Finite Group Databasebeta

← #13165744

Subgroups — #13165744

Order 1728.

368 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 54 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#11723037C211yesmaximal, characteristic
2#11723015C211yesmaximal, characteristic
3#11722796C211yesmaximal, characteristic
4#11722779C211yesmaximal, characteristic
5#11722778C211yesmaximal, characteristic
6C3×#63743C211yesmaximal, characteristic
7#11722724C211yesmaximal, characteristic
8#10593383—31nomaximal
9C2×#6293—21no—
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88815C2211yescharacteristic
13#88815C2211yescharacteristic
14#88807—21no—
15Dic3×C3⋊Dic3C2211yescharacteristic
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×Dic3⋊Dic3—21no—
20#63919—31no—
21#63870—31no—
22#63846—61no—
23#63840—61no—
24#63777—31no—
25#63743—31no—
26#63743D311yescharacteristic
27#63742—31no—
28#63742—31no—
29C2×C3⋊(C3⋊Dic3)D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3D411yescharacteristic
36#6293—42no—
37C3⋊#622Q812yescomplemented
38C3×#622Q812yescomplemented
39#4876—91nomaximal
40C2×#634—61no—
41C2×#627—61no—
42C2×#627—61no—
43C2×#627—121no—
44C2×#622—31no—
45C2×#622C2×D311yescharacteristic
46#3750—61no—
47C4×C3⋊Dic3—31no—
48C3×C4⋊Dic3—61no—
49C3×C4⋊Dic3—61no—
50C3×Dic3⋊C4—61no—
51C3×Dic3⋊C4—61no—
52C3×Dic3⋊C4—61no—
53C3×Dic3⋊C4—61no—
54#3724—31no—
55#3724—31no—
56#3724C2×D311yescharacteristic
57#3724C2×D311yescharacteristic
58Dic3⋊Dic3—21no—
59Dic3⋊Dic3—21no—
60Dic3⋊Dic3—61no—
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—121no—
65Dic32—61no—
66Dic32—61no—
67C22×C33C2×D411yescharacteristic
68C3⋊(C3⋊Dic3)—21no—
69C3×C3⋊Dic3—41no—
70C3×C3⋊Dic3—41no—
71C3×C3⋊Dic3C2×Q811yescharacteristic
72C3×C3⋊Dic3C2×Q811yescharacteristic
73C32×Dic3—21no—
74C32×Dic3—41no—
75C32×Dic3—41no—
76#890—91no—
77#889—181no—
78#888—91no—
79#888—181no—
80C3×#107—91no—
81#817—91no—
82#815—91no—
83#808—91no—
84#807—91no—
85C2×C32×C4—31no—
86C2×C32×C4—61no—
87C2×C32×C4—61no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3—31no—
91C2×C3⋊Dic3—61no—
92C2×C3⋊Dic3—61no—
93C2×C3⋊Dic3—121no—
94C2×C3⋊Dic3—121no—
95C2×C3⋊Dic3C22×D311yescharacteristic
96#634—122no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—121no—
112C2×C3×Dic3—121no—
113#627—122no—
114#627—122no—
115#627—124no—
116#622—32no—
117#622Dic612yescomplemented
118#501—271noSylow
119C2×C33C4⋊Q811yescharacteristic
120C2×C33AGL(1,ℤ/8)11yescharacteristic
121C2×C33#13711yescharacteristic
122C2×Dic6—181no—
123C2×Dic6—181no—
124C2×Dic6—181no—
125C2×Dic6—361no—
126C2×C3×C8—91no—
127C3×C4⋊C4—91no—
128C3×C4⋊C4—91no—
129C3×C4⋊C4—181no—
130C3×C4⋊C4—181no—
131C4⋊Dic3—61no—
132C4⋊Dic3—61no—
133C4⋊Dic3—91no—
134C4⋊Dic3—91no—
135C4⋊Dic3—181no—
136Dic3⋊C4—61no—
137Dic3⋊C4—61no—
138Dic3⋊C4—61no—
139Dic3⋊C4—61no—
140Dic3⋊C4—181no—
141Dic3⋊C4—181no—
142Dic3⋊C4—181no—
143Dic3⋊C4—181no—
144Dic3⋊C4—361no—
145C4×Dic3—91no—
146C4×Dic3—181no—
147C4×Dic3—181no—
148C2×C3⋊C8—91no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32D3×D411yescharacteristic
154C32×C4—61no—
155C32×C4—121no—
156C32×C4—121no—
157C3⋊Dic3—41no—
158C3⋊Dic3—41no—
159C3⋊Dic3—62no—
160C3⋊Dic3—61no—
161C3⋊Dic3—121no—
162C3⋊Dic3—122no—
163C3⋊Dic3—241no—
164C3⋊Dic3C2×Dic611yescharacteristic
165C3⋊Dic3D3×Q811yescharacteristic
166C3×Dic3—22no—
167C3×Dic3—41no—
168C3×Dic3—41no—
169C3×Dic3—41no—
170C3×Dic3—42no—
171C3×Dic3—41no—
172C3×Dic3—41no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—81no—
178C3×Dic3—121no—
179C3×Dic3—121no—
180C3×Dic3—122no—
181C3×Dic3—122no—
182C3×Dic3—121no—
183C3×Dic3—121no—
184C4⋊Q8—271no—
185#125—271no—
186#107—91no—
187#107—271no—
188#106—271no—
189#106—271no—
190#97—271no—
191C33#50111yescharacteristic, complemented, Sylow
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—91no—
199C2×C3×C4—121no—
200C2×C3×C4—121no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—91no—
212C2×Dic3—121no—
213C2×Dic3—121no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216C2×Dic3—361no—
217C2×Dic3—361no—
218C2×Dic3GO+(4,𝔽2)11yescharacteristic
219Dic6—362no—
220Dic6—362no—
221Dic6—362no—
222Dic6—364no—
223C3×C8—92no—
224C3⋊C8—92no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32#88811yescharacteristic
238C2×C32#91111yescharacteristic
239C2×C32#91811yescharacteristic
240C2×Q8—541no—
241C2×C8—91no—
242C2×C8—271no—
243C4⋊C4—91no—
244C4⋊C4—91no—
245C4⋊C4—181no—
246C4⋊C4—181no—
247C4⋊C4—271no—
248C4⋊C4—271no—
249C4⋊C4—541no—
250C42—271no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3C2×GO+(4,𝔽2)11yescharacteristic
256C3×C4—62no—
257C3×C4—91no—
258C3×C4—91no—
259C3×C4—121no—
260C3×C4—121no—
261C3×C4—121no—
262C3×C4—122no—
263C3×C4—121no—
264C3×C4—121no—
265C3×C4—241no—
266C3×C4—361no—
267C3×C4—361no—
268Dic3—21no—
269Dic3—41no—
270Dic3—41no—
271Dic3—61no—
272Dic3—61no—
273Dic3—61no—
274Dic3—61no—
275Dic3—121no—
276Dic3—121no—
277Dic3—122no—
278Dic3—122no—
279Dic3—121no—
280Dic3—121no—
281Dic3—182no—
282Dic3—181no—
283Dic3—361no—
284Dic3—362no—
285Dic3—721no—
286C32—21no—
287C32—21no—
288C32—41no—
289C32—41no—
290C32#487611yescharacteristic, complemented
291Q8—1082no—
292C2×C4—31no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—91no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4—271no—
299C8—92no—
300C8—272no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3#6422211yescharacteristic
314C2×C3#6422411yescharacteristic
315C2×C3#6423911yescharacteristic
316C22D3×GO+(4,𝔽2)11yescharacteristic
317C4—61no—
318C4—91no—
319C4—91no—
320C4—121no—
321C4—121no—
322C4—361no—
323C4—361no—
324C4—541no—
325C3—21no—
326C3—21no—
327C3—41no—
328C3—41no—
329C3#1059338311yescharacteristic, complemented
330C2#1172412511yescharacteristic
331C2#1172414011yescharacteristic
332C2#1172414311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#501
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
37. C3⋊#622Q8semidirect
38. C3×#622Q8semidirect
117. #622Dic6semidirect
191. C33#501semidirect
290. C32#4876semidirect
329. C3#10593383semidirect

Maximal subgroups

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

subgroupindex
#117230372
#117230152
#117227962
#117227792
#117227782
C3×#637432
#117227242
#105933833
#48769