Finite Group Databasebeta

← #13156628

Subgroups — #13156628

Order 1728.

858 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 114 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
1#11723697C211yesmaximal, characteristic, complemented
2#11723440C211yesmaximal, characteristic, complemented
3C2×#88791C212yesmaximal, complemented
4C2×C3×#3717C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6#10591621—31nomaximal
7#10590881—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C32×Dic6C2211yescharacteristic
11C2×#6295—22no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88804—22no—
15#88791—24no—
16C3×#3717—24no—
17#64156—31no—
18C2×#3745—32no—
19C2×C3×#215—32no—
20C2×C3×#204—32no—
21#63914—31no—
22#63911—32no—
23#63864—32no—
24#63864—61no—
25C2×#3717D312yescomplemented
26C2×#3714—32no—
27C2×#3714—32no—
28C2×#3714—62no—
29C3×#815—32no—
30#63555D311yescharacteristic, complemented
31C22×C3⋊(C3⋊D3)—21no—
32C22×C3×C3⋊D3—22no—
33C2×C33×C4C2311yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—22no—
35C2×C3×C3⋊Dic3D412yes—
36C3×C4⋊(C3⋊D3)—22no—
37C3×C4⋊(C3⋊D3)—22no—
38C2×C32×Dic3—21no—
39C32×Dic6—22no—
40#6295—44no—
41C32×C3⋊C8D414yescomplemented
42#5176—92no—
43#4712—91no—
44C2×C32×Q8—31no—
45C2×C4⋊(C3⋊D3)—31no—
46C2×C4⋊(C3⋊D3)—32no—
47C2×C4⋊(C3⋊D3)—62no—
48C2×C4⋊(C3⋊D3)—61no—
49C2×C4⋊(C3⋊D3)—121no—
50C2×C4⋊(C3⋊D3)C2×D312yes—
51C2×C3×D12—32no—
52C2×C3×D12—32no—
53C2×C3×D12—62no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6C2×D312yes—
56C2×#626—64no—
57C2×#625—62no—
58C2×C32×C8—32no—
59#3752—62no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61#3745—64no—
62C3×#215—64no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—22no—
66C2×C3×C3⋊C8C2×D312yes—
67C2×C3×C3⋊C8C2×D312yes—
68C3×#204—64no—
69#3722—64no—
70#3722—64no—
71#3717—24no—
72#3714—68no—
73#3714—64no—
74#3714—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3⋊(C3⋊D3)—41no—
77C2×C3⋊(C3⋊D3)—42no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—44no—
80C33×C4C2×D412yes—
81C3×C3⋊Dic3—22no—
82C32×Dic3—41no—
83C2×C3×SD16—92no—
84#947—92no—
85#947—181no—
86C2×#215—32no—
87#940—92no—
88C2×#213—92no—
89C2×D24—92no—
90C2×D24—92no—
91C2×D24—182no—
92C2×#204—32no—
93#875—91no—
94#874—91no—
95C3×#97—91no—
96#815—32no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3—61no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—62no—
101C22×C3⋊D3—122no—
102C22×C3⋊D3—122no—
103C22×C3×D3—62no—
104C22×C3×D3—62no—
105C22×C3×D3—122no—
106C32×Q8—62no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C22×D312yes—
113C2×C3⋊Dic3D1212yes—
114C4⋊(C3⋊D3)—22no—
115C4⋊(C3⋊D3)—22no—
116C4⋊(C3⋊D3)—62no—
117C4⋊(C3⋊D3)—64no—
118C4⋊(C3⋊D3)—62no—
119C4⋊(C3⋊D3)—62no—
120C4⋊(C3⋊D3)—122no—
121C4⋊(C3⋊D3)—122no—
122C4⋊(C3⋊D3)—124no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—62no—
127C3×D12—64no—
128C3×D12—62no—
129C3×D12—62no—
130C3×D12—62no—
131C3×D12—62no—
132C3×D12—122no—
133C3×Dic6—22no—
134C3×Dic6—22no—
135C3×Dic6—42no—
136#626—128no—
137#625—124no—
138C32×C8—34no—
139C3×C3⋊C8—24no—
140C3×C3⋊C8D1214yescomplemented
141C3×C3⋊C8#6714yescomplemented
142#496—271noSylow
143C2×C33#12711yescharacteristic
144C2×C33AGL(1,ℤ/8)12yes—
145C3⋊(C3⋊D3)—81no—
146C3×C3⋊D3—82no—
147C2×C3×Q8—32no—
148C2×C3×Q8—61no—
149C2×C3×D4—92no—
150C2×#67—184no—
151C2×D12—32no—
152C2×D12—32no—
153C2×D12—62no—
154C2×D12—91no—
155C2×D12—92no—
156C2×D12—182no—
157C2×D12—181no—
158C2×D12—182no—
159C2×D12—361no—
160C2×Dic6D3211yescharacteristic
161C3×SD16—184no—
162C2×C3×C8—32no—
163C2×C3×C8—32no—
164C2×C3×C8—62no—
165C3×C42—91no—
166#215—64no—
167#213—184no—
168(C2×D3)⋊C4—182no—
169(C2×D3)⋊C4—184no—
170(C2×D3)⋊C4—184no—
171C4×Dic3—32no—
172C4×Dic3—61no—
173C2×C3⋊C8D3212yes—
174D24—184no—
175D24—184no—
176D24—188no—
177#204—64no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32C2×D1211yescharacteristic
182C22×C32D3×D412yes—
183C2×C3⋊D3—42no—
184C2×C3⋊D3—44no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—122no—
187C2×C3⋊D3—124no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—122no—
190C2×C3⋊D3—122no—
191C2×C3⋊D3—124no—
192C2×C3⋊D3—124no—
193C2×C3⋊D3—244no—
194C2×C3⋊D3—244no—
195C2×C3×D3—124no—
196C2×C3×D3—122no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—122no—
200C2×C3×D3—244no—
201C32×C4—22no—
202C32×C4—22no—
203C32×C4—22no—
204C32×C4—42no—
205C32×C4—121no—
206C32×C4C2×D1212yes—
207C32×C4D3×D412yes—
208C32×C4C2×#6712yes—
209C3⋊Dic3—22no—
210C3×Dic3—42no—
211C3×Dic3—42no—
212C3×Dic3—64no—
213C3×Dic3—122no—
214C2×SD16—92no—
215C2×D8—272no—
216#127—271no—
217#124—271no—
218#97—91no—
219C33#49611yescharacteristic, complemented, Sylow
220C23×C3—182no—
221C22×D3—62no—
222C22×D3—62no—
223C22×D3—122no—
224C22×D3—182no—
225C22×D3—182no—
226C22×D3—181no—
227C22×D3—362no—
228C22×D3—362no—
229C3×Q8—62no—
230C3×Q8—62no—
231C3×Q8—122no—
232C3×D4—182no—
233C3×D4—182no—
234C2×C3×C4—21no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—92no—
240C2×C3×C4C2×D3212yes—
241C2×C3×C4C2×D3211yescharacteristic
242#67—368no—
243C2×Dic3—21no—
244C2×Dic3—34no—
245C2×Dic3—62no—
246D12—64no—
247D12—62no—
248D12—62no—
249D12—62no—
250D12—62no—
251D12—122no—
252D12—184no—
253D12—182no—
254D12—182no—
255D12—182no—
256D12—362no—
257D12—362no—
258D12—364no—
259D12—364no—
260Dic6—22no—
261C3×C8—34no—
262C3×C8—34no—
263C3×C8—64no—
264C3⋊C8#62614yescomplemented
265C2×C32—24no—
266C2×C32—22no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—41no—
270C2×C32—42no—
271C2×C32#87411yescharacteristic
272C2×C32#90812yes—
273C2×C32#91414yes—
274C2×C32#94012yes—
275C3⋊D3—82no—
276C3⋊D3—242no—
277C3⋊D3—244no—
278C3⋊D3—244no—
279C3⋊D3—242no—
280C3⋊D3—241no—
281C3×D3—244no—
282C3×D3—242no—
283C3×D3—242no—
284C2×Q8—31no—
285C2×D4—92no—
286C2×D4—271no—
287C2×D4—542no—
288SD16—184no—
289D8—544no—
290C2×C8—32no—
291C22⋊C4—542no—
292C42—91no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—41no—
296C22×C3—364no—
297C22×C3—362no—
298C22×C3D3×D1212yes—
299C22×C3#381111yescharacteristic
300C2×D3—124no—
301C2×D3—124no—
302C2×D3—122no—
303C2×D3—124no—
304C2×D3—122no—
305C2×D3—244no—
306C2×D3—364no—
307C2×D3—362no—
308C2×D3—364no—
309C2×D3—364no—
310C2×D3—364no—
311C2×D3—362no—
312C2×D3—362no—
313C2×D3—361no—
314C2×D3—724no—
315C2×D3—724no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—42no—
320C3×C4—122no—
321C3×C4—122no—
322C3×C4—182no—
323C3×C4D3×D1212yes—
324C3×C4C2×#62612yes—
325C3×C4D3×#6712yes—
326Dic3—41no—
327Dic3—64no—
328Dic3—122no—
329C32—21no—
330C32—22no—
331C32—41no—
332C32#471211yescharacteristic, complemented
333C32#517612yescomplemented
334C23—182no—
335C23—541no—
336Q8—62no—
337D4—182no—
338D4—182no—
339D4—542no—
340D4—1084no—
341C2×C4—61no—
342C2×C4—92no—
343C2×C4D3311yescharacteristic
344C8—34no—
345C2×C3—21no—
346C2×C3—22no—
347C2×C3—22no—
348C2×C3—24no—
349C2×C3—41no—
350C2×C3—42no—
351C2×C3—722no—
352C2×C3#6379414yes—
353C2×C3#6391112yes—
354C2×C3#6394112yes—
355C2×C3#6397911yescharacteristic
356D3—244no—
357D3—242no—
358D3—242no—
359D3—722no—
360D3—724no—
361D3—724no—
362D3—721no—
363D3—722no—
364C22—362no—
365C22—364no—
366C22—1081no—
367C22—1082no—
368C22#8895511yescharacteristic
369C4—121no—
370C4—182no—
371C4D3×#62612yes—
372C3—22no—
373C3—21no—
374C3—41no—
375C3#1059088112yescomplemented
376C3#1059162111yescharacteristic, complemented
377C2—722no—
378C2—2161no—
379C2#1172288112yes—
380C2#1172303011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#496
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723697C2semidirect
2. #11723440C2semidirect
3. C2×#88791C2semidirect
4. C2×C3×#3717C2semidirect
5. C3×#63555C2semidirect
25. C2×#3717D3semidirect
30. #63555D3semidirect
41. C32×C3⋊C8D4semidirect
140. C3×C3⋊C8D12semidirect
141. C3×C3⋊C8#67semidirect
219. C33#496semidirect
264. C3⋊C8#626semidirect
332. C32#4712semidirect
333. C32#5176semidirect
375. C3#10590881semidirect
376. C3#10591621semidirect

Maximal subgroups

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

subgroupindex
#117236972
#117234402
C2×#887912
C2×C3×#37172
C3×#635552
#105916213
#105908813