Finite Group Databasebeta

← #13159996

Subgroups — #13159996

Order 1728.

852 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 176 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×D3×C8C211yesmaximal, characteristic
2#11723426C211yesmaximal, characteristic
3#11723425C211yesmaximal, characteristic
4C2×#88800C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7#11722055C211yesmaximal, characteristic, complemented
8#10592815—31nomaximal
9#10590848—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C32×D3×C8—22no—
18#88807—21no—
19#88806—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88800—22no—
23C22×C32×C8—31no—
24#64133—31no—
25#64133—31no—
26C2×#3746—31no—
27C2×C3×D3×C8D314yes—
28#63857—34no—
29#63856—34no—
30C2×#3719—34no—
31#63645—31no—
32#63643—31no—
33#63643—34no—
34#63643—64no—
35#63643D311yescharacteristic, complemented
36C3×#816—34no—
37#63575—34no—
38C22×C32×D3D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3D411yescharacteristic
49C32×C4×D3—21no—
50C32×C4×D3—21no—
51C33×C8—21no—
52C32×C3⋊C8D412yes—
53#5116—94no—
54#4813—91no—
55C22×C32×C4—31no—
56C2×#634—31no—
57C2×#634—34no—
58C2×#634—64no—
59C2×#634C2×D311yescharacteristic
60C2×C3×C4×D3C2×D314yes—
61C2×C3×Dic6—34no—
62C2×C32×C8—24no—
63C2×C32×C8—31no—
64C2×C32×C8—62no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C2×D311yescharacteristic
67#3757—61no—
68#3757—61no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73#3750—61no—
74#3750—61no—
75#3746—62no—
76C3×C4⋊Dic3—34no—
77C2×C3×C3⋊C8C2×D314yes—
78C3×D3×C8—28no—
79#3724—64no—
80Dic3⋊Dic3—64no—
81(C2×D3)⋊Dic3—64no—
82(C2×D3)⋊Dic3—64no—
83#3719—68no—
84C22×C33C2×D411yescharacteristic
85C2×C32×D3—21no—
86C2×C32×D3—21no—
87C2×C32×D3Q1612yescomplemented
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—41no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—21no—
95C22×C3×C8—34no—
96#946—91no—
97C2×#216—91no—
98#924—94no—
99#924—94no—
100C2×Dic12—94no—
101C2×D3×C8C3⋊D311yescharacteristic
102#897—91no—
103C3×#103—91no—
104#818—94no—
105#816—34no—
106#816—94no—
107#816—91no—
108#816—184no—
109#807—91no—
110C23×C32—31no—
111C22×C3×D3D1214yes—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—61no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—124no—
125C2×C3⋊Dic3—124no—
126#634—21no—
127#634—21no—
128#634—68no—
129#634—64no—
130#634—61no—
131#634—61no—
132#634—64no—
133#634—124no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3D1214yes—
137C3×C4×D3—24no—
138C3×C4×D3—24no—
139C3×Dic6—64no—
140C3×Dic6—64no—
141C32×C8—21no—
142C32×C8—24no—
143C32×C8—32no—
144C32×C8—44no—
145C3×C3⋊C8#6718yes—
146#467—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33C2×Q1611yescharacteristic
149C2×C33#13511yescharacteristic
150C32×D3—22no—
151C2×C3×Q8—91no—
152C22×C3×C4—34no—
153C2×C4×D3C2×C3⋊D311yescharacteristic
154C2×Dic6—34no—
155C2×Dic6—94no—
156C2×Dic6—91no—
157C2×Dic6—184no—
158C2×C3×C8—24no—
159C2×C3×C8—34no—
160C2×C3×C8—68no—
161C2×C3×C8D3214yes—
162C2×C3×C8C2×C3⋊D311yescharacteristic
163C3×C4⋊C4—91no—
164#217—184no—
165#217—184no—
166#216—182no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—34no—
170C4⋊Dic3—94no—
171C4⋊Dic3—91no—
172C4⋊Dic3—184no—
173C4⋊Dic3—181no—
174Dic3⋊C4—184no—
175Dic3⋊C4—184no—
176Dic3⋊C4—181no—
177C2×C3⋊C8C2×C3⋊D311yescharacteristic
178Dic12—188no—
179D3×C8—22no—
180C22×C32—24no—
181C22×C32—32no—
182C22×C32—61no—
183C22×C32—61no—
184C22×C32C2×D1214yes—
185C22×C32D3×D411yescharacteristic
186C2×C3×D3—24no—
187C2×C3×D3—24no—
188C2×C3×D3Dic1218yescomplemented
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—61no—
192C32×C4#23514yes—
193C32×C4#23511yescharacteristic
194C32×C4D3×D411yescharacteristic
195C32×C4C2×#6714yes—
196C3⋊Dic3—41no—
197C3⋊Dic3—41no—
198C3⋊Dic3—128no—
199C3⋊Dic3—121no—
200C3⋊Dic3—121no—
201C3⋊Dic3—124no—
202C3⋊Dic3—124no—
203C3⋊Dic3—244no—
204C3×Dic3—24no—
205C3×Dic3—124no—
206C3×Dic3—124no—
207C2×Q16—271no—
208C22×C8—31no—
209C22⋊Q8—271no—
210C22⋊Q8—271no—
211#107—271no—
212#103—91no—
213#103—271no—
214C33#46711yescharacteristic, complemented, Sylow
215C23×C3—34no—
216C22×D3C4⋊(C3⋊D3)11yescharacteristic
217C3×Q8—181no—
218C3×Q8—181no—
219C2×C3×C4—24no—
220C2×C3×C4—34no—
221C2×C3×C4—64no—
222C2×C3×C4—64no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—64no—
228C2×Dic3—64no—
229C2×Dic3—184no—
230C2×Dic3—184no—
231C2×Dic3—181no—
232C2×Dic3—181no—
233C2×Dic3—364no—
234C2×Dic3—364no—
235C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
236C4×D3—21no—
237C4×D3—21no—
238Dic6—64no—
239Dic6—64no—
240Dic6—188no—
241Dic6—184no—
242Dic6—184no—
243Dic6—181no—
244Dic6—181no—
245Dic6—364no—
246C3×C8—24no—
247C3×C8—21no—
248C3×C8—38no—
249C3×C8—44no—
250C3⋊C8#63812yes—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—62no—
255C2×C32#88411yescharacteristic
256C2×C32#90414yes—
257C2×C32C2×Dic1214yes—
258C2×C32#91611yescharacteristic
259C2×C32D3×Q1611yescharacteristic
260C2×C32#93014yes—
261C3×D3—28no—
262C2×Q8—91no—
263C2×Q8—271no—
264C22×C4—31no—
265Q16—542no—
266C2×C8—31no—
267C2×C8—62no—
268C2×C8D3×C3⋊D311yescharacteristic
269C4⋊C4—91no—
270C4⋊C4—271no—
271C4⋊C4—541no—
272C4⋊C4—541no—
273C22⋊C4—541no—
274C22⋊C4—541no—
275C22×C3—24no—
276C22×C3—38no—
277C22×C3—64no—
278C22×C3—64no—
279C22×C3D3×D1214yes—
280C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3#374612yescomplemented
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—64no—
287C3×C4—361no—
288C3×C4—361no—
289C3×C4#379814yes—
290C3×C4D3×#6714yes—
291C3×C4#382811yescharacteristic
292C3×C4C2×#63811yescharacteristic
293Dic3—21no—
294Dic3—124no—
295Dic3—124no—
296Dic3—368no—
297Dic3—364no—
298Dic3—361no—
299Dic3—361no—
300Dic3—364no—
301Dic3—724no—
302C32—24no—
303C32#481311yescharacteristic, complemented
304C32#511614yescomplemented
305C23—31no—
306Q8—181no—
307Q8—181no—
308Q8—541no—
309Q8—541no—
310C2×C4—31no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—541no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—21no—
319C8—32no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—68no—
324C2×C3D3×Dic1214yes—
325C2×C3#6380614yes—
326C2×C3#6390614yes—
327C2×C3#6411311yescharacteristic
328C2×C3C2×#374611yescharacteristic
329C2×C3#6413911yescharacteristic
330D3—22no—
331C22—32no—
332C22—61no—
333C22—61no—
334C22D3×C4⋊(C3⋊D3)11yescharacteristic
335C4—61no—
336C4—361no—
337C4—361no—
338C4—1081no—
339C4—1081no—
340C4#8902011yescharacteristic
341C4D3×#63811yescharacteristic
342C3—24no—
343C3#1059084814yescomplemented
344C3#1059281511yescharacteristic, complemented
345C2—62no—
346C2D3×#374611yescharacteristic
347C2#1172333511yescharacteristic
348C2#1172351411yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#467
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
4. C2×#88800C2semidirect
5. #11722186C2semidirect
6. C3×#63643C2semidirect
7. #11722055C2semidirect
35. #63643D3semidirect
87. C2×C32×D3Q16semidirect
188. C2×C3×D3Dic12semidirect
214. C33#467semidirect
283. C2×D3#3746semidirect
303. C32#4813semidirect
304. C32#5116semidirect
343. C3#10590848semidirect
344. C3#10592815semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×C82
#117234262
#117234252
C2×#888002
#117221862
C3×#636432
#117220552
#105928153
#105908483