Finite Group Databasebeta

← #13139251

Subgroups — #13139251

Order 1728.

341 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 53 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
1D8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2#11720855C211yesmaximal, characteristic, complemented
3#11720566C211yesmaximal, characteristic, complemented
4#63653⋊C3C211yesmaximal, characteristic, complemented
5#11720255C211yesmaximal, characteristic, complemented
6#11720248C211yesmaximal, characteristic, complemented
7#11720247C211yesmaximal, characteristic, complemented
8#10591036—31nomaximal
9#10591033—31nomaximal
10D4×(C3⋊D3)⋊C3—21no—
11#88648—21no—
12D8×H(3,𝔽3)C2211yescharacteristic
13#88530C2211yescharacteristic, complemented
14#3753⋊C3—21no—
15#3745⋊C3C2211yescharacteristic
16#3745⋊C3C2211yescharacteristic, complemented
17C8×(C3⋊D3)⋊C3C2211yescharacteristic
18#88438—21no—
19#88386C2211yescharacteristic, complemented
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic, complemented
21D8×C3⋊D3D311yescharacteristic, complemented
22C3×D3×D8—31no—
23#63810—31no—
24#63808—31no—
25#63653D311yescharacteristic, complemented
26C3×#827—31no—
27C3×#827—31no—
28C3×#827—61no—
29C3×#826—31no—
30#63549—31no—
31#63547—31no—
32#63546—31no—
33#63546—31no—
34#63544—31no—
35#63543—31no—
36C22×(C3⋊D3)⋊C3—41no—
37D4×H(3,𝔽3)—21no—
38#6229—21no—
39#638⋊C3—41no—
40C4⋊((C3⋊D3)⋊C3)—21no—
41C4⋊((C3⋊D3)⋊C3)—21no—
42C4×(C3⋊D3)⋊C3D411yescharacteristic
43#6198—41no—
44((C3⋊D3)⋊C3)⋊C4—41no—
45C8×H(3,𝔽3)C2311yescharacteristic
46(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
47#5147—91no—
48#4914—91no—
49#4911—91no—
50D4×C3⋊D3—21no—
51C3×D3×D4—61no—
52#3799—61no—
53#3796—61no—
54C32×D8—21no—
55C32×D8C2×D311yescharacteristic
56C32×D8C2×D311yescharacteristic
57#3753—21no—
58#3745C2×D311yescharacteristic
59#3745C2×D311yescharacteristic, complemented
60C8×C3⋊D3C2×D311yescharacteristic
61C3×#213—61no—
62C3×Dic12—31no—
63C3×Dic12—31no—
64C3×Dic12—61no—
65C3×D24—31no—
66C3×D24—31no—
67C3×D3×C8—31no—
68#3717—61no—
69#3715—61no—
70C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
71C3×C3⋊C16—31no—
72C3×C3⋊C16—31no—
73C3×C3⋊C16—31no—
74C3×C3⋊C16—61no—
75C22×H(3,𝔽3)—41no—
76C2×(C3⋊D3)⋊C3—41no—
77C2×(C3⋊D3)⋊C3—41no—
78C2×(C3⋊D3)⋊C3—82no—
79C2×(C3⋊D3)⋊C3D811yescharacteristic
80C4×H(3,𝔽3)C2×D411yescharacteristic
81#1084—41no—
82(C3⋊Dic3)⋊C3D811yescharacteristic
83C2×C3×D8—91no—
84#919—91no—
85#912—91no—
86D3×D8—31no—
87D3×D8—91no—
88#904—91no—
89C3×SD32—91no—
90C3×D16—91no—
91#828—91no—
92#827—31no—
93#827—31no—
94#827—61no—
95#827—91no—
96#826—31no—
97#826—91no—
98#826—91no—
99#826—91no—
100#812—91no—
101#800—91no—
102D48—91no—
103#798—91no—
104#798—91no—
105C22×C3⋊D3—41no—
106C22×C3×D3—121no—
107C32×D4—21no—
108C32×D4—21no—
109C32×D4—41no—
110#638—41no—
111C4⋊(C3⋊D3)—21no—
112C4⋊(C3⋊D3)—21no—
113C4×C3⋊D3#6711yescharacteristic
114C3×#67—121no—
115C3×D12—61no—
116C3×D12—61no—
117C3×C4×D3—31no—
118C3×Dic6—61no—
119C3×Dic6—61no—
120C3×Dic6—121no—
121#626—121no—
122#626—121no—
123#624—121no—
124D3×Dic3—121no—
125C32×C8—21no—
126C32×C8C22×D311yescharacteristic
127C32×C8C22×D311yescharacteristic
128C3⋊(C3⋊C8)#6711yescharacteristic
129C3×C3⋊C8—31no—
130#509—271noSylow
131C2×H(3,𝔽3)—81no—
132C2×H(3,𝔽3)C2×D811yescharacteristic
133(C3⋊D3)⋊C3—21no—
134(C3⋊D3)⋊C3—81no—
135(C3⋊D3)⋊C3—81no—
136C2×C3×D4—181no—
137#239—181no—
138#237—181no—
139D3×D4—61no—
140D3×D4—181no—
141#235—181no—
142C3×Q16—91no—
143C3×D8—31no—
144C3×D8—31no—
145C3×D8—61no—
146C3×D8—91no—
147C3×D8—91no—
148C3×D8—181no—
149C3×D8D3211yescharacteristic
150C2×C3×C8—91no—
151#215—181no—
152#214—181no—
153#213—61no—
154#213—181no—
155Dic12—31no—
156Dic12—31no—
157Dic12—61no—
158D24—31no—
159D24—31no—
160D24—91no—
161D24—91no—
162#204—181no—
163D3×C8—31no—
164D3×C8—91no—
165C3×C16—91no—
166C3×C16—91no—
167C3⋊C16—31no—
168C3⋊C16—31no—
169C3⋊C16—31no—
170C3⋊C16—61no—
171C3⋊C16—91no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32—81no—
175C2×C3⋊D3—41no—
176C2×C3⋊D3—41no—
177C2×C3⋊D3—82no—
178C2×C3⋊D3#21311yescharacteristic
179C2×C3×D3—31no—
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C2×C3×D3—242no—
183C32×C4—21no—
184C32×C4D3×D411yescharacteristic
185C32×C4C2×#6711yescharacteristic
186C3⋊Dic3#21311yescharacteristic
187C3×Dic3—31no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C3×Dic3—241no—
191#135—271no—
192C2×D8—91no—
193SD32—91no—
194SD32—271no—
195D16—91no—
196D16—271no—
197M32—271no—
198H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
199C23×C3—361no—
200C22×D3—121no—
201C22×D3—361no—
202C3×Q8—181no—
203C3×D4—21no—
204C3×D4—61no—
205C3×D4—61no—
206C3×D4—121no—
207C3×D4—181no—
208C3×D4—181no—
209C3×D4—361no—
210C2×C3×C4—91no—
211#67—121no—
212#67—361no—
213#67—361no—
214#67—361no—
215C2×Dic3—361no—
216D12—61no—
217D12—61no—
218D12—181no—
219D12—181no—
220D12—361no—
221C4×D3—31no—
222C4×D3—91no—
223C4×D3—361no—
224C4×D3—361no—
225Dic6—61no—
226Dic6—61no—
227Dic6—121no—
228C3×C8—31no—
229C3×C8—31no—
230C3×C8—61no—
231C3×C8—91no—
232C3×C8C2×D3211yescharacteristic
233C3⋊C8—31no—
234C3⋊C8—91no—
235C2×C32—21no—
236C2×C32—81no—
237C2×C32—81no—
238C2×C32—161no—
239C2×C32D3×D811yescharacteristic
240C2×C32C2×#21311yescharacteristic
241C3⋊D3—21no—
242C3⋊D3—81no—
243C3⋊D3—81no—
244C3×D3—61no—
245C3×D3—241no—
246C3×D3—241no—
247#41—541no—
248C2×D4—181no—
249Q16—91no—
250SD16—541no—
251D8—91no—
252D8—91no—
253D8—181no—
254D8((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C2×C8—91no—
256C16—91no—
257C16—91no—
258C22×C3—41no—
259C22×C3—91no—
260C22×C3—121no—
261C22×C3—121no—
262C22×C3—241no—
263C22×C3—361no—
264C22×C3—361no—
265C22×C3—722no—
266C2×D3—31no—
267C2×D3—91no—
268C2×D3—121no—
269C2×D3—121no—
270C2×D3—242no—
271C2×D3—361no—
272C2×D3—361no—
273C2×D3—722no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—61no—
277C3×C4—91no—
278C3×C4—361no—
279C3×C4D3×#6711yescharacteristic
280Dic3—31no—
281Dic3—91no—
282Dic3—121no—
283Dic3—121no—
284Dic3—241no—
285C32—21no—
286C32#491111yescharacteristic, complemented
287C32#514711yescharacteristic, complemented
288C23—361no—
289Q8—181no—
290D4—21no—
291D4—181no—
292D4—181no—
293D4—361no—
294D4—1081no—
295C2×C4—91no—
296C2×C4—1081no—
297C8—91no—
298C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—61no—
302C2×C3—81no—
303C2×C3—181no—
304C2×C3—241no—
305C2×C3—241no—
306C2×C3—481no—
307C2×C3—721no—
308C2×C3—721no—
309C2×C3D3×#21311yescharacteristic
310D3—61no—
311D3—181no—
312D3—241no—
313D3—241no—
314D3—721no—
315D3—721no—
316C22—41no—
317C22—91no—
318C22—361no—
319C22—361no—
320C22—722no—
321C4—91no—
322C4—361no—
323C4#8867611yescharacteristic
324C3—31no—
325C3—31no—
326C3—61no—
327C3#1059103111yescharacteristic
328C2—81no—
329C2—181no—
330C2—721no—
331C2—721no—
332C2#1172119411yescharacteristic
subgroupG/H
centerC2#11721194
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingD8×H(3,𝔽3)C22
socleC2×C3D3×#213
hypercenterD8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#509
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
1. D8×(C3⋊D3)⋊C3C2semidirect
2. #11720855C2semidirect
3. #11720566C2semidirect
4. #63653⋊C3C2semidirect
5. #11720255C2semidirect
6. #11720248C2semidirect
7. #11720247C2semidirect
13. #88530C22semidirect
16. #3745⋊C3C22semidirect
19. #88386C22semidirect
20. (C3⋊(C3⋊C16))⋊C3C22semidirect
21. D8×C3⋊D3D3semidirect
25. #63653D3semidirect
59. #3745C2×D3semidirect
70. C3⋊(C3⋊C16)C2×D3semidirect
198. H(3,𝔽3)#509semidirect
286. C32#4911semidirect
287. C32#5147semidirect

Maximal subgroups

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

subgroupindex
D8×(C3⋊D3)⋊C32
#117208552
#117205662
#63653⋊C32
#117202552
#117202482
#117202472
#105910363
#105910333