Finite Group Databasebeta

← #13159312

Subgroups — #13159312

Order 1728.

928 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723928C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic
3C2×#88800C211yesmaximal, characteristic
4C2×#88799C211yesmaximal, characteristic
5C3×#63637C211yesmaximal, characteristic
6C32×#810C211yesmaximal, characteristic
7#11722050C211yesmaximal, characteristic
8#10592494—31nomaximal
9#10590566—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20#88806—21no—
21#88800—22no—
22#88799—22no—
23#64153—31no—
24C2×#3756—31no—
25C2×#3746—31no—
26#64096—31no—
27#64096—34no—
28#64096—64no—
29#63873—34no—
30C2×#3719—34no—
31C2×#3719—34no—
32C32×#103—31no—
33#63641—31no—
34#63637D311yescharacteristic
35C3×#810—34no—
36C3×#810D314yes—
37#63572—34no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43#6308—21no—
44#6308D412yes—
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#4795—94no—
54#4795—91no—
55C2×C32×Q8—31no—
56C2×#634—31no—
57C2×#634—34no—
58C2×#634—64no—
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C3×Dic6C2×D314yes—
62C2×C32×C8—31no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66#3756—62no—
67#3750—61no—
68#3750—61no—
69#3750—61no—
70#3750—64no—
71#3750—68no—
72C4×C3⋊Dic3—31no—
73C4×C3⋊Dic3—34no—
74C4×C3⋊Dic3—61no—
75C4×C3⋊Dic3—64no—
76C4×C3⋊Dic3—64no—
77C4×C3⋊Dic3—68no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79#3746—62no—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8C2×D314yes—
82#3724—64no—
83#3724—64no—
84#3719—68no—
85#3719—68no—
86C22×C33C2×D411yescharacteristic
87C33×C4—41no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—22no—
91C3⋊(C3⋊Dic3)—21no—
92C3⋊(C3⋊Dic3)Q1612yescomplemented
93C3×C3⋊Dic3—41no—
94C32×Dic3—41no—
95#944—94no—
96#944—91no—
97C2×#216—94no—
98C2×#216—91no—
99C2×Dic12—94no—
100C2×Dic12—91no—
101#887—94no—
102#887—91no—
103#887—184no—
104C3×#103—34no—
105C3×#103—91no—
106Dic3⋊C8—94no—
107Dic3⋊C8—91no—
108#810—34no—
109#810C3⋊D311yescharacteristic
110C32×Q8—61no—
111C32×Q8—61no—
112C2×C32×C4—28no—
113C2×C32×C4—24no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—61no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—34no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—68no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—61no—
129#634—21no—
130#634—21no—
131#634—32no—
132#634—38no—
133#634—68no—
134#634—68no—
135#634—64no—
136#634—61no—
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—34no—
152C2×C3×Q8—91no—
153C2×Dic6—34no—
154C2×Dic6—94no—
155C2×Dic6—91no—
156C2×Dic6—184no—
157C2×Dic6C2×C3⋊D311yescharacteristic
158C2×C3×C8—34no—
159C2×C3×C8—91no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4D3214yes—
162C3×C4⋊C4C2×C3⋊D311yescharacteristic
163#216—182no—
164#216—188no—
165Dic3⋊C4—181no—
166Dic3⋊C4—181no—
167Dic3⋊C4—184no—
168Dic3⋊C4—184no—
169Dic3⋊C4—184no—
170Dic3⋊C4—181no—
171Dic3⋊C4—188no—
172C4×Dic3—94no—
173C4×Dic3—91no—
174C4×Dic3—184no—
175C4×Dic3—184no—
176C4×Dic3—181no—
177C4×Dic3—188no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8C2×C3⋊D311yescharacteristic
180Dic12—188no—
181Dic12—182no—
182C22×C32—24no—
183C22×C32D3×D414yes—
184C22×C32D3×D411yescharacteristic
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—48no—
188C32×C4—44no—
189C32×C4—41no—
190C32×C4—121no—
191C32×C4#23514yes—
192C32×C4#23511yescharacteristic
193C32×C4D3×D414yes—
194C32×C4D3×D411yescharacteristic
195C3⋊Dic3—32no—
196C3⋊Dic3—38no—
197C3⋊Dic3—41no—
198C3⋊Dic3—68no—
199C3⋊Dic3—616no—
200C3⋊Dic3—68no—
201C3⋊Dic3—62no—
202C3⋊Dic3—61no—
203C3⋊Dic3—64no—
204C3⋊Dic3—124no—
205C3×Dic3—44no—
206C3×Dic3—124no—
207C2×Q16—271no—
208C2×Q16—271no—
209C4⋊Q8—271no—
210C4×Q8—271no—
211C4⋊C8—271no—
212#103—31no—
213#103—91no—
214C33#46211yescharacteristic, complemented, Sylow
215C3×Q8—64no—
216C3×Q8—64no—
217C3×Q8—181no—
218C3×Q8—181no—
219C2×C3×C4—28no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—64no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—64no—
229C2×Dic3—94no—
230C2×Dic3—91no—
231C2×Dic3—94no—
232C2×Dic3—91no—
233C2×Dic3—184no—
234C2×Dic3—184no—
235C2×Dic3—188no—
236C2×Dic3—184no—
237C2×Dic3—181no—
238Dic6—21no—
239Dic6—21no—
240Dic6—64no—
241Dic6—64no—
242Dic6—98no—
243Dic6—92no—
244Dic6—188no—
245Dic6—188no—
246Dic6—184no—
247Dic6—181no—
248C3×C8—64no—
249C3×C8—181no—
250C3⋊C8—21no—
251C3⋊C8—64no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32#88414yes—
256C2×C32#88411yescharacteristic
257C2×C32#91514yes—
258C2×C32#91511yescharacteristic
259C2×C32D3×Q1614yes—
260C2×C32D3×Q1611yescharacteristic
261C2×Q8—31no—
262C2×Q8—91no—
263C2×Q8—271no—
264Q16—542no—
265Q16—542no—
266C2×C8—31no—
267C2×C8—91no—
268C4⋊C4—541no—
269C4⋊C4—541no—
270C4⋊C4—541no—
271C4⋊C4D3×C3⋊D311yescharacteristic
272C42—271no—
273C42—541no—
274C22×C3—24no—
275C22×C3#381114yes—
276C22×C3D4×C3⋊D311yescharacteristic
277C3×C4—24no—
278C3×C4—24no—
279C3×C4—48no—
280C3×C4—41no—
281C3×C4—44no—
282C3×C4—124no—
283C3×C4—361no—
284C3×C4#379814yes—
285C3×C4#380214yes—
286C3×C4#382811yescharacteristic
287C3×C4D4×C3⋊D311yescharacteristic
288Dic3—41no—
289Dic3—92no—
290Dic3—98no—
291Dic3—124no—
292Dic3—188no—
293Dic3—1816no—
294Dic3—182no—
295Dic3—181no—
296Dic3—188no—
297Dic3—184no—
298Dic3—364no—
299C32—24no—
300C32#479514yescomplemented
301C32#479511yescharacteristic, complemented
302Q8—61no—
303Q8—61no—
304Q8—181no—
305Q8—181no—
306Q8—272no—
307Q8—541no—
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#6391214yes—
321C2×C3#6393114yes—
322C2×C3#6394414yes—
323C2×C3#6409311yescharacteristic
324C2×C3#6412411yescharacteristic
325C2×C3Q16×C3⋊D311yescharacteristic
326C22#8903911yescharacteristic
327C4—41no—
328C4—121no—
329C4—272no—
330C4—361no—
331C4—542no—
332C4—541no—
333C4#8902011yescharacteristic
334C4#8902611yescharacteristic
335C3—24no—
336C3#1059056614yescomplemented
337C3#1059249411yescharacteristic, complemented
338C2#1172352711yescharacteristic
339C2#1172355411yescharacteristic
340C2#1172357711yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#462
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
92. C3⋊(C3⋊Dic3)Q16semidirect
214. C33#462semidirect
300. C32#4795semidirect
301. C32#4795semidirect
336. C3#10590566semidirect
337. C3#10592494semidirect

Maximal subgroups

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

subgroupindex
#117239282
#117234512
C2×#888002
C2×#887992
C3×#636372
C32×#8102
#117220502
#105924943
#105905663