Finite Group Databasebeta

← #13165737

Subgroups — #13165737

Order 1728.

396 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 60 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#11723032C211yesmaximal, characteristic
3#11722796C211yesmaximal, characteristic
4#11722778C211yesmaximal, characteristic
5#11722778C211yesmaximal, characteristic
6C3×#63742C211yesmaximal, characteristic
7Dic3×#622C211yesmaximal, characteristic
8#10593384—31nomaximal
9C2×#6293—21no—
10C2×C3×#627—21no—
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88815C2211yescharacteristic
14#88815C2211yescharacteristic
15#88806—21no—
16Dic3×C3⋊Dic3C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20#63919—31no—
21#63873—31no—
22#63840—61no—
23#63839—61no—
24C4×#622—31no—
25#63742—31no—
26#63742—31no—
27#63742—31no—
28#63742D311yescharacteristic
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×#627—42no—
38C3⋊#622Q812yescomplemented
39C3×#622Q812yescomplemented
40#4855—91nomaximal
41C2×#634—61no—
42C2×C3×Dic6—61no—
43C2×C3×Dic6—61no—
44C2×#627—21no—
45C2×#627—61no—
46C2×#627—61no—
47C2×#627—121no—
48C2×#622—31no—
49C2×#622C2×D311yescharacteristic
50#3750—61no—
51C4×C3⋊Dic3—31no—
52C3×C4⋊Dic3—61no—
53C3×Dic3⋊C4—61no—
54C3×Dic3⋊C4—61no—
55C3×Dic3⋊C4—61no—
56#3724—31no—
57#3724—31no—
58#3724—61no—
59#3724—61no—
60#3724—121no—
61#3724C2×D311yescharacteristic
62#3724C2×D311yescharacteristic
63Dic3⋊Dic3—21no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66Dic32—61no—
67Dic32—61no—
68C22×C33C2×D411yescharacteristic
69C3⋊(C3⋊Dic3)SD1612yes—
70C3×C3⋊Dic3—41no—
71C3×C3⋊Dic3—41no—
72C3×C3⋊Dic3C2×Q811yescharacteristic
73C3×C3⋊Dic3C2×Q811yescharacteristic
74C32×Dic3—41no—
75C32×Dic3—41no—
76C32×Dic3SD1612yes—
77#944—181no—
78#888—91no—
79#888—91no—
80#888—181no—
81C3×#106—91no—
82#817—91no—
83C8×Dic3—91no—
84#808—91no—
85#808—91no—
86C2×C32×C4—31no—
87C2×C32×C4—61no—
88C2×C32×C4—61no—
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3—31no—
92C2×C3⋊Dic3—61no—
93C2×C3⋊Dic3—61no—
94C2×C3⋊Dic3—121no—
95C2×C3⋊Dic3—121no—
96C2×C3⋊Dic3C22×D311yescharacteristic
97#634—122no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—121no—
113C2×C3×Dic3—121no—
114C3×Dic6—122no—
115C3×Dic6—122no—
116#627—42no—
117#627—122no—
118#627—122no—
119#627—124no—
120#622—32no—
121#622Dic612yescomplemented
122#498—271noSylow
123C2×C33C4⋊Q811yescharacteristic
124C2×C33C2×SD1611yescharacteristic
125C2×C33C2×SD1611yescharacteristic
126C2×C3×Q8—181no—
127C2×Dic6—61no—
128C2×Dic6—61no—
129C2×Dic6—181no—
130C2×Dic6—181no—
131C2×Dic6—181no—
132C2×Dic6—361no—
133C2×C3×C8—91no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—181no—
137C4⋊Dic3—61no—
138C4⋊Dic3—91no—
139C4⋊Dic3—91no—
140Dic3⋊C4—61no—
141Dic3⋊C4—61no—
142Dic3⋊C4—61no—
143Dic3⋊C4—181no—
144Dic3⋊C4—181no—
145Dic3⋊C4—181no—
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—361no—
149C4×Dic3—91no—
150C4×Dic3—181no—
151C4×Dic3—181no—
152C2×C3⋊C8—91no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32D3×D411yescharacteristic
158C32×C4—32no—
159C32×C4—121no—
160C32×C4—121no—
161C3⋊Dic3—32no—
162C3⋊Dic3—41no—
163C3⋊Dic3—41no—
164C3⋊Dic3—62no—
165C3⋊Dic3—62no—
166C3⋊Dic3—122no—
167C3⋊Dic3—122no—
168C3⋊Dic3C2×Dic611yescharacteristic
169C3⋊Dic3D3×Q811yescharacteristic
170C3×Dic3—22no—
171C3×Dic3—22no—
172C3×Dic3—41no—
173C3×Dic3—41no—
174C3×Dic3—42no—
175C3×Dic3—41no—
176C3×Dic3—41no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—61no—
180C3×Dic3—61no—
181C3×Dic3—81no—
182C3×Dic3—121no—
183C3×Dic3—121no—
184C3×Dic3—122no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C3×Dic3—241no—
188C4⋊Q8—271no—
189C4⋊Q8—271no—
190#106—91no—
191#106—271no—
192#106—271no—
193#106—271no—
194C4×C8—271no—
195C33#49811yescharacteristic, complemented, Sylow
196C3×Q8—362no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—91no—
204C2×C3×C4—121no—
205C2×C3×C4—121no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—91no—
217C2×Dic3—121no—
218C2×Dic3—121no—
219C2×Dic3—181no—
220C2×Dic3—181no—
221C2×Dic3—361no—
222C2×Dic3—361no—
223C2×Dic3GO+(4,𝔽2)11yescharacteristic
224Dic6—122no—
225Dic6—122no—
226Dic6—362no—
227Dic6—362no—
228Dic6—362no—
229Dic6—364no—
230C3×C8—92no—
231C3⋊C8—92no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32#88811yescharacteristic
245C2×C32D3×SD1611yescharacteristic
246C2×C32D3×SD1611yescharacteristic
247C2×Q8—181no—
248C2×Q8—541no—
249C2×C8—91no—
250C2×C8—271no—
251C4⋊C4—91no—
252C4⋊C4—91no—
253C4⋊C4—181no—
254C4⋊C4—271no—
255C4⋊C4—271no—
256C4⋊C4—541no—
257C42—271no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C2×GO+(4,𝔽2)11yescharacteristic
263C3×C4—62no—
264C3×C4—62no—
265C3×C4—91no—
266C3×C4—91no—
267C3×C4—121no—
268C3×C4—121no—
269C3×C4—122no—
270C3×C4—121no—
271C3×C4—121no—
272C3×C4—241no—
273C3×C4—361no—
274C3×C4—361no—
275Dic3—41no—
276Dic3—41no—
277Dic3—61no—
278Dic3—61no—
279Dic3—61no—
280Dic3—61no—
281Dic3—92no—
282Dic3—121no—
283Dic3—121no—
284Dic3—122no—
285Dic3—121no—
286Dic3—121no—
287Dic3—182no—
288Dic3—182no—
289Dic3—241no—
290Dic3—362no—
291Dic3—362no—
292Dic3#377512yes—
293C32—21no—
294C32—21no—
295C32—41no—
296C32—41no—
297C32#485511yescharacteristic, complemented
298Q8—362no—
299Q8—1082no—
300C2×C4—31no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4—181no—
306C2×C4—271no—
307C8—92no—
308C8—272no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3#6422211yescharacteristic
322C2×C3#6422711yescharacteristic
323C2×C3C2×#377511yescharacteristic
324C22D3×GO+(4,𝔽2)11yescharacteristic
325C4—32no—
326C4—91no—
327C4—91no—
328C4—121no—
329C4—121no—
330C4—272no—
331C4—361no—
332C4—361no—
333C3—21no—
334C3—21no—
335C3—41no—
336C3—41no—
337C3#1059338411yescharacteristic, complemented
338C2#1172412511yescharacteristic
339C2D3×#377511yescharacteristic
340C2#1172412911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#498
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
38. C3⋊#622Q8semidirect
39. C3×#622Q8semidirect
121. #622Dic6semidirect
195. C33#498semidirect
297. C32#4855semidirect
337. C3#10593384semidirect

Maximal subgroups

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

subgroupindex
#117230372
#117230322
#117227962
#117227782
C3×#637422
Dic3×#6222
#105933843
#48559