Finite Group Databasebeta

← #13138942

Subgroups — #13138942

Order 1728.

545 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 79 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×C8×#272C211yesmaximal, characteristic, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic, complemented
3#11721159C212yesmaximal, complemented
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#11720318C212yesmaximal, complemented
6#10590832—31nomaximal
7#10590831—31nomaximal
8C2×C4×#272C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6200—22no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C8×#272—22no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#3745⋊C3—22no—
16#88450—22no—
17C2×#3745D311yescharacteristic, complemented
18C2×C3×D24—31no—
19C2×C3×D3×C8—31no—
20C2×C3×D3×C8—31no—
21C2×C3×D3×C8—61no—
22#63909—32no—
23#63908—32no—
24#63645D311yescharacteristic, complemented
25C3×#818—31no—
26#63564—32no—
27#63563—32no—
28C22×#272D411yescharacteristic
29C22×(C3⋊D3)⋊C3—22no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084D411yescharacteristic
32C4×#272—21no—
33C4×#272—21no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37#6200—44no—
38C8×H(3,𝔽3)D412yescomplemented
39#6180—21no—
40#5157—91no—
41#5115—91no—
42#4883—91no—
43C2×C4⋊(C3⋊D3)C2×D312yes—
44C2×C3×D12—32no—
45C2×C3×C4×D3—31no—
46C2×C3×C4×D3—31no—
47C2×C3×C4×D3—61no—
48C2×#626—62no—
49C2×#625—62no—
50C2×C32×C8—21no—
51C2×C32×C8C2×D311yescharacteristic
52C2×C32×C8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)C2×D312yes—
54#3745—22no—
55C3×C4⋊Dic3—32no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—61no—
59C3×D24—62no—
60C3×D3×C8—64no—
61C3×D3×C8—62no—
62C3×D3×C8—62no—
63#3722—62no—
64(C2×D3)⋊Dic3—62no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C2×#272—21no—
67C2×#272—21no—
68C2×#272D812yescomplemented
69C2×(C3⋊D3)⋊C3—42no—
70C2×(C3⋊D3)⋊C3—42no—
71C2×(C3⋊D3)⋊C3—42no—
72C4×H(3,𝔽3)C2×D411yescharacteristic
73C4×H(3,𝔽3)#4111yescharacteristic
74#1084—21no—
75(C3⋊Dic3)⋊C3—42no—
76C2×C3×D8—91no—
77C22×C3×C8—91no—
78#938—92no—
79#930—92no—
80C2×D24—31no—
81C2×D24—91no—
82C2×D3×C8—31no—
83C2×D3×C8—31no—
84C2×D3×C8—61no—
85#895—92no—
86C3×#107—91no—
87#832—92no—
88#821—92no—
89#818—31no—
90#818—91no—
91#809—92no—
92C22×C3⋊D3—22no—
93C22×C3×D3—31no—
94C22×C3×D3—31no—
95C22×C3×D3—61no—
96C22×C3×D3—62no—
97C2×C32×C4—21no—
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—22no—
101C4⋊(C3⋊D3)—22no—
102C4⋊(C3⋊D3)—22no—
103C2×C3×Dic3—31no—
104C2×C3×Dic3—31no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—62no—
107C3×D12—62no—
108C3×D12—62no—
109C3×C4×D3—62no—
110C3×C4×D3—62no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×C4×D3—61no—
114C3×C4×D3—61no—
115#626—124no—
116#625—124no—
117C32×C8—22no—
118C32×C8D1212yescomplemented
119C32×C8#6712yescomplemented
120C3×C3⋊C8—62no—
121C3×C3⋊C8—61no—
122C3×C3⋊C8—61no—
123#466—271noSylow
124C2×H(3,𝔽3)#12111yescharacteristic
125C2×H(3,𝔽3)C2×D811yescharacteristic
126C2×H(3,𝔽3)#13511yescharacteristic
127#272—22no—
128(C3⋊D3)⋊C3—82no—
129C2×C3×D4—92no—
130C22×C3×C4—91no—
131C2×#67—182no—
132C2×#67—182no—
133C2×D12—32no—
134C2×D12—92no—
135C2×D12—182no—
136C2×C4×D3—31no—
137C2×C4×D3—31no—
138C2×C4×D3—61no—
139C3×D8—182no—
140C2×C3×C8—31no—
141C2×C3×C8—31no—
142C2×C3×C8—61no—
143C2×C3×C8—91no—
144C2×C3×C8—182no—
145C2×C3×C8D3211yescharacteristic
146C3×C4⋊C4—92no—
147#217—182no—
148(C2×D3)⋊C4—182no—
149(C2×D3)⋊C4—182no—
150C4⋊Dic3—32no—
151C4⋊Dic3—92no—
152C2×C3⋊C8—31no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—61no—
155D24—62no—
156D24—182no—
157D3×C8—62no—
158D3×C8—62no—
159D3×C8—64no—
160C22×C32—21no—
161C22×C32D3×D411yescharacteristic
162C22×C32D3×D411yescharacteristic
163C2×C3⋊D3—42no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3—42no—
166C2×C3×D3—32no—
167C2×C3×D3—32no—
168C2×C3×D3—62no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C2×C3×D3—61no—
172C2×C3×D3—61no—
173C2×C3×D3—61no—
174C2×C3×D3—61no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C32×C4—21no—
179C32×C4—21no—
180C32×C4C2×D1211yescharacteristic
181C32×C4#23711yescharacteristic
182C32×C4#23911yescharacteristic
183C32×C4C2×#6711yescharacteristic
184C3⋊Dic3—42no—
185C3×Dic3—62no—
186C3×Dic3—61no—
187C3×Dic3—61no—
188C3×Dic3—122no—
189C2×D8—91no—
190C22×C8—91no—
191#121—272no—
192#107—91no—
193#102—272no—
194H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
195C23×C3—91no—
196C23×C3—182no—
197C22×D3—31no—
198C22×D3—31no—
199C22×D3—61no—
200C22×D3—62no—
201C22×D3—182no—
202C3×D4—182no—
203C3×D4—182no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—61no—
207C2×C3×C4—91no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4—182no—
211C2×C3×C4C2×D3211yescharacteristic
212#67—364no—
213#67—364no—
214C2×Dic3—31no—
215C2×Dic3—31no—
216C2×Dic3—61no—
217C2×Dic3—62no—
218C2×Dic3—182no—
219D12—62no—
220D12—62no—
221D12—182no—
222D12—182no—
223D12—364no—
224C4×D3—61no—
225C4×D3—61no—
226C4×D3—61no—
227C4×D3—61no—
228C4×D3—62no—
229C4×D3—62no—
230C3×C8—32no—
231C3×C8—32no—
232C3×C8—62no—
233C3×C8—181no—
234C3×C8#62612yes—
235C3⋊C8—61no—
236C3⋊C8—61no—
237C3⋊C8—62no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32#89511yescharacteristic
242C2×C32D3×D811yescharacteristic
243C2×C32D3×D811yescharacteristic
244C2×C32#91211yescharacteristic
245C2×C32#91911yescharacteristic
246C2×C32#93811yescharacteristic
247C3⋊D3—82no—
248C3×D3—64no—
249C3×D3—62no—
250C3×D3—62no—
251C3×D3—242no—
252C2×D4—92no—
253C2×D4—542no—
254C22×C4—91no—
255D8—182no—
256C2×C8—91no—
257C2×C8—182no—
258C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C4⋊C4—92no—
260C22⋊C4—542no—
261C22×C3—31no—
262C22×C3—31no—
263C22×C3—61no—
264C22×C3—92no—
265C22×C3—181no—
266C22×C3—181no—
267C22×C3—362no—
268C22×C3—362no—
269C22×C3—362no—
270C22×C3#380211yescharacteristic
271C2×D3—32no—
272C2×D3—32no—
273C2×D3—62no—
274C2×D3—61no—
275C2×D3—61no—
276C2×D3—61no—
277C2×D3—61no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—122no—
281C2×D3—122no—
282C2×D3—122no—
283C2×D3—362no—
284C2×D3—362no—
285C2×D3—362no—
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—61no—
291C3×C4—61no—
292C3×C4—181no—
293C3×C4—362no—
294C3×C4#379611yescharacteristic
295C3×C4C2×#62611yescharacteristic
296Dic3—61no—
297Dic3—61no—
298Dic3—62no—
299Dic3—122no—
300Dic3—362no—
301C32—21no—
302C32#488311yescharacteristic, complemented
303C32#515711yescharacteristic, complemented
304C23—91no—
305C23—182no—
306D4—182no—
307D4—182no—
308D4—1084no—
309C2×C4—91no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—182no—
313C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
314C8—181no—
315C8#620012yescomplemented
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—61no—
325C2×C3—182no—
326C2×C3—722no—
327C2×C3#6379711yescharacteristic
328C2×C3#6381011yescharacteristic
329C2×C3#6391611yescharacteristic
330D3—62no—
331D3—62no—
332D3—64no—
333D3—242no—
334D3—722no—
335C22—92no—
336C22—181no—
337C22—181no—
338C22—362no—
339C22—362no—
340C22—362no—
341C22#8865511yescharacteristic
342C4—181no—
343C4—362no—
344C4#8864911yescharacteristic
345C4C2×#620011yescharacteristic
346C3—31no—
347C3—31no—
348C3—61no—
349C3#1059084411yescharacteristic
350C2—182no—
351C2—722no—
352C2#1172083811yescharacteristic
353C2#1172084411yescharacteristic
354C2#1172115911yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
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×C8×#272C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #11721159C2semidirect
4. #63645⋊C3C2semidirect
5. #11720318C2semidirect
17. C2×#3745D3semidirect
24. #63645D3semidirect
38. C8×H(3,𝔽3)D4semidirect
68. C2×#272D8semidirect
118. C32×C8D12semidirect
119. C32×C8#67semidirect
194. H(3,𝔽3)#466semidirect
302. C32#4883semidirect
303. C32#5157semidirect
315. C8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
C2×#3745⋊C32
#117211592
#63645⋊C32
#117203182
#105908323
#105908313