Finite Group Databasebeta

← #13138028

Subgroups — #13138028

Order 1728.

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

normal subgroups only

This group has 357 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#64099⋊C3C211yesmaximal, characteristic
2#11721083C211yesmaximal, characteristic
3#11720532C211yesmaximal, characteristic
4#63634⋊C3C211yesmaximal, characteristic
5#11720326C211yesmaximal, characteristic
6#11720314C211yesmaximal, characteristic
7#11720311C211yesmaximal, characteristic
8#10590494—31nomaximal
9#10590493—31nomaximal
10C2×#6229C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)—21no—
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#88447—21no—
21#64099D311yescharacteristic
22C3×#890—31no—
23#63843—31no—
24Dic3×Dic6—31no—
25#63634D311yescharacteristic
26C3×#810—31no—
27C3×#810—31no—
28C3×#810—61no—
29C3×#807—31no—
30#63573—31no—
31#63573—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63566—31no—
35#63565—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39#6229—21no—
40#6229Q812yes—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#6180—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47#5063—91no—
48#4877—91no—
49#4799—91no—
50C2×C3×Dic6—31no—
51C2×C3×Dic6—31no—
52C2×C3×Dic6—61no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)—21no—
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—61no—
65C3×Dic3⋊C4—61no—
66C3×C4×Dic3—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic32—61no—
74Dic32—61no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)—41no—
77C4×H(3,𝔽3)C2×Q811yescharacteristic
78C4×H(3,𝔽3)#4111yescharacteristic
79#1084—22no—
80(C3⋊Dic3)⋊C3—21no—
81(C3⋊Dic3)⋊C3—21no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84C3×#125—91no—
85Q8×Dic3—91no—
86#890—31no—
87#890—91no—
88#887—91no—
89C4×Dic6—91no—
90C3×#107—91no—
91C3×#103—91no—
92#835—91no—
93#817—91no—
94#816—91no—
95Dic3⋊C8—91no—
96Dic3⋊C8—91no—
97#810—31no—
98#810—31no—
99#810—61no—
100#810—91no—
101#808—91no—
102#808—91no—
103#807—31no—
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
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—32no—
124C3×Dic6—32no—
125C3×Dic6—62no—
126C3×Dic6—61no—
127C3×Dic6—61no—
128C3×Dic6—121no—
129C3⋊(C3⋊C8)—21no—
130C3×C3⋊C8—61no—
131C3×C3⋊C8—61no—
132C3×C3⋊C8—61no—
133C3×C3⋊C8—121no—
134#479—271noSylow
135C2×H(3,𝔽3)C22⋊Q811yescharacteristic
136C2×H(3,𝔽3)#13511yescharacteristic
137C2×H(3,𝔽3)#13711yescharacteristic
138C2×C3×Q8—91no—
139C2×Dic6—31no—
140C2×Dic6—31no—
141C2×Dic6—61no—
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—181no—
150C3×C4⋊C4—181no—
151C3×C4⋊C4D3211yescharacteristic
152C3×C42—91no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—61no—
156C4⋊Dic3—91no—
157C4⋊Dic3—91no—
158C4⋊Dic3—181no—
159C4⋊Dic3—181no—
160Dic3⋊C4—61no—
161Dic3⋊C4—181no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164C4×Dic3—31no—
165C4×Dic3—91no—
166C4×Dic3—181no—
167C4×Dic3—181no—
168C4×Dic3—181no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—61no—
173C2×C3⋊C8—91no—
174C22×C32—21no—
175C22×C32D3×D411yescharacteristic
176C22×C32C2×#6711yescharacteristic
177C32×C4—21no—
178C32×C4—21no—
179C32×C4—41no—
180C32×C4—41no—
181C32×C4—81no—
182C32×C4#23511yescharacteristic
183C32×C4D3×Q811yescharacteristic
184C32×C4D3×Q811yescharacteristic
185C32×C4#23911yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—21no—
188C3⋊Dic3—41no—
189C3⋊Dic3—41no—
190C3×Dic3—62no—
191C3×Dic3—61no—
192C3×Dic3—61no—
193C3×Dic3—62no—
194C3×Dic3—122no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197#125—91no—
198C4×Q8—271no—
199#107—91no—
200#106—271no—
201C4⋊C8—271no—
202#103—91no—
203#103—271no—
204H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
205C3×Q8—92no—
206C3×Q8—181no—
207C2×C3×C4—21no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—121no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—31no—
221C2×Dic3—31no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—91no—
227C2×Dic3—91no—
228C2×Dic3—121no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231Dic6—32no—
232Dic6—32no—
233Dic6—62no—
234Dic6—61no—
235Dic6—61no—
236Dic6—121no—
237C3×C8—181no—
238C3×C8—181no—
239C3⋊C8—61no—
240C3⋊C8—61no—
241C3⋊C8—61no—
242C3⋊C8—121no—
243C3⋊C8—181no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32#89611yescharacteristic
248C2×C32#91211yescharacteristic
249C2×C32#91811yescharacteristic
250C2×C32#94611yescharacteristic
251C2×C32#95011yescharacteristic
252C2×C32#95111yescharacteristic
253C2×Q8—91no—
254C2×C8—91no—
255C2×C8—91no—
256C4⋊C4—91no—
257C4⋊C4—91no—
258C4⋊C4—181no—
259C4⋊C4—181no—
260C4⋊C4—541no—
261C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
262C42—91no—
263C42—541no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—61no—
267C22×C3D3×#6711yescharacteristic
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—41no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—121no—
276C3×C4—121no—
277C3×C4—181no—
278C3×C4—181no—
279C3×C4—182no—
280C3×C4—241no—
281C3×C4—361no—
282C3×C4—361no—
283C3×C4#379711yescharacteristic
284C3×C4#379911yescharacteristic
285Dic3—62no—
286Dic3—62no—
287Dic3—61no—
288Dic3—61no—
289Dic3—122no—
290Dic3—121no—
291Dic3—121no—
292Dic3—181no—
293Dic3—181no—
294Dic3—361no—
295Dic3—361no—
296C32—21no—
297C32#487711yescharacteristic, complemented
298C32#506311yescharacteristic, complemented
299Q8—92no—
300Q8—181no—
301C2×C4—21no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C8—181no—
309C8—181no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3#6388811yescharacteristic
320C2×C3#6393311yescharacteristic
321C2×C3#6394311yescharacteristic
322C22#8867611yescharacteristic
323C4—41no—
324C4—181no—
325C4—182no—
326C4—181no—
327C4—361no—
328C4—361no—
329C4#8865111yescharacteristic
330C4#8865211yescharacteristic
331C3—31no—
332C3—31no—
333C3—61no—
334C3#1059049211yescharacteristic
335C2#1172114811yescharacteristic
336C2#1172120111yescharacteristic
337C2#1172123711yescharacteristic
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)#479
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
204. H(3,𝔽3)#479semidirect
297. C32#4877semidirect
298. C32#5063semidirect

Maximal subgroups

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

subgroupindex
#64099⋊C32
#117210832
#117205322
#63634⋊C32
#117203262
#117203142
#117203112
#105904943
#105904933