
317C2×C3D3×#21611yescharacteristic
318C22#8867611yescharacteristic
319C4—41no—
320C4—92no—
321C4—92no—
322C4—361no—
323C4—361no—
324C4—361no—
325C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
326C4#8865111yescharacteristic
327C3—31no—
328C3—31no—
329C3—61no—
330C3#1059041111yescharacteristic
331C2#1172107911yescharacteristic
332C2#1172119411yescharacteristic
333C2#1172123611yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#500
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
45. #6180Q8semidirect
46. (C3⋊(C3⋊C8))⋊C3Q8semidirect
127. C3⋊(C3⋊C8)Dic6semidirect
202. H(3,𝔽3)#500semidirect
293. C32#4874semidirect
294. C32#5078semidirect

Maximal subgroups

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

subgroupindex
#64097⋊C32
#117211162
#117205292
#63634⋊C32
#117203152
#117203102
#117203062
#105904153
#105904113