Finite Group Databasebeta

← #13137906

Subgroups — #13137906

Order 1728.

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

normal subgroups only

This group has 351 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#64097⋊C3C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic
3#11720529C211yesmaximal, characteristic
4#63635⋊C3C211yesmaximal, characteristic
5#11720314C211yesmaximal, characteristic
6#11720312C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic
8#10590546—31nomaximal
9#10590420—31nomaximal
10C2×#634⋊C3—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21#64097D311yescharacteristic
22C3×#888—31no—
23#63872—31no—
24#63869—31no—
25#63635D311yescharacteristic
26C3×#808—31no—
27C3×#807—31no—
28C3×#807—31no—
29C3×#807—61no—
30#63576—31no—
31#63575—31no—
32#63573—31no—
33#63573—31no—
34#63555—31no—
35#63554—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#6180Q812yescomplemented
45(C3⋊(C3⋊C8))⋊C3Q812yescomplemented
46#5081—91no—
47#4876—91no—
48#4856—91no—
49C2×#634—21no—
50C2×C3×Dic6—61no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)C2×D311yescharacteristic
56#3750—21no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—61no—
64C3×Dic3⋊C4—61no—
65C3×C4×Dic3—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—61no—
70#3724—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C4×H(3,𝔽3)—41no—
76C4×H(3,𝔽3)C2×Q811yescharacteristic
77C4×H(3,𝔽3)C2×Q811yescharacteristic
78#1084—41no—
79(C3⋊Dic3)⋊C3—21no—
80(C3⋊Dic3)⋊C3—21no—
81(C3⋊Dic3)⋊C3—41no—
82(C3⋊Dic3)⋊C3—41no—
83C3×C4⋊Q8—91no—
84#890—91no—
85#890—91no—
86#888—31no—
87#888—91no—
88#870—91no—
89C3×#107—91no—
90C3×#106—91no—
91#818—91no—
92#817—91no—
93#815—91no—
94#815—91no—
95#808—31no—
96#808—91no—
97#808—91no—
98#808—91no—
99#807—31no—
100#807—31no—
101#807—61no—
102#807—91no—
103#807—91no—
104#803—91no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3#6711yescharacteristic
114C2×C3⋊Dic3#6711yescharacteristic
115#634—42no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—121no—
123C3×Dic6—122no—
124C3⋊(C3⋊C8)Dic612yescomplemented
125C3×C3⋊C8—32no—
126C3×C3⋊C8—32no—
127C3×C3⋊C8—32no—
128C3×C3⋊C8—62no—
129#501—271noSylow
130C2×H(3,𝔽3)C4⋊Q811yescharacteristic
131C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
132C2×H(3,𝔽3)#13711yescharacteristic
133C2×C3×Q8—181no—
134C2×Dic6—61no—
135C2×Dic6—181no—
136C2×C3×C8—91no—
137C2×C3×C8—91no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—91no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4D3211yescharacteristic
146C3×C42—91no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—31no—
151C4⋊Dic3—61no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154C4⋊Dic3—181no—
155C4⋊Dic3—181no—
156Dic3⋊C4—61no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162C4×Dic3—31no—
163C4×Dic3—91no—
164C2×C3⋊C8—31no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—91no—
169C22×C32—21no—
170C22×C32D3×D411yescharacteristic
171C22×C32C2×#6711yescharacteristic
172C32×C4—21no—
173C32×C4—21no—
174C32×C4—41no—
175C32×C4—41no—
176C32×C4—81no—
177C32×C4C2×Dic611yescharacteristic
178C32×C4D3×Q811yescharacteristic
179C32×C4D3×Q811yescharacteristic
180C32×C4D3×Q811yescharacteristic
181C3⋊Dic3—21no—
182C3⋊Dic3—21no—
183C3⋊Dic3—41no—
184C3⋊Dic3—41no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—121no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C3×Dic3—121no—
191C3×Dic3—241no—
192C4⋊Q8—91no—
193#125—271no—
194#107—91no—
195#107—271no—
196#106—91no—
197#106—271no—
198#97—271no—
199H(3,𝔽3)#50111yescharacteristic, complemented, Sylow
200C3×Q8—362no—
201C2×C3×C4—21no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4—121no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4C2×D3211yescharacteristic
214C2×Dic3—31no—
215C2×Dic3—31no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—91no—
221C2×Dic3—91no—
222C2×Dic3—121no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225Dic6—122no—
226Dic6—362no—
227C3×C8—92no—
228C3×C8—92no—
229C3⋊C8—32no—
230C3⋊C8—32no—
231C3⋊C8—32no—
232C3⋊C8—62no—
233C3⋊C8—92no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32#88811yescharacteristic
238C2×C32#91411yescharacteristic
239C2×C32#91511yescharacteristic
240C2×C32#93211yescharacteristic
241C2×C32#94211yescharacteristic
242C2×C32#94411yescharacteristic
243C2×Q8—181no—
244C2×C8—91no—
245C2×C8—91no—
246C4⋊C4—91no—
247C4⋊C4—91no—
248C4⋊C4—91no—
249C4⋊C4—181no—
250C4⋊C4—541no—
251C4⋊C4—541no—
252C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
253C42—91no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—61no—
257C22×C3D3×#6711yescharacteristic
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—41no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—121no—
266C3×C4—121no—
267C3×C4—181no—
268C3×C4—181no—
269C3×C4—241no—
270C3×C4—361no—
271C3×C4—361no—
272C3×C4—361no—
273C3×C4D3×Dic611yescharacteristic
274C3×C4#379711yescharacteristic
275Dic3—61no—
276Dic3—61no—
277Dic3—121no—
278Dic3—121no—
279Dic3—121no—
280Dic3—121no—
281Dic3—181no—
282Dic3—181no—
283Dic3—241no—
284Dic3—361no—
285Dic3—361no—
286C32—21no—
287C32#485611yescharacteristic, complemented
288C32#508111yescharacteristic, complemented
289Q8—362no—
290C2×C4—21no—
291C2×C4—91no—
292C2×C4—91no—
293C2×C4—181no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
297C8—92no—
298C8—92no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3#6383911yescharacteristic
309C2×C3#6393411yescharacteristic
310C2×C3#6394611yescharacteristic
311C22#8867611yescharacteristic
312C4—41no—
313C4—181no—
314C4—181no—
315C4—361no—
316C4—361no—
317C4—361no—
318C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
319C4#8865111yescharacteristic
320C3—31no—
321C3—31no—
322C3—61no—
323C3#1059042111yescharacteristic
324C2#1172107911yescharacteristic
325C2#1172119611yescharacteristic
326C2#1172123811yescharacteristic
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)#501
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
44. #6180Q8semidirect
45. (C3⋊(C3⋊C8))⋊C3Q8semidirect
124. C3⋊(C3⋊C8)Dic6semidirect
199. H(3,𝔽3)#501semidirect
287. C32#4856semidirect
288. C32#5081semidirect

Maximal subgroups

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

subgroupindex
#64097⋊C32
#117210932
#117205292
#63635⋊C32
#117203142
#117203122
#117203062
#105905463
#105904203