Finite Group Databasebeta

← #13138973

Subgroups — #13138973

Order 1728.

545 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88529C211yesmaximal, characteristic, complemented
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#11721156C212yesmaximal, complemented
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#11720324C212yesmaximal, complemented
6#10590844—31nomaximal
7#10590831—31nomaximal
8C2×C4⋊#272C2212yes—
9C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×#6200—22no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12#88529—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C8×(C3⋊D3)⋊C3—22no—
16#88451—22no—
17C2×C8×C3⋊D3D311yescharacteristic, complemented
18C2×C3×D24—31no—
19C2×C3×D24—31no—
20C2×C3×D24—61no—
21C2×C3×D3×C8—31no—
22#63916—32no—
23#63908—32no—
24#63645D311yescharacteristic, complemented
25C3×#818—31no—
26#63563—32no—
27#63562—32no—
28C22×#272—22no—
29C22×(C3⋊D3)⋊C3D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C4⋊#272—22no—
32C4⋊#272—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3D411yescharacteristic
35C4×(C3⋊D3)⋊C3—21no—
36C4×(C3⋊D3)⋊C3—21no—
37#6200—44no—
38C8×H(3,𝔽3)D412yescomplemented
39(C3⋊(C3⋊C8))⋊C3—21no—
40#5157—91no—
41#5115—91no—
42#4883—91no—
43C2×C4×C3⋊D3C2×D311yescharacteristic
44C2×C3×D12—32no—
45C2×C3×D12—32no—
46C2×C3×D12—62no—
47C2×C3×C4×D3—31no—
48C2×#626—62no—
49C2×#625—62no—
50C2×C32×C8—21no—
51C2×C32×C8C2×D311yescharacteristic
52C2×C32×C8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)C2×D312yes—
54C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
55C8×C3⋊D3—22no—
56C3×C4⋊Dic3—32no—
57C2×C3×C3⋊C8—31no—
58C3×D24—64no—
59C3×D24—62no—
60C3×D24—62no—
61C3×D3×C8—62no—
62(C2×D3)⋊Dic3—62no—
63(C2×D3)⋊Dic3—62no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—42no—
66C2×#272—42no—
67C2×#272—42no—
68C2×(C3⋊D3)⋊C3—21no—
69C2×(C3⋊D3)⋊C3—21no—
70C2×(C3⋊D3)⋊C3D812yescomplemented
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C4×H(3,𝔽3)#4111yescharacteristic
73(C3⋊Dic3)⋊C3—21no—
74(C3⋊Dic3)⋊C3—42no—
75C2×C3×D8—91no—
76C22×C3×C8—91no—
77#938—92no—
78#930—92no—
79C2×D24—31no—
80C2×D24—31no—
81C2×D24—61no—
82C2×D3×C8—31no—
83C2×D3×C8—91no—
84#895—92no—
85C3×#107—91no—
86#832—92no—
87#821—92no—
88#818—31no—
89#818—91no—
90#809—92no—
91C22×C3⋊D3D1211yescharacteristic
92C22×C3×D3—31no—
93C22×C3×D3—62no—
94C22×C3×D3—62no—
95C22×C3×D3—122no—
96C2×C32×C4—21no—
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3D1211yescharacteristic
101C4×C3⋊D3—21no—
102C4×C3⋊D3—21no—
103C2×C3×Dic3—31no—
104C2×C3×Dic3—62no—
105C3×D12—64no—
106C3×D12—62no—
107C3×D12—62no—
108C3×D12—62no—
109C3×D12—62no—
110C3×D12—122no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113#626—124no—
114#625—124no—
115C32×C8—22no—
116C32×C8D1212yescomplemented
117C32×C8#6712yescomplemented
118C3⋊(C3⋊C8)—21no—
119C3×C3⋊C8—61no—
120#466—271noSylow
121C2×H(3,𝔽3)#12111yescharacteristic
122C2×H(3,𝔽3)C2×D811yescharacteristic
123C2×H(3,𝔽3)#13511yescharacteristic
124#272—82no—
125(C3⋊D3)⋊C3—22no—
126C2×C3×D4—92no—
127C22×C3×C4—91no—
128C2×#67—182no—
129C2×#67—182no—
130C2×D12—32no—
131C2×D12—32no—
132C2×D12—62no—
133C2×D12—182no—
134C2×C4×D3—31no—
135C2×C4×D3—91no—
136C3×D8—182no—
137C2×C3×C8—31no—
138C2×C3×C8—31no—
139C2×C3×C8—61no—
140C2×C3×C8—91no—
141C2×C3×C8—182no—
142C2×C3×C8D3211yescharacteristic
143C3×C4⋊C4—92no—
144#217—182no—
145(C2×D3)⋊C4—182no—
146(C2×D3)⋊C4—182no—
147C4⋊Dic3—32no—
148C4⋊Dic3—92no—
149C2×C3⋊C8—31no—
150C2×C3⋊C8—91no—
151D24—62no—
152D24—62no—
153D24—64no—
154D3×C8—62no—
155D3×C8—182no—
156C22×C32—21no—
157C22×C32C2×D1211yescharacteristic
158C22×C32D3×D411yescharacteristic
159C2×C3⋊D3—21no—
160C2×C3⋊D3—21no—
161C2×C3⋊D3D2412yescomplemented
162C2×C3×D3—32no—
163C2×C3×D3—61no—
164C2×C3×D3—61no—
165C2×C3×D3—124no—
166C2×C3×D3—122no—
167C2×C3×D3—122no—
168C2×C3×D3—122no—
169C2×C3×D3—122no—
170C2×C3×D3—122no—
171C2×C3×D3—122no—
172C2×C3×D3—242no—
173C2×C3×D3—242no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4C2×D1211yescharacteristic
177C32×C4#23511yescharacteristic
178C32×C4#23911yescharacteristic
179C32×C4C2×#6711yescharacteristic
180C3⋊Dic3—21no—
181C3⋊Dic3—42no—
182C3×Dic3—61no—
183C3×Dic3—122no—
184C2×D8—91no—
185C22×C8—91no—
186#121—272no—
187#107—91no—
188#102—272no—
189H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
190C23×C3—91no—
191C23×C3—182no—
192C22×D3—31no—
193C22×D3—62no—
194C22×D3—62no—
195C22×D3—91no—
196C22×D3—122no—
197C3×D4—182no—
198C3×D4—182no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—91no—
203C2×C3×C4—181no—
204C2×C3×C4—181no—
205C2×C3×C4—182no—
206C2×C3×C4C2×D3211yescharacteristic
207#67—364no—
208#67—364no—
209C2×Dic3—31no—
210C2×Dic3—62no—
211C2×Dic3—91no—
212C2×Dic3—182no—
213D12—62no—
214D12—62no—
215D12—62no—
216D12—62no—
217D12—64no—
218D12—122no—
219D12—364no—
220C4×D3—61no—
221C4×D3—61no—
222C4×D3—181no—
223C4×D3—181no—
224C3×C8—32no—
225C3×C8—32no—
226C3×C8—62no—
227C3×C8—181no—
228C3×C8#62612yes—
229C3⋊C8—61no—
230C3⋊C8—181no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32#89511yescharacteristic
235C2×C32C2×D2411yescharacteristic
236C2×C32#90411yescharacteristic
237C2×C32D3×D811yescharacteristic
238C2×C32#91911yescharacteristic
239C2×C32#93011yescharacteristic
240C3⋊D3—22no—
241C3×D3—62no—
242C3×D3—244no—
243C3×D3—242no—
244C3×D3—242no—
245C2×D4—92no—
246C2×D4—542no—
247C22×C4—91no—
248D8—182no—
249C2×C8—91no—
250C2×C8—182no—
251C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C4⋊C4—92no—
253C22⋊C4—542no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—61no—
257C22×C3—92no—
258C22×C3—181no—
259C22×C3—181no—
260C22×C3—362no—
261C22×C3—362no—
262C22×C3—362no—
263C22×C3D3×D1211yescharacteristic
264C2×D3—32no—
265C2×D3—61no—
266C2×D3—61no—
267C2×D3—92no—
268C2×D3—122no—
269C2×D3—122no—
270C2×D3—122no—
271C2×D3—122no—
272C2×D3—122no—
273C2×D3—122no—
274C2×D3—124no—
275C2×D3—181no—
276C2×D3—181no—
277C2×D3—242no—
278C2×D3—242no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—61no—
284C3×C4—61no—
285C3×C4—181no—
286C3×C4—362no—
287C3×C4#379911yescharacteristic
288C3×C4C2×#62611yescharacteristic
289Dic3—61no—
290Dic3—122no—
291Dic3—181no—
292Dic3—362no—
293C32—21no—
294C32#488311yescharacteristic, complemented
295C32#511511yescharacteristic, complemented
296C23—91no—
297C23—182no—
298D4—182no—
299D4—182no—
300D4—1084no—
301C2×C4—91no—
302C2×C4—181no—
303C2×C4—181no—
304C2×C4—182no—
305C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C8—181no—
307C8#620012yescomplemented
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—61no—
315C2×C3—61no—
316C2×C3—61no—
317C2×C3—182no—
318C2×C3—722no—
319C2×C3D3×D2411yescharacteristic
320C2×C3#6380811yescharacteristic
321C2×C3#6390911yescharacteristic
322D3—62no—
323D3—182no—
324D3—242no—
325D3—242no—
326D3—244no—
327C22—92no—
328C22—181no—
329C22—181no—
330C22—362no—
331C22—362no—
332C22—362no—
333C22#8865411yescharacteristic
334C4—181no—
335C4—362no—
336C4#8865211yescharacteristic
337C4C2×#620011yescharacteristic
338C3—31no—
339C3—31no—
340C3—61no—
341C3#1059083211yescharacteristic
342C2—182no—
343C2—722no—
344C2#1172084711yescharacteristic
345C2#1172085011yescharacteristic
346C2#1172115611yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#466
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88529C2semidirect
2. C2×C8×(C3⋊D3)⋊C3C2semidirect
3. #11721156C2semidirect
4. #63645⋊C3C2semidirect
5. #11720324C2semidirect
17. C2×C8×C3⋊D3D3semidirect
24. #63645D3semidirect
38. C8×H(3,𝔽3)D4semidirect
70. C2×(C3⋊D3)⋊C3D8semidirect
116. C32×C8D12semidirect
117. C32×C8#67semidirect
161. C2×C3⋊D3D24semidirect
189. H(3,𝔽3)#466semidirect
294. C32#4883semidirect
295. C32#5115semidirect
307. C8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
C2×C8×(C3⋊D3)⋊C32
#117211562
#63645⋊C32
#117203242
#105908443
#105908313