Finite Group Databasebeta

← #13161177

Subgroups — #13161177

Order 1728.

888 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 166 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×#88863C211yesmaximal, characteristic
2C32×#944C211yesmaximal, characteristic
3#11723399C211yesmaximal, characteristic
4C2×#88799C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592789—31nomaximal
9#10591575—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13#88863—22no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88806—21no—
21Dic3×C3⋊Dic3—21no—
22#88799—22no—
23C32×C4⋊Q8—31no—
24C2×#3756—31no—
25C2×#3756—34no—
26C2×#3756—31no—
27C2×#3756—64no—
28C4×#634—31no—
29C3×#944D314yes—
30#63844—34no—
31C2×#3719—34no—
32#63662—31no—
33#63662D311yescharacteristic
34C4⋊(C3⋊(C3⋊C8))—31no—
35C3×#835—34no—
36Dic3⋊(C3⋊C8)—34no—
37#63565—34no—
38C33×Q8—21no—
39C33×Q8—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308D412yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53#5090—94no—
54#4795—91no—
55C2×C32×Q8—24no—
56C2×C32×Q8—31no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×Dic6C2×D314yes—
63C32×C4⋊C4—61no—
64C32×C4⋊C4—61no—
65C32×C42—31no—
66#3756—62no—
67#3756—62no—
68#3756—68no—
69#3756—616no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71#3750—61no—
72C4×C3⋊Dic3—61no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C3×C4⋊Dic3—34no—
78C3×Dic3⋊C4—24no—
79C3×Dic3⋊C4—24no—
80C3×C4×Dic3C2×D314yes—
81C2×C3×C3⋊C8—34no—
82#3724—64no—
83Dic32—64no—
84#3719—68no—
85C22×C33C2×D411yescharacteristic
86C33×C4—41no—
87C33×C4C2×D411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—22no—
90C3×C3⋊Dic3—41no—
91C32×Dic3—21no—
92C32×Dic3—41no—
93C32×Dic3Q1612yescomplemented
94C3×C4⋊Q8—34no—
95#944C3⋊D311yescharacteristic
96C2×#216—94no—
97C2×#216—94no—
98C2×#216—91no—
99C2×#216—184no—
100C2×Dic12—91no—
101#887—91no—
102C4×Dic6—94no—
103C3×#103—91no—
104#835—34no—
105#835—94no—
106Dic3⋊C8—91no—
107#810—91no—
108C4⋊(C3⋊C8)—94no—
109C32×Q8—28no—
110C32×Q8—21no—
111C32×Q8—24no—
112C32×Q8—21no—
113C32×Q8—24no—
114C32×Q8—44no—
115C32×Q8—61no—
116C32×Q8—61no—
117C2×C32×C4—21no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—31no—
121C2×C32×C4—31no—
122C2×C32×C4—44no—
123C2×C32×C4—61no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—124no—
130#634—32no—
131#634—38no—
132#634—68no—
133#634—61no—
134#634—64no—
135#634—124no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—64no—
138C2×C3×Dic3#6714yes—
139C2×C3×Dic3#6714yes—
140C3×Dic6—24no—
141C3×Dic6—24no—
142C3⋊(C3⋊C8)—21no—
143C3⋊(C3⋊C8)—68no—
144C3⋊(C3⋊C8)—64no—
145C3⋊(C3⋊C8)—61no—
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×Q8D3214yes—
154C2×C3×Q8C2×C3⋊D311yescharacteristic
155C2×Dic6—94no—
156C2×Dic6—91no—
157C2×Dic6—184no—
158C2×Dic6C2×C3⋊D311yescharacteristic
159C2×C3×C8—91no—
160C3×C4⋊C4—64no—
161C3×C4⋊C4—64no—
162C3×C4⋊C4—91no—
163C3×C42—34no—
164#216—188no—
165#216—188no—
166#216—182no—
167#216—1816no—
168C4⋊Dic3—34no—
169Dic3⋊C4—21no—
170Dic3⋊C4—21no—
171Dic3⋊C4—184no—
172Dic3⋊C4—181no—
173C4×Dic3—184no—
174C4×Dic3—181no—
175C4×Dic3C2×C3⋊D311yescharacteristic
176C2×C3⋊C8—34no—
177C2×C3⋊C8—94no—
178C2×C3⋊C8—91no—
179C2×C3⋊C8—184no—
180Dic12—182no—
181C22×C32—24no—
182C22×C32D3×D411yescharacteristic
183C22×C32C2×#6714yes—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—32no—
187C32×C4—48no—
188C32×C4—41no—
189C32×C4—44no—
190C32×C4—61no—
191C32×C4—121no—
192C32×C4#23511yescharacteristic
193C32×C4D3×D414yes—
194C32×C4D3×D411yescharacteristic
195C32×C4#23714yes—
196C3⋊Dic3—41no—
197C3⋊Dic3—62no—
198C3⋊Dic3—68no—
199C3⋊Dic3—128no—
200C3×Dic3—24no—
201C3×Dic3—44no—
202C3×Dic3—124no—
203C3×Dic3#21618yescomplemented
204C2×Q16—271no—
205C2×Q16—271no—
206C4⋊Q8—31no—
207C4×Q8—271no—
208C4⋊C8—271no—
209#103—91no—
210#103—271no—
211C33#46211yescharacteristic, complemented, Sylow
212C3×Q8—28no—
213C3×Q8—24no—
214C3×Q8—24no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—44no—
218C3×Q8—64no—
219C3×Q8—64no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—34no—
224C2×C3×C4—34no—
225C2×C3×C4—44no—
226C2×C3×C4—64no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—64no—
232C2×Dic3—184no—
233C2×Dic3—181no—
234C2×Dic3—364no—
235C2×Dic3#63811yescharacteristic
236C2×Dic3#63811yescharacteristic
237Dic6—21no—
238Dic6—21no—
239Dic6—98no—
240Dic6—92no—
241Dic6—188no—
242Dic6—184no—
243Dic6—181no—
244Dic6—364no—
245C3×C8—181no—
246C3⋊C8—64no—
247C3⋊C8—188no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32#88411yescharacteristic
254C2×C32#91511yescharacteristic
255C2×C32D3×Q1611yescharacteristic
256C2×C32#93814yes—
257C2×C32#94214yes—
258C2×C32C2×#21614yes—
259C2×Q8—31no—
260C2×Q8—271no—
261C2×Q8D3×C3⋊D311yescharacteristic
262Q16—542no—
263Q16—542no—
264C2×C8—91no—
265C2×C8—271no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—91no—
269C4⋊C4—541no—
270C42—31no—
271C42—541no—
272C22×C3—24no—
273C22×C3D3×#6714yes—
274C22×C3C2×#63811yescharacteristic
275C3×C4—24no—
276C3×C4—24no—
277C3×C4—38no—
278C3×C4—48no—
279C3×C4—41no—
280C3×C4—44no—
281C3×C4—64no—
282C3×C4—124no—
283C3×C4—361no—
284C3×C4#380414yes—
285C3×C4#381114yes—
286C3×C4D4×C3⋊D311yescharacteristic
287C3×C4#383011yescharacteristic
288Dic3—21no—
289Dic3—41no—
290Dic3—124no—
291Dic3—182no—
292Dic3—188no—
293Dic3—368no—
294Dic3#375612yescomplemented
295C32—24no—
296C32#479511yescharacteristic, complemented
297C32#509014yescomplemented
298Q8—21no—
299Q8—21no—
300Q8—61no—
301Q8—61no—
302Q8—272no—
303Q8—541no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—61no—
308C2×C4—181no—
309C2×C4—541no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—181no—
312C8—541no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3D3×#21614yes—
317C2×C3#6394614yes—
318C2×C3#6395814yes—
319C2×C3#6414711yescharacteristic
320C2×C3#6415111yescharacteristic
321C2×C3C2×#375611yescharacteristic
322C22D3×#63811yescharacteristic
323C4—32no—
324C4—41no—
325C4—61no—
326C4—121no—
327C4—361no—
328C4—542no—
329C4#8902811yescharacteristic
330C4#8903911yescharacteristic
331C3—24no—
332C3#1059157514yescomplemented
333C3#1059278911yescharacteristic, complemented
334C2D3×#375611yescharacteristic
335C2#1172357911yescharacteristic
336C2#1172359911yescharacteristic
subgroupG/H
centerC22D3×#638
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
93. C32×Dic3Q16semidirect
203. C3×Dic3#216semidirect
211. C33#462semidirect
294. Dic3#3756semidirect
296. C32#4795semidirect
297. C32#5090semidirect
332. C3#10591575semidirect
333. C3#10592789semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C32×#9442
#117233992
C2×#887992
C3×#636622
Dic3⋊(C3⋊(C3⋊C8))2
#117220372
#105927893
#105915753