Finite Group Databasebeta

← #13139242

Subgroups — #13139242

Order 1728.

391 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 49 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×#272C211yesmaximal, characteristic, complemented
2#11720844C211yesmaximal, characteristic, complemented
3#63654⋊C3C211yesmaximal, characteristic, complemented
4#63653⋊C3C211yesmaximal, characteristic, complemented
5#11720254C211yesmaximal, characteristic, complemented
6#11720253C211yesmaximal, characteristic, complemented
7#11720252C211yesmaximal, characteristic, complemented
8#10591033—31nomaximal
9#10591031—31nomaximal
10D4×#272—21no—
11#88649—21no—
12D8×H(3,𝔽3)C2211yescharacteristic
13#88545—21no—
14#88529C2211yescharacteristic
15C8×#272C2211yescharacteristic
16#3745⋊C3C2211yescharacteristic, complemented
17#3746⋊C3C2211yescharacteristic, complemented
18#88431—21no—
19(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic, complemented
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic, complemented
21C3×D3×D8—31no—
22C3×D3×D8—31no—
23C3×D3×D8—61no—
24#63808—31no—
25#63807—31no—
26#63654D311yescharacteristic, complemented
27#63653D311yescharacteristic, complemented
28C3×#827—31no—
29C3×#826—31no—
30#63548—31no—
31#63547—31no—
32#63546—31no—
33#63545—31no—
34#63543—31no—
35#63543—31no—
36C22×#272—41no—
37D4×H(3,𝔽3)—21no—
38#6235—41no—
39C4⋊#272—21no—
40C4×#272D411yescharacteristic
41C4⋊((C3⋊D3)⋊C3)—21no—
42#634⋊C3—21no—
43#6200—41no—
44((C3⋊D3)⋊C3)⋊C4—41no—
45C8×H(3,𝔽3)C2311yescharacteristic
46#6180D411yescharacteristic
47#5147—91no—
48#4914—91no—
49#4911—91no—
50C3×D3×D4—61no—
51C3×D3×D4—61no—
52C3×D3×D4—62no—
53#3799—61no—
54#3795—61no—
55C32×D8—21no—
56C32×D8C2×D311yescharacteristic
57C32×D8C2×D311yescharacteristic
58#3746C2×D311yescharacteristic, complemented
59#3745C2×D311yescharacteristic, complemented
60C3×#213—61no—
61C3×#213—61no—
62C3×#213—62no—
63C3×Dic12—31no—
64C3×D24—31no—
65C3×D24—31no—
66C3×D24—31no—
67C3×D24—61no—
68C3×D3×C8—31no—
69C3×D3×C8—31no—
70C3×D3×C8—61no—
71#3717—61no—
72#3716—61no—
73C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
74C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
75C3×C3⋊C16—31no—
76C3×C3⋊C16—31no—
77C22×H(3,𝔽3)—41no—
78C2×#272—41no—
79C2×#272—82no—
80C2×#272D811yescharacteristic
81C2×(C3⋊D3)⋊C3—41no—
82C4×H(3,𝔽3)C2×D411yescharacteristic
83#1084D811yescharacteristic
84(C3⋊Dic3)⋊C3—41no—
85C2×C3×D8—91no—
86#919—91no—
87#912—91no—
88D3×D8—31no—
89D3×D8—31no—
90D3×D8—61no—
91#904—91no—
92C3×SD32—91no—
93C3×D16—91no—
94#828—91no—
95#827—31no—
96#827—91no—
97#827—91no—
98#826—31no—
99#826—91no—
100#826—91no—
101#826—91no—
102#812—91no—
103#800—91no—
104D48—91no—
105#798—91no—
106#798—91no—
107C22×C3×D3—121no—
108C22×C3×D3—121no—
109C22×C3×D3—122no—
110C32×D4—22no—
111C32×D4—21no—
112C32×D4—21no—
113C4⋊(C3⋊D3)—21no—
114#634—21no—
115C3×#67—121no—
116C3×#67—121no—
117C3×#67—122no—
118C3×D12—62no—
119C3×D12—61no—
120C3×D12—61no—
121C3×D12—61no—
122C3×C4×D3—31no—
123C3×C4×D3—31no—
124C3×C4×D3—61no—
125C3×Dic6—61no—
126#626—121no—
127#625—121no—
128#624—121no—
129D3×Dic3—121no—
130C32×C8—21no—
131C32×C8C22×D311yescharacteristic
132C32×C8C22×D311yescharacteristic
133C3×C3⋊C8—31no—
134C3×C3⋊C8—31no—
135C3×C3⋊C8—61no—
136#509—271noSylow
137C2×H(3,𝔽3)—81no—
138C2×H(3,𝔽3)C2×D811yescharacteristic
139#272—21no—
140#272—81no—
141(C3⋊D3)⋊C3—81no—
142C2×C3×D4—181no—
143#239—181no—
144#237—181no—
145D3×D4—61no—
146D3×D4—61no—
147D3×D4—62no—
148#235—181no—
149C3×Q16—91no—
150C3×D8—31no—
151C3×D8—31no—
152C3×D8—61no—
153C3×D8—91no—
154C3×D8—91no—
155C3×D8—181no—
156C3×D8D3211yescharacteristic
157C2×C3×C8—91no—
158#215—181no—
159#214—181no—
160#213—61no—
161#213—61no—
162#213—62no—
163Dic12—31no—
164Dic12—91no—
165D24—31no—
166D24—31no—
167D24—31no—
168D24—61no—
169D24—91no—
170#204—181no—
171D3×C8—31no—
172D3×C8—31no—
173D3×C8—61no—
174C3×C16—91no—
175C3×C16—91no—
176C3⋊C16—31no—
177C3⋊C16—31no—
178C3⋊C16—91no—
179C3⋊C16—91no—
180C22×C32—42no—
181C22×C32—41no—
182C22×C32—41no—
183C2×C3⋊D3—41no—
184C2×C3×D3—31no—
185C2×C3×D3—31no—
186C2×C3×D3—61no—
187C2×C3×D3—122no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C2×C3×D3—244no—
192C2×C3×D3—242no—
193C2×C3×D3—242no—
194C32×C4—21no—
195C32×C4D3×D411yescharacteristic
196C32×C4D3×D411yescharacteristic
197C3⋊Dic3—41no—
198C3×Dic3—31no—
199C3×Dic3—31no—
200C3×Dic3—61no—
201C3×Dic3—121no—
202#135—271no—
203C2×D8—91no—
204SD32—91no—
205SD32—271no—
206D16—91no—
207D16—271no—
208M32—271no—
209H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
210C23×C3—361no—
211C22×D3—121no—
212C22×D3—121no—
213C22×D3—122no—
214C3×Q8—181no—
215C3×D4—21no—
216C3×D4—61no—
217C3×D4—61no—
218C3×D4—62no—
219C3×D4—181no—
220C3×D4—181no—
221C3×D4—361no—
222C2×C3×C4—91no—
223#67—121no—
224#67—121no—
225#67—122no—
226#67—361no—
227#67—361no—
228C2×Dic3—361no—
229D12—61no—
230D12—61no—
231D12—62no—
232D12—61no—
233D12—181no—
234D12—361no—
235C4×D3—31no—
236C4×D3—31no—
237C4×D3—61no—
238C4×D3—361no—
239C4×D3—361no—
240Dic6—61no—
241Dic6—181no—
242C3×C8—31no—
243C3×C8—31no—
244C3×C8—61no—
245C3×C8—91no—
246C3×C8C2×D3211yescharacteristic
247C3⋊C8—31no—
248C3⋊C8—31no—
249C3⋊C8—61no—
250C2×C32—21no—
251C2×C32—82no—
252C2×C32—81no—
253C2×C32—81no—
254C2×C32D3×D811yescharacteristic
255C2×C32D3×D811yescharacteristic
256C3⋊D3—81no—
257C3×D3—62no—
258C3×D3—61no—
259C3×D3—61no—
260C3×D3—242no—
261C3×D3—241no—
262C3×D3—241no—
263C3×D3—241no—
264#41—541no—
265C2×D4—181no—
266Q16—91no—
267SD16—541no—
268D8—91no—
269D8—91no—
270D8—181no—
271D8((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C2×C8—91no—
273C16—91no—
274C16—91no—
275C22×C3—41no—
276C22×C3—91no—
277C22×C3—121no—
278C22×C3—121no—
279C22×C3—122no—
280C22×C3—361no—
281C22×C3—361no—
282C22×C3—722no—
283C2×D3—31no—
284C2×D3—31no—
285C2×D3—61no—
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—122no—
289C2×D3—121no—
290C2×D3—242no—
291C2×D3—242no—
292C2×D3—244no—
293C2×D3—361no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—61no—
297C3×C4—91no—
298C3×C4—361no—
299C3×C4#381111yescharacteristic
300Dic3—31no—
301Dic3—31no—
302Dic3—61no—
303Dic3—121no—
304Dic3—361no—
305C32—21no—
306C32#491111yescharacteristic, complemented
307C32#491411yescharacteristic, complemented
308C23—361no—
309Q8—181no—
310D4—21no—
311D4—181no—
312D4—181no—
313D4—361no—
314D4—1081no—
315C2×C4—91no—
316C2×C4—1081no—
317C8—91no—
318C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—61no—
322C2×C3—81no—
323C2×C3—181no—
324C2×C3—241no—
325C2×C3—241no—
326C2×C3—242no—
327C2×C3—721no—
328C2×C3—721no—
329C2×C3#6392611yescharacteristic
330D3—61no—
331D3—61no—
332D3—62no—
333D3—241no—
334D3—241no—
335D3—242no—
336D3—241no—
337D3—721no—
338C22—41no—
339C22—91no—
340C22—361no—
341C22—361no—
342C22—722no—
343C4—91no—
344C4—361no—
345C4#8867711yescharacteristic
346C3—31no—
347C3—31no—
348C3—61no—
349C3#1059103611yescharacteristic
350C2—81no—
351C2—181no—
352C2—721no—
353C2—721no—
354C2#1172119011yescharacteristic
subgroupG/H
centerC2#11721190
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingD8×H(3,𝔽3)C22
socleC2×C3#63926
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×#272C2semidirect
2. #11720844C2semidirect
3. #63654⋊C3C2semidirect
4. #63653⋊C3C2semidirect
5. #11720254C2semidirect
6. #11720253C2semidirect
7. #11720252C2semidirect
16. #3745⋊C3C22semidirect
17. #3746⋊C3C22semidirect
19. (C3⋊(C3⋊C16))⋊C3C22semidirect
20. (C3⋊(C3⋊C16))⋊C3C22semidirect
26. #63654D3semidirect
27. #63653D3semidirect
58. #3746C2×D3semidirect
59. #3745C2×D3semidirect
73. C3⋊(C3⋊C16)C2×D3semidirect
74. C3⋊(C3⋊C16)C2×D3semidirect
209. H(3,𝔽3)#509semidirect
306. C32#4911semidirect
307. C32#4914semidirect

Maximal subgroups

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

subgroupindex
D8×#2722
#117208442
#63654⋊C32
#63653⋊C32
#117202542
#117202532
#117202522
#105910333
#105910313