Finite Group Databasebeta

← #13138816

Subgroups — #13138816

Order 1728.

372 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 77 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#11721116C211yesmaximal, characteristic
2#11721082C211yesmaximal, characteristic
3#11720550C211yesmaximal, characteristic
4#63643⋊C3C211yesmaximal, characteristic
5#63641⋊C3C211yesmaximal, characteristic
6#11720309C211yesmaximal, characteristic
7#11720307C211yesmaximal, characteristic
8#10590816—31nomaximal
9#10590802—31nomaximal
10C2×#634⋊C3C2211yescharacteristic
11C2×#6196—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88448—21no—
19#88448—21no—
20#88447—21no—
21#63919—31no—
22#63918—31no—
23#63844—31no—
24#63843—31no—
25#63643D311yescharacteristic
26#63641D311yescharacteristic
27C3×#817—31no—
28C3×#817—31no—
29C3×#817—61no—
30C3×#816—31no—
31C3×Dic3⋊C8—31no—
32#63574—31no—
33#63573—31no—
34#63567—31no—
35#63565—31no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—21no—
38C2×#1084—21no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43#634⋊C3—21no—
44#634⋊C3Q812yescomplemented
45#6196—42no—
46C8×H(3,𝔽3)—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#4854—91no—
49#4851—91no—
50#4791—91no—
51C2×#634C2×D311yescharacteristic
52C2×C3×Dic6—31no—
53C2×#627—61no—
54C2×#627—61no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×C4⋊Dic3—61no—
68C3×C4×Dic3—31no—
69C2×C3×C3⋊C8—31no—
70#3724—61no—
71#3724—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74Dic32—61no—
75Dic32—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)C2×Q811yescharacteristic
78C4×H(3,𝔽3)#4111yescharacteristic
79#1084—41no—
80#1084—41no—
81(C3⋊Dic3)⋊C3—22no—
82(C3⋊Dic3)⋊C3—21no—
83(C3⋊Dic3)⋊C3—41no—
84(C3⋊Dic3)⋊C3SD1612yes—
85Q8×Dic3—91no—
86#888—91no—
87#888—91no—
88#887—91no—
89#869—91no—
90C4×Dic6—91no—
91C3×#106—91no—
92C3×C4⋊C8—91no—
93C3×#103—91no—
94#835—91no—
95#817—31no—
96#817—31no—
97#817—61no—
98#817—91no—
99#816—31no—
100#816—91no—
101#816—91no—
102Dic3⋊C8—31no—
103Dic3⋊C8—91no—
104#810—91no—
105#808—91no—
106#808—91no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3D1211yescharacteristic
113C2×C3⋊Dic3D1211yescharacteristic
114#634—21no—
115#634Dic612yescomplemented
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—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—121no—
125C2×C3×Dic3—121no—
126C3×Dic6—32no—
127C3×Dic6—61no—
128#627—122no—
129#627—122no—
130C32×C8—22no—
131C32×C8—21no—
132C32×C8—21no—
133C3⋊(C3⋊C8)—21no—
134C3×C3⋊C8—61no—
135#475—271noSylow
136C2×H(3,𝔽3)C22⋊Q811yescharacteristic
137C2×H(3,𝔽3)C2×SD1611yescharacteristic
138C2×H(3,𝔽3)#13711yescharacteristic
139C2×C3×Q8—91no—
140C2×Dic6—31no—
141C2×Dic6—91no—
142C2×Dic6—181no—
143C2×Dic6—181no—
144C2×Dic6—181no—
145C2×C3×C8—31no—
146C2×C3×C8—31no—
147C2×C3×C8—61no—
148C2×C3×C8—91no—
149C2×C3×C8D3211yescharacteristic
150C3×C4⋊C4—91no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—91no—
153C3×C42—91no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—61no—
160C4⋊Dic3—61no—
161C4⋊Dic3—91no—
162C4⋊Dic3—181no—
163C4⋊Dic3—181no—
164Dic3⋊C4—181no—
165Dic3⋊C4—181no—
166Dic3⋊C4—181no—
167Dic3⋊C4—181no—
168C4×Dic3—31no—
169C4×Dic3—91no—
170C4×Dic3—181no—
171C4×Dic3—181no—
172C4×Dic3—181no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—91no—
175C22×C32—21no—
176C22×C32C2×D1211yescharacteristic
177C22×C32D3×D411yescharacteristic
178C32×C4—21no—
179C32×C4—21no—
180C32×C4C2×Dic611yescharacteristic
181C32×C4#23711yescharacteristic
182C32×C4#23711yescharacteristic
183C32×C4D3×Q811yescharacteristic
184C3⋊Dic3—22no—
185C3⋊Dic3—21no—
186C3⋊Dic3—41no—
187C3⋊Dic3#20412yes—
188C3×Dic3—32no—
189C3×Dic3—62no—
190C3×Dic3—61no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196C3×Dic3—241no—
197C3×Dic3—241no—
198C4⋊Q8—271no—
199C4×Q8—271no—
200#106—91no—
201#106—271no—
202C4⋊C8—91no—
203#103—91no—
204#103—271no—
205H(3,𝔽3)#47511yescharacteristic, complemented, Sylow
206C3×Q8—92no—
207C3×Q8—181no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
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×C4—181no—
217C2×C3×C4C2×D3211yescharacteristic
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—91no—
227C2×Dic3—91no—
228C2×Dic3—121no—
229C2×Dic3—121no—
230C2×Dic3—181no—
231C2×Dic3—181no—
232Dic6—32no—
233Dic6—61no—
234Dic6—92no—
235Dic6—181no—
236Dic6—362no—
237Dic6—362no—
238Dic6—362no—
239C3×C8—21no—
240C3×C8—61no—
241C3×C8—61no—
242C3×C8—62no—
243C3×C8—181no—
244C3⋊C8—61no—
245C3⋊C8—181no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32#87811yescharacteristic
250C2×C32#89711yescharacteristic
251C2×C32C2×#20411yescharacteristic
252C2×C32#90911yescharacteristic
253C2×C32D3×SD1611yescharacteristic
254C2×C32#91811yescharacteristic
255C2×Q8—91no—
256C2×Q8—541no—
257C2×C8—91no—
258C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C4⋊C4—91no—
260C4⋊C4—91no—
261C4⋊C4—91no—
262C4⋊C4—541no—
263C4⋊C4—541no—
264C42—91no—
265C42—541no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—61no—
269C22×C3D3×D1211yescharacteristic
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—92no—
277C3×C4—181no—
278C3×C4—182no—
279C3×C4—361no—
280C3×C4—361no—
281C3×C4—361no—
282C3×C4D3×Dic611yescharacteristic
283C3×C4#380511yescharacteristic
284Dic3—32no—
285Dic3—61no—
286Dic3—62no—
287Dic3—92no—
288Dic3—121no—
289Dic3—121no—
290Dic3—121no—
291Dic3—121no—
292Dic3—121no—
293Dic3—182no—
294Dic3—181no—
295Dic3—241no—
296Dic3—241no—
297Dic3—361no—
298C32—21no—
299C32#479111yescharacteristic, complemented
300C32#485111yescharacteristic, complemented
301Q8—92no—
302Q8—181no—
303Q8—1082no—
304C2×C4—91no—
305C2×C4—91no—
306C2×C4—181no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C8—21no—
312C8—181no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—61no—
320C2×C3—61no—
321C2×C3—61no—
322C2×C3D3×#20411yescharacteristic
323C2×C3#6380111yescharacteristic
324C2×C3#6389311yescharacteristic
325C22#8865411yescharacteristic
326C4—92no—
327C4—181no—
328C4—182no—
329C4—361no—
330C4—361no—
331C4—361no—
332C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
333C4#8867211yescharacteristic
334C3—31no—
335C3—31no—
336C3—61no—
337C3#1059080311yescharacteristic
338C2#1172084611yescharacteristic
339C2#1172085311yescharacteristic
340C2#1172113311yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#475
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
44. #634⋊C3Q8semidirect
115. #634Dic6semidirect
205. H(3,𝔽3)#475semidirect
299. C32#4791semidirect
300. C32#4851semidirect

Maximal subgroups

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

subgroupindex
#117211162
#117210822
#117205502
#63643⋊C32
#63641⋊C32
#117203092
#117203072
#105908163
#105908023