Finite Group Databasebeta

← #13167363

Subgroups — #13167363

Order 1728.

446 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 49 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
3#11723645C211yesmaximal, characteristic, complemented
4C3×#63768C211yesmaximal, characteristic, complemented
5#11722783C212yesmaximal, complemented
6#10593287—31nomaximal
7C2×#157⋊D3—22no—
8C4⋊#1113C2212yes—
9C4×#1113C2211yescharacteristic
10C3×C4⋊#157C2212yes—
11#88939—22no—
12C3×#3745—21no—
13C3×#3743—21no—
14C3×C8×C3⋊D3C2211yescharacteristic
15#88810—21no—
16#88809—21no—
17#88808C2211yescharacteristic
18#64230—32no—
19#63797—31no—
20D3×D24—61no—
21#63791—31no—
22D3×ΓL(1,𝔽25)—61no—
23#63768D311yescharacteristic, 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#4865—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—
47#3743—21no—
48C8×C3⋊D3C2×D311yescharacteristic
49C3×D24—61no—
50C3×D24—61no—
51C3×D24—121no—
52C3×ΓL(1,𝔽25)—61no—
53C3×ΓL(1,𝔽25)—61no—
54C3×ΓL(1,𝔽25)—121no—
55C3×D3×C8—61no—
56C3×D3×C8—61no—
57#3714—62no—
58#3713—61no—
59#3712—61no—
60#3711—61no—
61#3711—61no—
62#3710—31no—
63D3×C3⋊C8—61no—
64C2×C3×C3⋊D3—21no—
65C2×C3×C3⋊D3—42no—
66C2×C3×C3⋊D3C2×D411yescharacteristic
67#1113—22no—
68#1113—42no—
69#1113—42no—
70C3×#157—42no—
71C33×C4C2×D411yescharacteristic
72C3×C3⋊Dic3—21no—
73C3×C3⋊Dic3#4111yescharacteristic
74D3×D8—181no—
75D3×M16—181no—
76C2×D24—91no—
77C2×ΓL(1,𝔽25)—91no—
78#895—92no—
79C3×#106—91no—
80#809—92no—
81C2×D32—31no—
82C2×D32—61no—
83C2×D32—62no—
84C2×D32—122no—
85C2×#157—22no—
86GO+(4,𝔽2)—124no—
87C4⋊(C3⋊D3)—22no—
88C4×C3⋊D3—21no—
89C4×C3⋊D3C22×D311yescharacteristic
90C3×D12—62no—
91C3×D12—62no—
92C3×D12—122no—
93C3×C4×D3—61no—
94C3×C4×D3—61no—
95C3×C4×D3—61no—
96C3×C4×D3—61no—
97C3×C4×D3—121no—
98#626—122no—
99#625—62no—
100#624—31no—
101#624—61no—
102D3×Dic3—61no—
103D3×Dic3—61no—
104D3×Dic3—61no—
105C32×C8—21no—
106C32×C8—21no—
107C32×C8—41no—
108C32×C8—41no—
109C32×C8D1211yescharacteristic, complemented
110C3⋊(C3⋊C8)—21no—
111C3⋊(C3⋊C8)D1211yescharacteristic, complemented
112C3×C3⋊C8—61no—
113C3×C3⋊C8—61no—
114C3×C3⋊C8—61no—
115C3×C3⋊C8—61no—
116C3×C3⋊C8—121no—
117#469—271noSylow
118C2×C33#12111yescharacteristic
119C3×C3⋊D3—41no—
120C3×C3⋊D3—82no—
121C3×C3⋊D3AGL(1,ℤ/8)12yes—
122D3×D4—182no—
123C2×D12—92no—
124C2×D12—182no—
125C2×C4×D3—91no—
126C2×C4×D3—181no—
127C3×D8—181no—
128C3×M16—181no—
129C2×C3×C8—91no—
130C3×C4⋊C4—92no—
131#213—182no—
132(C2×D3)⋊C4—182no—
133#208—181no—
134C2×C3⋊C8—91no—
135D24—61no—
136D24—61no—
137D24—121no—
138D24—181no—
139ΓL(1,𝔽25)—61no—
140ΓL(1,𝔽25)—61no—
141ΓL(1,𝔽25)—121no—
142ΓL(1,𝔽25)—181no—
143ΓL(1,𝔽25)—181no—
144D3×C8—61no—
145D3×C8—61no—
146D3×C8—181no—
147C2×C3⋊D3—21no—
148C2×C3⋊D3—42no—
149C2×C3⋊D3D3×D411yescharacteristic
150C2×C3×D3—61no—
151C2×C3×D3—61no—
152C2×C3×D3—61no—
153C2×C3×D3—61no—
154C2×C3×D3—121no—
155C2×C3×D3—122no—
156C2×C3×D3—122no—
157C2×C3×D3—242no—
158D32—62no—
159D32—122no—
160D32—122no—
161D32—122no—
162D32—242no—
163D32—242no—
164#157—42no—
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—41no—
168C32×C4—41no—
169C32×C4C2×D1211yescharacteristic
170C3⋊Dic3—21no—
171C3⋊Dic3#23911yescharacteristic
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—61no—
176C3×Dic3—121no—
177C2×D8—271no—
178C2×M16—271no—
179#121—272no—
180#106—91no—
181#102—272no—
182C33#46911yescharacteristic, complemented, Sylow
183C22×D3—91no—
184C22×D3—181no—
185C22×D3—182no—
186C22×D3—362no—
187C3×D4—182no—
188C2×C3×C4—91no—
189C2×C3×C4—181no—
190C2×C3×C4—182no—
191#67—362no—
192C2×Dic3—91no—
193C2×Dic3—181no—
194D12—62no—
195D12—62no—
196D12—122no—
197D12—182no—
198D12—364no—
199C4×D3—61no—
200C4×D3—61no—
201C4×D3—61no—
202C4×D3—61no—
203C4×D3—121no—
204C4×D3—181no—
205C4×D3—181no—
206C4×D3—181no—
207C3×C8—21no—
208C3×C8—21no—
209C3×C8—41no—
210C3×C8—41no—
211C3×C8—91no—
212C3×C8—181no—
213C3×C8GO+(4,𝔽2)11yescharacteristic, complemented
214C3⋊C8—61no—
215C3⋊C8—61no—
216C3⋊C8—61no—
217C3⋊C8—61no—
218C3⋊C8—121no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32#89511yescharacteristic
224C3⋊D3—41no—
225C3⋊D3—82no—
226C3⋊D3#91412yes—
227C3×D3—62no—
228C3×D3—62no—
229C3×D3—121no—
230C3×D3—121no—
231C3×D3—241no—
232C3×D3—242no—
233C3×D3—242no—
234C3×D3—482no—
235C2×D4—272no—
236C2×D4—542no—
237C22×C4—271no—
238D8—181no—
239D8—541no—
240M16—181no—
241M16—541no—
242C2×C8—91no—
243C2×C8—271no—
244C4⋊C4—92no—
245C22⋊C4—542no—
246C22×C3—91no—
247C22×C3—181no—
248C22×C3—362no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—61no—
252C2×D3—61no—
253C2×D3—121no—
254C2×D3—122no—
255C2×D3—122no—
256C2×D3—182no—
257C2×D3—242no—
258C2×D3—362no—
259C2×D3—362no—
260C2×D3—362no—
261C2×D3—722no—
262C2×D3—722no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—91no—
268C3×C4—181no—
269C3×C4—362no—
270C3×C4C2×GO+(4,𝔽2)11yescharacteristic
271Dic3—61no—
272Dic3—61no—
273Dic3—61no—
274Dic3—61no—
275Dic3—121no—
276C32—21no—
277C32—21no—
278C32—41no—
279C32—41no—
280C32#486511yescharacteristic, complemented
281C23—271no—
282C23—542no—
283D4—182no—
284D4—542no—
285D4—1084no—
286C2×C4—91no—
287C2×C4—181no—
288C2×C4—182no—
289C2×C4—271no—
290C2×C4—541no—
291C8—91no—
292C8—181no—
293C8#157⋊D311yescharacteristic, complemented
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—92no—
299C2×C3—361no—
300C2×C3—722no—
301C2×C3#6423011yescharacteristic
302D3—62no—
303D3—62no—
304D3—121no—
305D3—121no—
306D3—241no—
307D3—242no—
308D3—242no—
309D3—482no—
310C22—91no—
311C22—181no—
312C22—362no—
313C22—542no—
314C22—1082no—
315C22—1082no—
316C4—91no—
317C4—181no—
318C4—362no—
319C4C2×#157⋊D311yescharacteristic
320C3—21no—
321C3—21no—
322C3—41no—
323C3—41no—
324C3#1059328711yescharacteristic, complemented
325C2—92no—
326C2—361no—
327C2—722no—
328C2#1172434311yescharacteristic
subgroupG/H
centerC2#11724343
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#121
hypercenterC8#157⋊D3
hypocenterC33#469
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. #11723645C2semidirect
4. C3×#63768C2semidirect
5. #11722783C2semidirect
23. #63768D3semidirect
35. C33×C8D4semidirect
37. C3×C3⋊(C3⋊C8)D4semidirect
109. C32×C8D12semidirect
111. C3⋊(C3⋊C8)D12semidirect
182. C33#469semidirect
213. C3×C8GO+(4,𝔽2)semidirect
280. C32#4865semidirect
293. C8#157⋊D3semidirect
324. C3#10593287semidirect

Maximal subgroups

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

subgroupindex
#117243432
#117236472
#117236452
C3×#637682
#117227832
#105932873
#48659