Finite Group Databasebeta

← #13167509

Subgroups — #13167509

Order 1728.

462 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 57 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×#88942C212yesmaximal
2C4⋊#6296C211yesmaximal, characteristic, complemented
3#11723685C211yesmaximal, characteristic
4C3×#63779C211yesmaximal, characteristic, complemented
5#11722794C212yesmaximal, complemented
6#10593438—31nomaximal
7C2×#6296C2212yes—
8C2×#6295C2211yescharacteristic
9C2×#6294—21no—
10C2×C3×#622C2212yes—
11#88942—24no—
12C3×#3752—21no—
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15#88815—21no—
16#88815C2211yescharacteristic
17C2×#3775—32no—
18#63919—31no—
19#63918—61no—
20#63902—31no—
21#63880—61no—
22#63779D311yescharacteristic, complemented
23#63740—32no—
24C22×C3×C3⋊D3—21no—
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3—21no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30#6296—22no—
31#6296—22no—
32#6295—21no—
33#6295D412yes—
34#6294—42no—
35C3×#622—22no—
36#4841—91nomaximal
37C2×#627—32no—
38C2×#627—62no—
39C2×#626—61no—
40C2×#625—31no—
41C2×#624—61no—
42C2×D3×Dic3—61no—
43C2×D3×Dic3—61no—
44C2×#622C2×D312yes—
45#3775—64no—
46#3752—21no—
47C4⋊(C3⋊Dic3)—21no—
48C4×C3⋊Dic3C2×D311yescharacteristic
49C3×(C2×D3)⋊C4—61no—
50C3×(C2×D3)⋊C4—61no—
51C3×(C2×D3)⋊C4—121no—
52C3×C4⋊Dic3—61no—
53C3×C4⋊Dic3—61no—
54C3×C4⋊Dic3—121no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—61no—
57#3724—31no—
58#3724—61no—
59Dic3⋊Dic3—61no—
60Dic3⋊Dic3—62no—
61C22×C33C2×D411yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3—41no—
65C33×C4—21no—
66C3×C3⋊Dic3—42no—
67C3×C3⋊Dic3—41no—
68C3×C3⋊Dic3SD1612yescomplemented
69C3×C3⋊Dic3C2×D411yescharacteristic
70C3×C3⋊Dic3#4111yescharacteristic
71C2×#204—92no—
72#888—181no—
73#67⋊C4—181no—
74C4×D12—91no—
75#869—91no—
76C3×C4⋊C8—91no—
77#816—92no—
78C22×C3⋊D3—21no—
79C22×C3×D3—61no—
80C22×C3×D3—61no—
81C22×C3×D3—121no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4D1211yescharacteristic
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3D1211yescharacteristic
90C2×C3⋊Dic3C22×D311yescharacteristic
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—122no—
100C2×C3×Dic3—121no—
101#627—62no—
102#627—62no—
103#627—122no—
104#627—122no—
105#626—62no—
106#626—62no—
107#625—32no—
108#625—61no—
109#624—122no—
110D3×Dic3—122no—
111D3×Dic3—122no—
112#622—22no—
113#461—271noSylow
114C2×C33#12111yescharacteristic
115C2×C33C2×SD1611yescharacteristic
116C2×C33#13711yescharacteristic
117C3×C3⋊D3—42no—
118C2×#67—181no—
119C22×Dic3—181no—
120C2×D12—91no—
121C2×C4×D3—181no—
122C2×C4×D3—181no—
123C2×Dic6—92no—
124C2×Dic6—182no—
125C2×C3×C8—92no—
126C3×C4⋊C4—181no—
127C3×C22⋊C4—181no—
128C3×C42—91no—
129(C2×D3)⋊C4—61no—
130(C2×D3)⋊C4—61no—
131(C2×D3)⋊C4—121no—
132C4⋊Dic3—61no—
133C4⋊Dic3—61no—
134C4⋊Dic3—91no—
135C4⋊Dic3—121no—
136C4⋊Dic3—181no—
137Dic3⋊C4—181no—
138Dic3⋊C4—182no—
139C4×Dic3—61no—
140C4×Dic3—61no—
141#204—184no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—41no—
145C22×C32—41no—
146C22×C32C2×D1211yescharacteristic
147C2×C3⋊D3—22no—
148C2×C3⋊D3—41no—
149C2×C3⋊D3—41no—
150C2×C3×D3—62no—
151C2×C3×D3—62no—
152C2×C3×D3—122no—
153C2×C3×D3—121no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—241no—
158C2×C3×D3—241no—
159C32×C4—22no—
160C32×C4—21no—
161C32×C4—41no—
162C32×C4—42no—
163C32×C4—81no—
164C3⋊Dic3—41no—
165C3⋊Dic3—42no—
166C3⋊Dic3#20412yescomplemented
167C3⋊Dic3D3×D411yescharacteristic
168C3⋊Dic3#23911yescharacteristic
169C3×Dic3—62no—
170C3×Dic3—62no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—121no—
176C3×Dic3—122no—
177C3×Dic3—122no—
178C3×Dic3—121no—
179C3×Dic3—242no—
180C3×Dic3—241no—
181C2×SD16—272no—
182C4⋊Q8—271no—
183C4×D4—271no—
184C4⋊C8—91no—
185#103—272no—
186C33#46111yescharacteristic, complemented, Sylow
187C23×C3—181no—
188C22×D3—61no—
189C22×D3—61no—
190C22×D3—121no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—181no—
198C2×C3×C4—182no—
199C2×C3×C4GO+(4,𝔽2)11yescharacteristic
200#67—182no—
201#67—182no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—62no—
207C2×Dic3—61no—
208C2×Dic3—62no—
209C2×Dic3—61no—
210C2×Dic3—122no—
211C2×Dic3—121no—
212C2×Dic3—362no—
213D12—92no—
214D12—181no—
215C4×D3—362no—
216C4×D3—362no—
217Dic6—182no—
218Dic6—182no—
219Dic6—362no—
220Dic6—362no—
221C3×C8—182no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32#89511yescharacteristic
235C2×C32C2×#20411yescharacteristic
236C2×C32#90911yescharacteristic
237C3⋊D3—42no—
238C3×D3—122no—
239C3×D3—122no—
240C3×D3—242no—
241C2×Q8—272no—
242C2×D4—271no—
243C22×C4—541no—
244SD16—544no—
245C2×C8—92no—
246C4⋊C4—181no—
247C4⋊C4—271no—
248C4⋊C4—541no—
249C22⋊C4—181no—
250C42—91no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—182no—
256C22×C3—361no—
257C22×C3—361no—
258C22×C3C2×GO+(4,𝔽2)11yescharacteristic
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—122no—
262C2×D3—121no—
263C2×D3—121no—
264C2×D3—121no—
265C2×D3—121no—
266C2×D3—241no—
267C2×D3—241no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—42no—
272C3×C4—81no—
273C3×C4—92no—
274C3×C4—91no—
275C3×C4—91no—
276C3×C4—362no—
277C3×C4—361no—
278Dic3—61no—
279Dic3—61no—
280Dic3—62no—
281Dic3—61no—
282Dic3—61no—
283Dic3—62no—
284Dic3—122no—
285Dic3—121no—
286Dic3—121no—
287Dic3—122no—
288Dic3—241no—
289Dic3—242no—
290C32—21no—
291C32—21no—
292C32—41no—
293C32—41no—
294C32#484111yescharacteristic, complemented
295C23—181no—
296Q8—542no—
297Q8—542no—
298D4—272no—
299D4—541no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—181no—
303C2×C4—182no—
304C2×C4—1082no—
305C2×C4#157⋊D311yescharacteristic
306C8—182no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—362no—
320C2×C3#6422711yescharacteristic
321C2×C3#6423011yescharacteristic
322C2×C3#6423911yescharacteristic
323D3—122no—
324D3—122no—
325D3—242no—
326C22—182no—
327C22—361no—
328C22—361no—
329C22C2×#157⋊D311yescharacteristic
330C4—21no—
331C4—92no—
332C4—91no—
333C4—91no—
334C4—361no—
335C4—362no—
336C3—21no—
337C3—21no—
338C3—41no—
339C3—41no—
340C3#1059343811yescharacteristic, complemented
341C2—362no—
342C2#1172433911yescharacteristic
343C2#1172434311yescharacteristic
344C2#1172435211yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#6295C22
hypercenterC2×C4#157⋊D3
hypocenterC33#461
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4⋊#6296C2semidirect
4. C3×#63779C2semidirect
5. #11722794C2semidirect
22. #63779D3semidirect
68. C3×C3⋊Dic3SD16semidirect
166. C3⋊Dic3#204semidirect
186. C33#461semidirect
294. C32#4841semidirect
340. C3#10593438semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
C4⋊#62962
#117236852
C3×#637792
#117227942
#105934383
#48419