Finite Group Databasebeta

← #13159995

Subgroups — #13159995

Order 1728.

800 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 156 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×C32×Dic12C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic
3#11723397C211yesmaximal, characteristic
4C2×#88798C211yesmaximal, characteristic
5#11722184C211yesmaximal, characteristic
6C3×#63643C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592881—31nomaximal
9#10590798—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×Dic12—22no—
18#88806—21no—
19#88806—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#88798—22no—
23C2×C32×Q16—31no—
24Q8×C3⋊Dic3—31no—
25#64153—31no—
26C2×#3756—31no—
27C2×C3×Dic12D314yes—
28#63873—34no—
29#63843—34no—
30C2×#3718—34no—
31#63662—31no—
32#63643D311yescharacteristic
33#63641—31no—
34#63641—34no—
35#63641—64no—
36C3×#816—34no—
37#63565—34no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C3×#634—21no—
44C3×#634—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—21no—
47C32×Dic6—21no—
48C32×Dic6—21no—
49C32×Dic6—21no—
50C32×Dic6D412yes—
51C33×C8—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53#5182—94no—
54#4795—91no—
55C2×C32×Q8—31no—
56C2×C32×Q8—31no—
57C2×#634C2×D311yescharacteristic
58C2×C3×Dic6—34no—
59C2×C3×Dic6C2×D314yes—
60C2×C3×Dic6C2×D314yes—
61C32×Q16—62no—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D311yescharacteristic
65#3756—62no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—61no—
69#3750—61no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—61no—
73C4×C3⋊Dic3—64no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C3×C4⋊Dic3—34no—
78C3×Dic12—28no—
79#3724—64no—
80#3724—64no—
81Dic3⋊Dic3—64no—
82Dic32—64no—
83#3718—68no—
84C22×C33C2×D411yescharacteristic
85C33×C4C2×D411yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—21no—
88C3⋊(C3⋊Dic3)Q1612yescomplemented
89C3×C3⋊Dic3—41no—
90C3×C3⋊Dic3—41no—
91C32×Dic3—22no—
92C32×Dic3—41no—
93C2×C3×Q16—34no—
94Q8×Dic3—94no—
95#944—94no—
96#944—91no—
97C2×#216—94no—
98C2×#216—91no—
99C2×Dic12C3⋊D311yescharacteristic
100#887—91no—
101C3×#103—91no—
102#835—94no—
103#816—34no—
104Dic3⋊C8—94no—
105Dic3⋊C8—91no—
106Dic3⋊C8—184no—
107#810—91no—
108C32×Q8—32no—
109C32×Q8—61no—
110C32×Q8—61no—
111C32×Q8—61no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—34no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—64no—
125#634—21no—
126#634—21no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—64no—
131C3×Dic6—24no—
132C3×Dic6—24no—
133C3×Dic6—24no—
134C3×Dic6—64no—
135C3×Dic6—64no—
136C3×Dic6#6718yes—
137C32×C8—21no—
138C32×C8—24no—
139C32×C8—44no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—64no—
142C3⋊(C3⋊C8)—124no—
143#462—271noSylow
144C2×C33#12111yescharacteristic
145C2×C33C2×Q1611yescharacteristic
146C2×C33#13711yescharacteristic
147C2×C3×Q8—34no—
148C2×C3×Q8—34no—
149C2×C3×Q8—91no—
150C2×Dic6—34no—
151C2×Dic6C2×C3⋊D311yescharacteristic
152C2×Dic6C2×C3⋊D311yescharacteristic
153C3×Q16—68no—
154C2×C3×C8—24no—
155C2×C3×C8D3214yes—
156C2×C3×C8C2×C3⋊D311yescharacteristic
157C3×C4⋊C4—91no—
158#216—188no—
159#216—182no—
160C4⋊Dic3—34no—
161C4⋊Dic3—184no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165Dic3⋊C4—184no—
166Dic3⋊C4—184no—
167C4×Dic3—94no—
168C4×Dic3—91no—
169C4×Dic3—184no—
170C4×Dic3—181no—
171C4×Dic3—184no—
172C2×C3⋊C8—94no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—184no—
175Dic12—22no—
176C22×C32—24no—
177C22×C32D3×D414yes—
178C22×C32D3×D411yescharacteristic
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—62no—
182C32×C4—121no—
183C32×C4#23511yescharacteristic
184C32×C4D3×D411yescharacteristic
185C32×C4#23714yes—
186C32×C4C2×#6714yes—
187C3⋊Dic3—32no—
188C3⋊Dic3—38no—
189C3⋊Dic3—41no—
190C3⋊Dic3—41no—
191C3⋊Dic3—68no—
192C3⋊Dic3—61no—
193C3⋊Dic3—64no—
194C3⋊Dic3—124no—
195C3×Dic3—28no—
196C3×Dic3—44no—
197C3×Dic3—124no—
198C3×Dic3—124no—
199C2×Q16—31no—
200C2×Q16—271no—
201C4⋊Q8—271no—
202C4×Q8—271no—
203C4⋊C8—271no—
204#103—91no—
205#103—271no—
206C33#46211yescharacteristic, complemented, Sylow
207C3×Q8—38no—
208C3×Q8—64no—
209C3×Q8—64no—
210C3×Q8—64no—
211C3×Q8—181no—
212C3×Q8—181no—
213C2×C3×C4—24no—
214C2×C3×C4—64no—
215C2×C3×C4—64no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—64no—
223C2×Dic3—64no—
224C2×Dic3—94no—
225C2×Dic3—91no—
226C2×Dic3—94no—
227C2×Dic3—91no—
228C2×Dic3—184no—
229C2×Dic3—184no—
230Dic6—21no—
231Dic6—21no—
232Dic6—21no—
233Dic6—64no—
234Dic6—64no—
235Dic6#63812yes—
236C3×C8—24no—
237C3×C8—21no—
238C3×C8—44no—
239C3⋊C8—184no—
240C3⋊C8—181no—
241C3⋊C8—364no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32—24no—
245C2×C32#88411yescharacteristic
246C2×C32#91511yescharacteristic
247C2×C32D3×Q1614yes—
248C2×C32D3×Q1611yescharacteristic
249C2×C32#91814yes—
250C2×C32#93914yes—
251C2×Q8—31no—
252C2×Q8—31no—
253C2×Q8—91no—
254Q16—62no—
255Q16—542no—
256C2×C8—271no—
257C2×C8D3×C3⋊D311yescharacteristic
258C4⋊C4—91no—
259C4⋊C4—541no—
260C4⋊C4—541no—
261C4⋊C4—541no—
262C42—271no—
263C42—541no—
264C22×C3—24no—
265C22×C3#380214yes—
266C22×C3D4×C3⋊D311yescharacteristic
267C3×C4—24no—
268C3×C4—24no—
269C3×C4—68no—
270C3×C4—124no—
271C3×C4—361no—
272C3×C4—361no—
273C3×C4#379514yes—
274C3×C4D3×#6714yes—
275C3×C4#383011yescharacteristic
276C3×C4C2×#63811yescharacteristic
277Dic3—22no—
278Dic3—41no—
279Dic3—98no—
280Dic3—92no—
281Dic3—124no—
282Dic3—124no—
283Dic3—188no—
284Dic3—184no—
285Dic3—181no—
286Dic3—364no—
287C32—24no—
288C32#479511yescharacteristic, complemented
289C32#518214yescomplemented
290Q8—32no—
291Q8—61no—
292Q8—61no—
293Q8—61no—
294Q8—181no—
295Q8—181no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—271no—
301C2×C4—271no—
302C2×C4C2×D3×C3⋊D311yescharacteristic
303C8—21no—
304C8—541no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3#6380214yes—
309C2×C3#6381114yes—
310C2×C3#6388514yes—
311C2×C3Q16×C3⋊D311yescharacteristic
312C2×C3#6412711yescharacteristic
313C2×C3#6414811yescharacteristic
314C22#8902611yescharacteristic
315C4—62no—
316C4—121no—
317C4—272no—
318C4—361no—
319C4—361no—
320C4—541no—
321C4#8901411yescharacteristic
322C4D3×#63811yescharacteristic
323C3—24no—
324C3#1059079814yescomplemented
325C3#1059288111yescharacteristic, complemented
326C2#1172333011yescharacteristic
327C2#1172334611yescharacteristic
328C2#1172347411yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#462
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
88. C3⋊(C3⋊Dic3)Q16semidirect
206. C33#462semidirect
288. C32#4795semidirect
289. C32#5182semidirect
324. C3#10590798semidirect
325. C3#10592881semidirect

Maximal subgroups

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

subgroupindex
C2×C32×Dic122
#117234512
#117233972
C2×#887982
#117221842
C3×#636432
#117220372
#105928813
#105907983