Finite Group Databasebeta

← #13161070

Subgroups — #13161070

Order 1728.

1136 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 158 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×#88861C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3C32×#940C211yesmaximal, characteristic, complemented
4#11723439C211yesmaximal, characteristic, complemented
5C2×#88794C211yesmaximal, characteristic, complemented
6C2×#88789C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10592773—31nomaximal
9#10591469—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×#67—22no—
14C2×C32×D12C2211yescharacteristic
15#88861—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3753—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20#88803—22no—
21#88794—22no—
22#88789—22no—
23C32×#127—31no—
24C2×#3754—31no—
25C2×#3754—34no—
26C2×#3754—31no—
27C2×#3754—64no—
28C2×#3753—31no—
29C2×#3753D311yescharacteristic, complemented
30#64084—31no—
31C3×#940D314yescomplemented
32C2×C3×#213—34no—
33#63860—34no—
34C2×#3716—34no—
35C2×#3713—34no—
36#63630—31no—
37#63554—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C32×#67—44no—
50C2×C32×Dic3D412yes—
51C32×D12—21no—
52C32×D12—21no—
53C3⋊(C3⋊(C3⋊C8))D412yescomplemented
54C3×C3⋊(C3⋊C8)D412yescomplemented
55#5154—91no—
56#5074—94no—
57C2×C32×D4—24no—
58C2×C32×D4—31no—
59C2×C32×D4—62no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C2×D311yescharacteristic
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×#634—31no—
64C2×#634—34no—
65C2×#634—64no—
66C2×C3×#67—28no—
67C2×C3×D12—34no—
68C2×C3×D12C2×D314yes—
69C32×C42—31no—
70#3754—62no—
71#3754—68no—
72#3754—616no—
73#3754—62no—
74#3753—22no—
75#3753—62no—
76#3752—62no—
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—64no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#213—68no—
82C3×C4×Dic3C2×D314yes—
83C2×C3×C3⋊C8—34no—
84(C2×D3)⋊Dic3—68no—
85#3716—68no—
86#3713—68no—
87C22×C33—41no—
88C22×C33—42no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—42no—
92C2×C32×D3—41no—
93C2×C32×D3—42no—
94C33×C4C2×D411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3⋊(C3⋊Dic3)—41no—
97C32×Dic3—22no—
98C2×C3×D8—91no—
99C3×#127—34no—
100#940C3⋊D311yescharacteristic, complemented
101#936—91no—
102C2×#214—94no—
103C2×#214—94no—
104C2×#214—91no—
105C2×#214—184no—
106C2×#213—34no—
107C2×#213—94no—
108C2×#213—91no—
109C2×#204—91no—
110#875—94no—
111#815—91no—
112#803—94no—
113C23×C32—21no—
114C23×C32—24no—
115C23×C32—44no—
116C23×C32—61no—
117C22×C3⋊D3—21no—
118C22×C3×D3—24no—
119C22×C3×D3—64no—
120C32×D4—28no—
121C32×D4—24no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—24no—
125C32×D4—44no—
126C32×D4—61no—
127C32×D4—61no—
128C32×D4—124no—
129C2×C32×C4—24no—
130C2×C32×C4—32no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—124no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138#634—68no—
139#634—64no—
140#634—61no—
141#634—61no—
142#634—64no—
143#634—124no—
144C3×#67—416no—
145C2×C3×Dic3#6718yes—
146C3×D12—24no—
147C3×D12—24no—
148C3×D12—64no—
149C3×D12—64no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153C3⋊(C3⋊C8)#6712yescomplemented
154C3×C3⋊C8—38no—
155#496—271noSylow
156C2×C33—81no—
157C2×C33#12711yescharacteristic
158C2×C33AGL(1,ℤ/8)12yes—
159C3×C3⋊D3—81no—
160C32×D3—81no—
161C2×C3×D4—24no—
162C2×C3×D4—34no—
163C2×C3×D4—68no—
164C2×C3×D4—91no—
165C2×C3×D4D3214yes—
166C2×C3×D4C2×C3⋊D311yescharacteristic
167C2×#67—22no—
168C2×D12—34no—
169C2×D12C2×C3⋊D311yescharacteristic
170C2×Dic6—94no—
171C2×Dic6—91no—
172C2×Dic6—184no—
173C3×D8—182no—
174C2×C3×C8—91no—
175C3×C42—34no—
176#217—182no—
177#214—188no—
178#214—188no—
179#214—182no—
180#214—1816no—
181#213—68no—
182#213—182no—
183#213—188no—
184(C2×D3)⋊C4—188no—
185C4×Dic3C2×C3⋊D311yescharacteristic
186C2×C3⋊C8—34no—
187C2×C3⋊C8—94no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—184no—
190#204—182no—
191C22×C32—24no—
192C22×C32—48no—
193C22×C32—42no—
194C22×C32—44no—
195C22×C32—41no—
196C22×C32—48no—
197C22×C32—88no—
198C22×C32—122no—
199C22×C32—121no—
200C22×C32D3×D411yescharacteristic
201C22×C32C2×#6714yes—
202C2×C3⋊D3—42no—
203C2×C3⋊D3—41no—
204C2×C3×D3—48no—
205C2×C3×D3—44no—
206C2×C3×D3—128no—
207C2×C3×D3—124no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—62no—
211C32×C4D3×D414yes—
212C32×C4D3×D414yes—
213C32×C4D3×D411yescharacteristic
214C32×C4C2×#6711yescharacteristic
215C3⋊Dic3—128no—
216C3⋊Dic3—124no—
217C3⋊Dic3—121no—
218C3×Dic3—28no—
219C2×SD16—271no—
220C2×SD16—271no—
221C2×D8—91no—
222C2×D8—271no—
223#127—31no—
224#124—271no—
225#97—271no—
226C33#49611yescharacteristic, complemented, Sylow
227C23×C3—24no—
228C23×C3—21no—
229C23×C3—44no—
230C23×C3—64no—
231C23×C3—181no—
232C22×D3—21no—
233C22×D3—64no—
234C3×D4—28no—
235C3×D4—24no—
236C3×D4—24no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—44no—
240C3×D4—64no—
241C3×D4—64no—
242C3×D4—1216no—
243C3×D4—181no—
244C3×D4—181no—
245C2×C3×C4—24no—
246C2×C3×C4—38no—
247C2×C3×C4C2×D3214yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249#67—44no—
250C2×Dic3—184no—
251C2×Dic3—181no—
252C2×Dic3—364no—
253C2×Dic3#63812yes—
254D12—21no—
255D12—21no—
256D12—64no—
257D12—64no—
258Dic6—188no—
259Dic6—184no—
260Dic6—184no—
261Dic6—181no—
262Dic6—181no—
263Dic6—364no—
264C3×C8—92no—
265C3⋊C8—38no—
266C3⋊C8—98no—
267C3⋊C8—92no—
268C3⋊C8—188no—
269C2×C32—28no—
270C2×C32—24no—
271C2×C32—88no—
272C2×C32—81no—
273C2×C32—84no—
274C2×C32—241no—
275C2×C32#91112yes—
276C2×C32#93218yes—
277C2×C32#94014yes—
278C2×C32#94011yescharacteristic
279C3⋊D3—81no—
280C3×D3—84no—
281C3×D3—244no—
282C2×Q8—271no—
283C2×D4—31no—
284C2×D4—62no—
285C2×D4—91no—
286C2×D4D3×C3⋊D311yescharacteristic
287SD16—542no—
288SD16—542no—
289D8—182no—
290D8—542no—
291C2×C8—91no—
292C2×C8—271no—
293C22⋊C4—542no—
294C42—31no—
295C22×C3—24no—
296C22×C3—48no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—48no—
300C22×C3—44no—
301C22×C3—88no—
302C22×C3—128no—
303C22×C3—124no—
304C22×C3—362no—
305C22×C3—361no—
306C22×C3D3×#6714yes—
307C22×C3C2×#63811yescharacteristic
308C2×D3—42no—
309C2×D3—41no—
310C2×D3—128no—
311C2×D3—124no—
312C3×C4—24no—
313C3×C4—24no—
314C3×C4—68no—
315C3×C4D3×#6714yes—
316C3×C4#381114yes—
317C3×C4D4×C3⋊D311yescharacteristic
318C3×C4D4×C3⋊D311yescharacteristic
319Dic3—22no—
320Dic3—368no—
321Dic3—361no—
322Dic3—364no—
323C32—24no—
324C32#507414yescomplemented
325C32#515411yescharacteristic, complemented
326C23—21no—
327C23—61no—
328C23—181no—
329Q8—541no—
330Q8—541no—
331D4—21no—
332D4—21no—
333D4—61no—
334D4—61no—
335D4—124no—
336D4—181no—
337D4—181no—
338C2×C4—32no—
339C2×C4—541no—
340C2×C4C2×D3×C3⋊D311yescharacteristic
341C8—92no—
342C8—272no—
343C2×C3—28no—
344C2×C3—24no—
345C2×C3—88no—
346C2×C3—81no—
347C2×C3—84no—
348C2×C3—244no—
349C2×C3—721no—
350C2×C3#6393218yes—
351C2×C3#6397914yes—
352C2×C3#6414112yes—
353C2×C3#6414911yescharacteristic
354D3—81no—
355D3—244no—
356C22—42no—
357C22—41no—
358C22—122no—
359C22—121no—
360C22—362no—
361C22—361no—
362C22D3×#63811yescharacteristic
363C4—62no—
364C4—1081no—
365C4C3⋊D3×#6711yescharacteristic
366C4#8903911yescharacteristic
367C3—24no—
368C3#1059146914yescomplemented
369C3#1059277311yescharacteristic, complemented
370C2—81no—
371C2—241no—
372C2—721no—
373C2#1172355512yes—
374C2#1172363311yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#496
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88861C2semidirect
2. C2×C3×#3753C2semidirect
3. C32×#940C2semidirect
4. #11723439C2semidirect
5. C2×#88794C2semidirect
6. C2×#88789C2semidirect
7. #11722019C2semidirect
29. C2×#3753D3semidirect
31. C3×#940D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
54. C3×C3⋊(C3⋊C8)D4semidirect
100. #940C3⋊D3semidirect
153. C3⋊(C3⋊C8)#67semidirect
226. C33#496semidirect
324. C32#5074semidirect
325. C32#5154semidirect
368. C3#10591469semidirect
369. C3#10592773semidirect

Maximal subgroups

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

subgroupindex
C2×#888612
C2×C3×#37532
C32×#9402
#117234392
C2×#887942
C2×#887892
#117220192
#105927733
#105914693