Finite Group Databasebeta

← #13167361

Subgroups — #13167361

Order 1728.

464 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 51 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#11724343C212yesmaximal, complemented
2#11723647C211yesmaximal, characteristic, complemented
3C8×#1113C211yesmaximal, characteristic, complemented
4C3×#63769C211yesmaximal, characteristic, complemented
5#11722783C212yesmaximal, complemented
6#10593285—31nomaximal
7C2×#157⋊D3—22no—
8C4⋊#1113C2212yes—
9C4×#1113C2211yescharacteristic
10C3×C4⋊#157C2212yes—
11#88939—22no—
12C3×#3745—21no—
13C3×C8×C3⋊D3—21no—
14C3×C8×C3⋊D3C2211yescharacteristic
15#88810—21no—
16#88808—21no—
17#88808C2211yescharacteristic
18#64230—32no—
19#63797—31no—
20D3×D24—61no—
21D32×C8—31no—
22D32×C8—61no—
23#63769D311yescharacteristic, complemented
24#63729—32no—
25C2×#1113—22no—
26C2×#1113D411yescharacteristic
27C2×C3×#157—22no—
28#157⋊D3—44no—
29C3×C4⋊(C3⋊D3)—22no—
30C3×C4×C3⋊D3—21no—
31C3×C4×C3⋊D3C2311yescharacteristic
32#6295—22no—
33#6294—21no—
34#6294D411yescharacteristic
35C33×C8D411yescharacteristic, complemented
36C3×C3⋊(C3⋊C8)—21no—
37C3×C3⋊(C3⋊C8)D411yescharacteristic, complemented
38#4883—91nomaximal
39C2×GO+(4,𝔽2)—62no—
40#3802—32no—
41D3×D12—62no—
42C4×D32—31no—
43C4×D32—61no—
44C4⋊#157C2×D312yes—
45D32⋊C4—62no—
46#3745—21no—
47C8×C3⋊D3—21no—
48C8×C3⋊D3C2×D311yescharacteristic
49C3×D24—61no—
50C3×D24—61no—
51C3×D24—121no—
52C3×D3×C8—61no—
53C3×D3×C8—61no—
54C3×D3×C8—61no—
55C3×D3×C8—61no—
56C3×D3×C8—121no—
57#3714—62no—
58#3713—61no—
59#3710—31no—
60#3710—61no—
61D3×C3⋊C8—61no—
62D3×C3⋊C8—61no—
63D3×C3⋊C8—61no—
64C2×C3×C3⋊D3—21no—
65C2×C3×C3⋊D3—42no—
66C2×C3×C3⋊D3C2×D411yescharacteristic
67#1113—21no—
68#1113—42no—
69#1113—42no—
70#1113D812yescomplemented
71C3×#157—42no—
72C33×C4C2×D411yescharacteristic
73C3×C3⋊Dic3—21no—
74C3×C3⋊Dic3#4111yescharacteristic
75D3×D8—181no—
76C2×D24—91no—
77C2×D3×C8—91no—
78C2×D3×C8—181no—
79#895—92no—
80C3×#107—91no—
81#809—92no—
82C2×D32—31no—
83C2×D32—61no—
84C2×D32—62no—
85C2×D32—122no—
86C2×#157—22no—
87GO+(4,𝔽2)—124no—
88C4⋊(C3⋊D3)—22no—
89C4×C3⋊D3—21no—
90C4×C3⋊D3C22×D311yescharacteristic
91C3×D12—62no—
92C3×D12—62no—
93C3×D12—122no—
94C3×C4×D3—61no—
95C3×C4×D3—61no—
96C3×C4×D3—61no—
97C3×C4×D3—61no—
98C3×C4×D3—121no—
99#626—122no—
100#625—62no—
101#624—31no—
102#624—61no—
103D3×Dic3—61no—
104D3×Dic3—61no—
105D3×Dic3—61no—
106C32×C8—21no—
107C32×C8—21no—
108C32×C8—41no—
109C32×C8—41no—
110C32×C8D1211yescharacteristic, complemented
111C3⋊(C3⋊C8)—21no—
112C3⋊(C3⋊C8)D1211yescharacteristic, complemented
113C3×C3⋊C8—61no—
114C3×C3⋊C8—61no—
115C3×C3⋊C8—61no—
116C3×C3⋊C8—61no—
117C3×C3⋊C8—121no—
118#466—271noSylow
119C2×C33#12111yescharacteristic
120C3×C3⋊D3—22no—
121C3×C3⋊D3—82no—
122C3×C3⋊D3C2×D811yescharacteristic
123C3×C3⋊D3#13511yescharacteristic
124D3×D4—182no—
125C2×D12—92no—
126C2×D12—182no—
127C2×C4×D3—91no—
128C2×C4×D3—181no—
129C3×D8—181no—
130C2×C3×C8—91no—
131C2×C3×C8—181no—
132C3×C4⋊C4—92no—
133#213—182no—
134(C2×D3)⋊C4—182no—
135C2×C3⋊C8—91no—
136C2×C3⋊C8—181no—
137D24—61no—
138D24—61no—
139D24—121no—
140D24—181no—
141D3×C8—61no—
142D3×C8—61no—
143D3×C8—61no—
144D3×C8—61no—
145D3×C8—121no—
146D3×C8—181no—
147D3×C8—181no—
148D3×C8—181no—
149C2×C3⋊D3—21no—
150C2×C3⋊D3—42no—
151C2×C3⋊D3D3×D411yescharacteristic
152C2×C3×D3—61no—
153C2×C3×D3—61no—
154C2×C3×D3—61no—
155C2×C3×D3—61no—
156C2×C3×D3—121no—
157C2×C3×D3—122no—
158C2×C3×D3—122no—
159C2×C3×D3—242no—
160D32—32no—
161D32—62no—
162D32—62no—
163D32—61no—
164D32—122no—
165D32—122no—
166D32—242no—
167D32—242no—
168#157—42no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4C2×D1211yescharacteristic
174C3⋊Dic3—21no—
175C3⋊Dic3#23911yescharacteristic
176C3×Dic3—61no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—61no—
180C3×Dic3—121no—
181C2×D8—271no—
182C22×C8—271no—
183#121—272no—
184#107—91no—
185#102—272no—
186C33#46611yescharacteristic, complemented, Sylow
187C22×D3—91no—
188C22×D3—181no—
189C22×D3—182no—
190C22×D3—362no—
191C3×D4—182no—
192C2×C3×C4—91no—
193C2×C3×C4—181no—
194C2×C3×C4—182no—
195#67—362no—
196C2×Dic3—91no—
197C2×Dic3—181no—
198D12—62no—
199D12—62no—
200D12—122no—
201D12—182no—
202D12—364no—
203C4×D3—61no—
204C4×D3—61no—
205C4×D3—61no—
206C4×D3—61no—
207C4×D3—121no—
208C4×D3—181no—
209C4×D3—181no—
210C4×D3—181no—
211C3×C8—21no—
212C3×C8—21no—
213C3×C8—41no—
214C3×C8—41no—
215C3×C8—91no—
216C3×C8—181no—
217C3×C8GO+(4,𝔽2)11yescharacteristic, complemented
218C3⋊C8—61no—
219C3⋊C8—61no—
220C3⋊C8—61no—
221C3⋊C8—61no—
222C3⋊C8—121no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32#89511yescharacteristic
228C3⋊D3—22no—
229C3⋊D3—82no—
230C3⋊D3D3×D811yescharacteristic
231C3⋊D3#91911yescharacteristic
232C3×D3—61no—
233C3×D3—61no—
234C3×D3—61no—
235C3×D3—61no—
236C3×D3—62no—
237C3×D3—62no—
238C3×D3—122no—
239C3×D3—242no—
240C3×D3—242no—
241C3×D3—482no—
242C2×D4—272no—
243C2×D4—542no—
244C22×C4—271no—
245D8—181no—
246D8—541no—
247C2×C8—91no—
248C2×C8—181no—
249C2×C8—271no—
250C2×C8—541no—
251C4⋊C4—92no—
252C22⋊C4—542no—
253C22×C3—91no—
254C22×C3—181no—
255C22×C3—362no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—61no—
259C2×D3—61no—
260C2×D3—92no—
261C2×D3—121no—
262C2×D3—122no—
263C2×D3—122no—
264C2×D3—182no—
265C2×D3—181no—
266C2×D3—182no—
267C2×D3—242no—
268C2×D3—362no—
269C2×D3—362no—
270C2×D3—722no—
271C2×D3—722no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—91no—
277C3×C4—181no—
278C3×C4—362no—
279C3×C4C2×GO+(4,𝔽2)11yescharacteristic
280Dic3—61no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—121no—
285C32—21no—
286C32—21no—
287C32—41no—
288C32—41no—
289C32#488311yescharacteristic, complemented
290C23—271no—
291C23—542no—
292D4—182no—
293D4—542no—
294D4—1084no—
295C2×C4—91no—
296C2×C4—181no—
297C2×C4—182no—
298C2×C4—271no—
299C2×C4—541no—
300C8—91no—
301C8—181no—
302C8#157⋊D311yescharacteristic, complemented
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—91no—
308C2×C3—91no—
309C2×C3—182no—
310C2×C3—722no—
311C2×C3#6423011yescharacteristic
312D3—61no—
313D3—61no—
314D3—61no—
315D3—61no—
316D3—62no—
317D3—62no—
318D3—122no—
319D3—242no—
320D3—242no—
321D3—482no—
322C22—91no—
323C22—181no—
324C22—272no—
325C22—362no—
326C22—541no—
327C22—1082no—
328C22—1082no—
329C4—91no—
330C4—181no—
331C4—362no—
332C4C2×#157⋊D311yescharacteristic
333C3—21no—
334C3—21no—
335C3—41no—
336C3—41no—
337C3#1059328511yescharacteristic, complemented
338C2—91no—
339C2—91no—
340C2—182no—
341C2—722no—
342C2#1172434311yescharacteristic
subgroupG/H
centerC2#11724343
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#121
hypercenterC8#157⋊D3
hypocenterC33#466
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11724343C2semidirect
2. #11723647C2semidirect
3. C8×#1113C2semidirect
4. C3×#63769C2semidirect
5. #11722783C2semidirect
23. #63769D3semidirect
35. C33×C8D4semidirect
37. C3×C3⋊(C3⋊C8)D4semidirect
70. #1113D8semidirect
110. C32×C8D12semidirect
112. C3⋊(C3⋊C8)D12semidirect
186. C33#466semidirect
217. C3×C8GO+(4,𝔽2)semidirect
289. C32#4883semidirect
302. C8#157⋊D3semidirect
337. C3#10593285semidirect

Maximal subgroups

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

subgroupindex
#117243432
#117236472
C8×#11132
C3×#637692
#117227832
#105932853
#48839