Finite Group Databasebeta

← #13161259

Subgroups — #13161259

Order 1728.

928 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 168 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
1Q8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic
3C2×C32×#216C211yesmaximal, characteristic
4#11723451C211yesmaximal, characteristic
5#11722050C211yesmaximal, characteristic
6#11722040C211yesmaximal, characteristic
7#11722038C211yesmaximal, characteristic
8#10592881—31nomaximal
9#10591633—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3756—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#216—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88806—21no—
22#88806—21no—
23C2×C32×Q16—31no—
24Q8×C3⋊Dic3—31no—
25Q8×C3⋊Dic3—34no—
26Q8×C3⋊Dic3—64no—
27#64153—31no—
28C2×#3756D311yescharacteristic
29C2×C3×#216—34no—
30C2×C3×#216D314yes—
31#63873—34no—
32#63662—31no—
33#63643—31no—
34#63641—31no—
35#63572—34no—
36#63566—34no—
37#63566—34no—
38C33×Q8—21no—
39C33×Q8D412yes—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C32×C3⋊C8—21no—
53#5182—94no—
54#5182—91no—
55C2×C32×Q8—24no—
56C2×C32×Q8—31no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C3×Dic6C2×D314yes—
62C32×Q16—62no—
63C2×C32×C8—31no—
64#3756—22no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)—64no—
70C4⋊(C3⋊Dic3)—68no—
71#3750—61no—
72#3750—61no—
73C4×C3⋊Dic3—31no—
74C4×C3⋊Dic3—34no—
75C4×C3⋊Dic3—61no—
76C4×C3⋊Dic3—64no—
77C4×C3⋊Dic3—64no—
78C4×C3⋊Dic3—68no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×#216—28no—
81C3×#216—68no—
82C2×C3×C3⋊C8—34no—
83C2×C3×C3⋊C8C2×D314yes—
84#3724—64no—
85#3724—64no—
86C22×C33C2×D411yescharacteristic
87C33×C4—22no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)—41no—
92C3⋊(C3⋊Dic3)Q1612yescomplemented
93C3×C3⋊Dic3—41no—
94C32×Dic3—41no—
95C2×C3×Q16—34no—
96C2×C3×Q16—91no—
97Q8×Dic3—94no—
98Q8×Dic3—91no—
99Q8×Dic3—184no—
100#944—94no—
101#944—91no—
102C2×#216—34no—
103C2×#216C3⋊D311yescharacteristic
104#835—94no—
105#835—91no—
106#816—94no—
107#816—91no—
108Dic3⋊C8—94no—
109Dic3⋊C8—91no—
110C32×Q8—28no—
111C32×Q8—24no—
112C32×Q8—21no—
113C32×Q8—28no—
114C32×Q8—61no—
115C32×Q8—61no—
116C32×Q8#6718yes—
117C32×Q8#6712yes—
118C2×C32×C4—28no—
119C2×C32×C4—24no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—31no—
128C2×C3⋊Dic3—34no—
129C2×C3⋊Dic3—34no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—68no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—61no—
135#634—21no—
136#634—21no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—64no—
139C3×Dic6—24no—
140C3×Dic6—24no—
141C3×Dic6—64no—
142C3×Dic6—64no—
143C32×C8—61no—
144C3⋊(C3⋊C8)—21no—
145C3×C3⋊C8—24no—
146C3×C3⋊C8—64no—
147#462—271noSylow
148C2×C33#12111yescharacteristic
149C2×C33C2×Q1611yescharacteristic
150C2×C33#13711yescharacteristic
151C2×C3×Q8—24no—
152C2×C3×Q8—34no—
153C2×C3×Q8—91no—
154C2×C3×Q8D3214yes—
155C2×C3×Q8C2×C3⋊D311yescharacteristic
156C2×Dic6—34no—
157C2×Dic6C2×C3⋊D311yescharacteristic
158C3×Q16—68no—
159C3×Q16—182no—
160C2×C3×C8—34no—
161C2×C3×C8—91no—
162#216—22no—
163#216—68no—
164C4⋊Dic3—94no—
165C4⋊Dic3—91no—
166C4⋊Dic3—184no—
167C4⋊Dic3—184no—
168C4⋊Dic3—181no—
169C4⋊Dic3—188no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172Dic3⋊C4—184no—
173Dic3⋊C4—184no—
174C4×Dic3—94no—
175C4×Dic3—91no—
176C4×Dic3—184no—
177C4×Dic3—184no—
178C4×Dic3—181no—
179C4×Dic3—188no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8C2×C3⋊D311yescharacteristic
182C22×C32—24no—
183C22×C32D3×D414yes—
184C22×C32D3×D411yescharacteristic
185C32×C4—216no—
186C32×C4—28no—
187C32×C4—22no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—121no—
191C32×C4#23714yes—
192C32×C4#23711yescharacteristic
193C32×C4C2×#6714yes—
194C32×C4C2×#6711yescharacteristic
195C3⋊Dic3—32no—
196C3⋊Dic3—38no—
197C3⋊Dic3—41no—
198C3⋊Dic3—68no—
199C3⋊Dic3—61no—
200C3⋊Dic3—64no—
201C3⋊Dic3—124no—
202C3⋊Dic3—128no—
203C3⋊Dic3—124no—
204C3⋊Dic3—121no—
205C3×Dic3—44no—
206C3×Dic3—124no—
207C2×Q16—31no—
208C2×Q16—91no—
209C4⋊Q8—271no—
210C4×Q8—271no—
211C4⋊C8—271no—
212#103—271no—
213#103—271no—
214C33#46211yescharacteristic, complemented, Sylow
215C3×Q8—28no—
216C3×Q8—24no—
217C3×Q8—21no—
218C3×Q8—28no—
219C3×Q8—64no—
220C3×Q8—64no—
221C3×Q8—181no—
222C3×Q8—181no—
223C3×Q8#62518yes—
224C3×Q8#63812yes—
225C2×C3×C4—28no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—64no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3214yes—
232C2×C3×C4C22×C3⋊D311yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—64no—
235C2×Dic3—94no—
236C2×Dic3—91no—
237C2×Dic3—94no—
238C2×Dic3—91no—
239C2×Dic3—184no—
240C2×Dic3—184no—
241C2×Dic3—188no—
242C2×Dic3—184no—
243C2×Dic3—181no—
244Dic6—21no—
245Dic6—21no—
246Dic6—64no—
247Dic6—64no—
248C3×C8—64no—
249C3×C8—181no—
250C3⋊C8—21no—
251C3⋊C8—64no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32D3×Q1614yes—
256C2×C32D3×Q1611yescharacteristic
257C2×C32#91814yes—
258C2×C32#91811yescharacteristic
259C2×C32#93914yes—
260C2×C32#93911yescharacteristic
261C2×Q8—31no—
262C2×Q8—91no—
263C2×Q8D3×C3⋊D311yescharacteristic
264Q16—62no—
265Q16—182no—
266C2×C8—31no—
267C2×C8—91no—
268C4⋊C4—271no—
269C4⋊C4—541no—
270C4⋊C4—541no—
271C4⋊C4—541no—
272C42—271no—
273C42—541no—
274C22×C3—24no—
275C22×C3#381114yes—
276C22×C3D4×C3⋊D311yescharacteristic
277C3×C4—216no—
278C3×C4—22no—
279C3×C4—28no—
280C3×C4—24no—
281C3×C4—24no—
282C3×C4—124no—
283C3×C4—361no—
284C3×C4#380514yes—
285C3×C4C2×#62514yes—
286C3×C4#383011yescharacteristic
287C3×C4C2×#63811yescharacteristic
288Dic3—41no—
289Dic3—92no—
290Dic3—98no—
291Dic3—124no—
292Dic3—188no—
293Dic3—181no—
294Dic3—184no—
295Dic3—364no—
296Dic3—368no—
297Dic3—361no—
298Dic3—364no—
299C32—24no—
300C32#518214yescomplemented
301C32#518211yescharacteristic, complemented
302Q8—21no—
303Q8—61no—
304Q8—61no—
305Q8—181no—
306Q8—181no—
307Q8#629012yes—
308C2×C4—21no—
309C2×C4—61no—
310C2×C4—181no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—541no—
314C2×C4C2×D3×C3⋊D311yescharacteristic
315C8—61no—
316C8—181no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3#6394414yes—
321C2×C3#6394514yes—
322C2×C3#6398014yes—
323C2×C3Q16×C3⋊D311yescharacteristic
324C2×C3#6412711yescharacteristic
325C2×C3#6414811yescharacteristic
326C22#8903911yescharacteristic
327C4—22no—
328C4—121no—
329C4—272no—
330C4—361no—
331C4—541no—
332C4—1081no—
333C4#8902911yescharacteristic
334C4C2×#629011yescharacteristic
335C3—24no—
336C3#1059163314yescomplemented
337C3#1059288111yescharacteristic, complemented
338C2#1172357711yescharacteristic
339C2#1172357811yescharacteristic
340C2#1172363711yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#462
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
92. C3⋊(C3⋊Dic3)Q16semidirect
214. C33#462semidirect
300. C32#5182semidirect
301. C32#5182semidirect
336. C3#10591633semidirect
337. C3#10592881semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊Dic3)2
C2×C3×#37562
C2×C32×#2162
#117234512
#117220502
#117220402
#117220382
#105928813
#105916333