Finite Group Databasebeta

← #13166513

Subgroups — #13166513

Order 1728.

500 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#88989C211yesmaximal, characteristic, complemented
2C2×#88926C211yesmaximal, characteristic, complemented
3#11723512C211yesmaximal, characteristic
4C3×#63778C211yesmaximal, characteristic, complemented
5Dic3⋊#622C211yesmaximal, characteristic, complemented
6#11722726C211yesmaximal, characteristic, complemented
7Dic3×#622C211yesmaximal, characteristic, complemented
8#10593752—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6292C411yescharacteristic
11C2×#6289C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14#88989—22no—
15C2×C3×#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17#88926—22no—
18#88859C411yescharacteristic
19C32×Dic3⋊C4C411yescharacteristic
20#88803C411yescharacteristic
21Dic3×C3⋊Dic3C2211yescharacteristic
22C2×#3788—31no—
23C2×#3788—31no—
24#64102—31no—
25#63904—62no—
26#63778—31no—
27#63778D311yescharacteristic, complemented
28C4⋊#622—31no—
29C4×#622—31no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×C4×C3⋊D3—21no—
36C3×C4×C3⋊D3—21no—
37C2×C32×Dic3C2×C411yescharacteristic
38C2×C32×Dic3C2×C411yescharacteristic
39#6292—22no—
40#6289—21no—
41#6289—21no—
42C3⋊#622—21no—
43C3⋊#622D412yescomplemented
44C3×#622—21no—
45C3×#622—21no—
46#4729—91nomaximal
47C2×C4⋊(C3⋊D3)—31no—
48C2×C4×C3⋊D3—31no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×C3×C4×D3—62no—
51C2×#626—62no—
52C2×#624—62no—
53C2×#622—31no—
54C2×#622—31no—
55C2×#622C2×D311yescharacteristic
56C2×#622C2×D311yescharacteristic
57#3788—62no—
58#3788—62no—
59C32×C4⋊C4—31no—
60#3752—31no—
61#3752—31no—
62#3752—62no—
63#3752—122no—
64C4×C3⋊Dic3—31no—
65C3×Dic3⋊C4—22no—
66(C2×D3)⋊Dic3—62no—
67Dic32—62no—
68C22×C33C22×C411yescharacteristic
69C2×C3⋊(C3⋊D3)—21no—
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)—21no—
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3—21no—
75C33×C4—21no—
76C3⋊(C3⋊Dic3)—21no—
77C3×C3⋊Dic3C2×D411yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—21no—
80C32×Dic3M1612yescomplemented
81C4×#67—91no—
82C2×#208—91no—
83C2×ΓL(1,𝔽25)—91no—
84#893—182no—
85C3×C22⋊C8—91no—
86(C2×D3)⋊C8—91no—
87Dic3⋊C8—91no—
88C8×Dic3—91no—
89C22×C3⋊D3—31no—
90C22×C3⋊D3—62no—
91C22×C3⋊D3—122no—
92C22×C3⋊D3C4×D311yescharacteristic
93C22×C3×D3—62no—
94C2×C32×C4—22no—
95C2×C32×C4—31no—
96C2×C32×C4—31no—
97C2×C32×C4—42no—
98C2×C32×C4C4×D311yescharacteristic
99C2×C3⋊Dic3—31no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—122no—
102C2×C3⋊Dic3C22×D311yescharacteristic
103C4⋊(C3⋊D3)—62no—
104C4×C3⋊D3—21no—
105C4×C3⋊D3—21no—
106C4×C3⋊D3—61no—
107C4×C3⋊D3—61no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—62no—
111C3×C4×D3—122no—
112C3×C4×D3—122no—
113#626—68no—
114#624—122no—
115#624—122no—
116#622—21no—
117#622—21no—
118#622—32no—
119#622—61no—
120#436—271noSylow
121C2×C33C4×D411yescharacteristic
122C2×C33C2×M1611yescharacteristic
123C2×C33#13111yescharacteristic
124C3⋊(C3⋊D3)—41no—
125C3×C3⋊D3—41no—
126C22×C3×C4—91no—
127C2×#67—91no—
128C2×D12—182no—
129C2×C4×D3—62no—
130C2×C4×D3—91no—
131C2×C4×D3—182no—
132C2×C3×C8—91no—
133C2×C3×C8—91no—
134C3×C4⋊C4—62no—
135#217—91no—
136(C2×D3)⋊C4—91no—
137(C2×D3)⋊C4—182no—
138(C2×D3)⋊C4—182no—
139(C2×D3)⋊C4—362no—
140Dic3⋊C4#15711yescharacteristic
141C4×Dic3—91no—
142C4×Dic3—182no—
143#208—182no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—91no—
146ΓL(1,𝔽25)—182no—
147C22×C32—22no—
148C22×C32—42no—
149C22×C32C2×C4×D311yescharacteristic
150C2×C3⋊D3—21no—
151C2×C3⋊D3—21no—
152C2×C3⋊D3—21no—
153C2×C3⋊D3—61no—
154C2×C3⋊D3—64no—
155C2×C3⋊D3—61no—
156C2×C3⋊D3—61no—
157C2×C3⋊D3—124no—
158C2×C3⋊D3—122no—
159C2×C3⋊D3—122no—
160C2×C3⋊D3—242no—
161C2×C3⋊D3—242no—
162C2×C3×D3—64no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165C32×C4—21no—
166C32×C4—32no—
167C32×C4—42no—
168C32×C4—61no—
169C32×C4—82no—
170C3⋊Dic3—61no—
171C3⋊Dic3—122no—
172C3⋊Dic3—242no—
173C3⋊Dic3D3×D411yescharacteristic
174C3⋊Dic3#23711yescharacteristic
175C3×Dic3—24no—
176C3×Dic3—42no—
177C3×Dic3—62no—
178C3×Dic3—62no—
179C2×M16—271no—
180C2×M16—271no—
181C4×D4—271no—
182C4⋊C8—271no—
183C22⋊C8—91no—
184C22⋊C8—271no—
185C4×C8—271no—
186C33#43611yescharacteristic, complemented, Sylow
187C23×C3—91no—
188C22×D3—62no—
189C22×D3—91no—
190C22×D3—182no—
191C22×D3—362no—
192C2×C3×C4—22no—
193C2×C3×C4—42no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—91no—
197C2×C3×C4—181no—
198C2×C3×C4—181no—
199C2×C3×C4C2×#15711yescharacteristic
200#67—182no—
201C2×Dic3—62no—
202C2×Dic3—91no—
203C2×Dic3—182no—
204C2×Dic3—362no—
205C2×Dic3C2×#15711yescharacteristic
206C2×Dic3C2×#15711yescharacteristic
207D12—188no—
208C4×D3—122no—
209C4×D3—122no—
210C4×D3—181no—
211C4×D3—181no—
212C4×D3—362no—
213C4×D3—362no—
214C3×C8—181no—
215C3×C8—181no—
216C3⋊C8—92no—
217C3⋊C8—181no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—42no—
222C2×C32—42no—
223C2×C32—42no—
224C2×C32#67⋊C411yescharacteristic
225C2×C32#90111yescharacteristic
226C2×C32D3×M1611yescharacteristic
227C3⋊D3—41no—
228C3⋊D3—121no—
229C3⋊D3—124no—
230C3⋊D3—244no—
231C3×D3—124no—
232C2×D4—271no—
233C22×C4—91no—
234C22×C4—271no—
235M16—542no—
236M16—542no—
237C2×C8—91no—
238C2×C8—91no—
239C2×C8—271no—
240C2×C8—271no—
241C4⋊C4—31no—
242C22⋊C4—271no—
243C22⋊C4—271no—
244C42—271no—
245C22×C3—22no—
246C22×C3—42no—
247C22×C3—181no—
248C22×C3—181no—
249C22×C3—181no—
250C22×C3C22×#15711yescharacteristic
251C2×D3—64no—
252C2×D3—122no—
253C2×D3—122no—
254C2×D3—184no—
255C2×D3—181no—
256C2×D3—181no—
257C2×D3—181no—
258C2×D3—362no—
259C2×D3—362no—
260C2×D3—364no—
261C2×D3—722no—
262C2×D3—722no—
263C3×C4—21no—
264C3×C4—42no—
265C3×C4—64no—
266C3×C4—82no—
267C3×C4—91no—
268C3×C4—91no—
269C3×C4—122no—
270Dic3—21no—
271Dic3—62no—
272Dic3—62no—
273Dic3—181no—
274Dic3—362no—
275Dic3—722no—
276Dic3#378812yescomplemented
277C32—22no—
278C32—42no—
279C32#472911yescharacteristic, complemented
280C23—91no—
281C23—271no—
282D4—542no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—271no—
289C2×C4—541no—
290C2×C4—541no—
291C2×C4D3×#15711yescharacteristic
292C8—181no—
293C8—181no—
294C8—272no—
295C8—541no—
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—22no—
299C2×C3—42no—
300C2×C3—42no—
301C2×C3—42no—
302C2×C3—361no—
303C2×C3C2×#378811yescharacteristic
304C2×C3D4×#15711yescharacteristic
305C2×C3#6428911yescharacteristic
306D3—124no—
307D3—364no—
308D3—361no—
309D3—724no—
310C22—181no—
311C22—181no—
312C22—181no—
313C22—541no—
314C22—541no—
315C22—541no—
316C22C2×D3×#15711yescharacteristic
317C4—21no—
318C4—32no—
319C4—61no—
320C4—91no—
321C4—91no—
322C4—541no—
323C3—22no—
324C3—42no—
325C3#1059375211yescharacteristic, complemented
326C2—361no—
327C2—1081no—
328C2D3×#378811yescharacteristic
329C2#1172425211yescharacteristic
330C2#67⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#6292C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88989C2semidirect
2. C2×#88926C2semidirect
4. C3×#63778C2semidirect
5. Dic3⋊#622C2semidirect
6. #11722726C2semidirect
7. Dic3×#622C2semidirect
27. #63778D3semidirect
43. C3⋊#622D4semidirect
80. C32×Dic3M16semidirect
186. C33#436semidirect
276. Dic3#3788semidirect
279. C32#4729semidirect
325. C3#10593752semidirect

Maximal subgroups

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

subgroupindex
C2×#889892
C2×#889262
#117235122
C3×#637782
Dic3⋊#6222
#117227262
Dic3×#6222
#105937523
#47299