Finite Group Databasebeta

← #13161240

Subgroups — #13161240

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×#88862C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic, complemented
4#11723524C211yesmaximal, characteristic, complemented
5C2×#88791C211yesmaximal, characteristic, complemented
6C2×#88789C211yesmaximal, characteristic, complemented
7#11722020C211yesmaximal, characteristic, complemented
8#10592875—31nomaximal
9#10591619—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6291—22no—
15#88862—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3752—22no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#215—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88791—22no—
22#88789—22no—
23C2×C32×SD16—31no—
24#64156D311yescharacteristic, complemented
25C2×#3755—31no—
26C2×#3755—34no—
27C2×#3755—64no—
28#64149—31no—
29C2×#3753—31no—
30C2×#3745—31no—
31C3×#947—34no—
32C2×C3×#215D314yescomplemented
33#63911—34no—
34C2×#3714—34no—
35C2×#3713—34no—
36#63642—31no—
37#63554—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3—21no—
41C33×Q8—21no—
42C33×Q8—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C2×C3×C3⋊Dic3D412yes—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6291—44no—
53C3⋊(C3⋊(C3⋊C8))D412yescomplemented
54C32×C3⋊C8D412yescomplemented
55#5176—94no—
56#5058—91no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×D4—31no—
61C2×#638—62no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C3×D12—34no—
67C2×C3×D12C2×D314yes—
68C2×#626—68no—
69C32×SD16—62no—
70C2×C32×C8—31no—
71#3755—62no—
72#3755—68no—
73#3755—616no—
74#3753—62no—
75#3752—22no—
76C4×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—64no—
80#3745—62no—
81C3×#215—28no—
82C3×(C2×D3)⋊C4—68no—
83C3×C4×Dic3—34no—
84C2×C3×C3⋊C8C2×D314yes—
85#3714—68no—
86#3713—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—42no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—42no—
92C2×C32×D3—41no—
93C2×C32×D3—42no—
94C33×C4—41no—
95C33×C4C2×D411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—22no—
98C2×C3×SD16—34no—
99C3×#124—91no—
100#947—34no—
101C2×#215—94no—
102C2×#215—91no—
103C2×#215—184no—
104C2×#215C3⋊D311yescharacteristic, complemented
105#940—94no—
106C2×#213—94no—
107C2×#213—91no—
108C2×#213—91no—
109C2×D24—94no—
110#874—91no—
111#815—94no—
112#803—91no—
113C23×C32—61no—
114C22×C3⋊D3—21no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—124no—
118C22×C3×D3—24no—
119C22×C3×D3—64no—
120C32×Q8—28no—
121C32×Q8—24no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—24no—
125C32×Q8—44no—
126C32×D4—61no—
127C32×D4—61no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—124no—
135C2×C3⋊Dic3#6712yes—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—68no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—61no—
142C4⋊(C3⋊D3)—64no—
143C4⋊(C3⋊D3)—124no—
144C2×C3×Dic3—38no—
145C3×D12—24no—
146C3×D12—24no—
147C3×D12—64no—
148C3×D12—64no—
149#626—1216no—
150C32×C8—32no—
151C3⋊(C3⋊C8)—32no—
152C3⋊(C3⋊C8)—38no—
153C3⋊(C3⋊C8)—68no—
154C3×C3⋊C8#6718yescomplemented
155#496—271noSylow
156C2×C33#12711yescharacteristic
157C2×C33AGL(1,ℤ/8)12yes—
158C3⋊(C3⋊D3)—81no—
159C3×C3⋊D3—81no—
160C32×D3—81no—
161C2×C3×Q8—24no—
162C2×C3×Q8D3214yes—
163C2×C3×Q8C2×C3⋊D311yescharacteristic
164C2×C3×D4—34no—
165C2×C3×D4—91no—
166C2×#67—188no—
167C2×D12—34no—
168C2×D12—94no—
169C2×D12—91no—
170C2×D12—184no—
171C2×D12—182no—
172C2×D12C2×C3⋊D311yescharacteristic
173C3×SD16—68no—
174C2×C3×C8—34no—
175C3×C22⋊C4—182no—
176C3×C42—91no—
177#215—22no—
178#215—188no—
179#215—182no—
180#215—1816no—
181#213—182no—
182#213—188no—
183#213—182no—
184(C2×D3)⋊C4—68no—
185C4×Dic3—34no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189C2×C3⋊C8C2×C3⋊D311yescharacteristic
190D24—188no—
191C22×C32—24no—
192C22×C32—121no—
193C22×C32—122no—
194C22×C32D3×D414yes—
195C22×C32C2×#6711yescharacteristic
196C2×C3⋊D3—41no—
197C2×C3⋊D3—42no—
198C2×C3⋊D3—128no—
199C2×C3⋊D3—124no—
200C2×C3⋊D3—121no—
201C2×C3⋊D3—122no—
202C2×C3⋊D3—128no—
203C2×C3⋊D3—248no—
204C2×C3×D3—48no—
205C2×C3×D3—44no—
206C2×C3×D3—128no—
207C2×C3×D3—124no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—48no—
211C32×C4—44no—
212C32×C4—41no—
213C32×C4D3×D414yes—
214C32×C4D3×D411yescharacteristic
215C32×C4D3×D411yescharacteristic
216C32×C4C2×#6714yes—
217C3⋊Dic3—22no—
218C3×Dic3—68no—
219C2×SD16—31no—
220C2×SD16—271no—
221C2×D8—271no—
222C2×D8—271no—
223#127—271no—
224#124—91no—
225#97—271no—
226C33#49611yescharacteristic, complemented, Sylow
227C23×C3—64no—
228C23×C3—181no—
229C22×D3—21no—
230C22×D3—64no—
231C22×D3—184no—
232C22×D3—181no—
233C22×D3—364no—
234C3×Q8—28no—
235C3×Q8—24no—
236C3×Q8—24no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—44no—
240C3×D4—64no—
241C3×D4—64no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—24no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4—44no—
248C2×C3×C4—92no—
249C2×C3×C4C2×D3214yes—
250C2×C3×C4C22×C3⋊D311yescharacteristic
251#67—3616no—
252C2×Dic3—38no—
253D12—21no—
254D12—21no—
255D12—64no—
256D12—64no—
257D12—188no—
258D12—184no—
259D12—184no—
260D12—181no—
261D12—181no—
262D12—364no—
263D12—364no—
264C3×C8—38no—
265C3⋊C8—98no—
266C3⋊C8—92no—
267C3⋊C8—188no—
268C3⋊C8#63812yescomplemented
269C2×C32—28no—
270C2×C32—24no—
271C2×C32—241no—
272C2×C32#91418yes—
273C2×C32#94014yes—
274C2×C32#94011yescharacteristic
275C2×C32#94912yes—
276C3⋊D3—81no—
277C3⋊D3—248no—
278C3⋊D3—244no—
279C3⋊D3—241no—
280C3×D3—84no—
281C3×D3—244no—
282C2×Q8D3×C3⋊D311yescharacteristic
283C2×D4—31no—
284C2×D4—91no—
285C2×D4—271no—
286C2×D4—542no—
287SD16—62no—
288SD16—542no—
289D8—542no—
290D8—542no—
291C2×C8—31no—
292C2×C8—271no—
293C22⋊C4—182no—
294C42—91no—
295C22×C3—24no—
296C22×C3—128no—
297C22×C3—124no—
298C22×C3—362no—
299C22×C3—361no—
300C22×C3D3×#6714yes—
301C22×C3D4×C3⋊D311yescharacteristic
302C2×D3—42no—
303C2×D3—41no—
304C2×D3—128no—
305C2×D3—124no—
306C2×D3—368no—
307C2×D3—362no—
308C2×D3—361no—
309C2×D3—368no—
310C2×D3—364no—
311C2×D3—728no—
312C3×C4—24no—
313C3×C4—24no—
314C3×C4—48no—
315C3×C4—41no—
316C3×C4—44no—
317C3×C4—182no—
318C3×C4D3×#6714yes—
319C3×C4#381114yes—
320C3×C4D4×C3⋊D311yescharacteristic
321C3×C4C2×#63811yescharacteristic
322Dic3—68no—
323C32—24no—
324C32#505811yescharacteristic, complemented
325C32#517614yescomplemented
326C23—61no—
327C23—181no—
328C23—541no—
329Q8—21no—
330Q8—21no—
331D4—61no—
332D4—61no—
333D4—181no—
334D4—181no—
335D4—541no—
336D4—541no—
337D4—1084no—
338C2×C4—21no—
339C2×C4—92no—
340C2×C4C2×D3×C3⋊D311yescharacteristic
341C8—32no—
342C8—272no—
343C2×C3—28no—
344C2×C3—24no—
345C2×C3—244no—
346C2×C3—721no—
347C2×C3#6393918yes—
348C2×C3#6397914yes—
349C2×C3#6412312yes—
350C2×C3#6414911yescharacteristic
351D3—81no—
352D3—244no—
353D3—728no—
354D3—721no—
355D3—724no—
356C22—122no—
357C22—121no—
358C22—362no—
359C22—361no—
360C22—1082no—
361C22—1081no—
362C22C3⋊D3×#6711yescharacteristic
363C4—41no—
364C4—182no—
365C4D3×#63811yescharacteristic
366C4#8903911yescharacteristic
367C3—24no—
368C3#1059161914yescomplemented
369C3#1059287511yescharacteristic, complemented
370C2—241no—
371C2—721no—
372C2—2161no—
373C2#1172357012yes—
374C2#1172363311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#496
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88862C2semidirect
2. C3×#64156C2semidirect
3. C2×C32×#215C2semidirect
4. #11723524C2semidirect
5. C2×#88791C2semidirect
6. C2×#88789C2semidirect
7. #11722020C2semidirect
24. #64156D3semidirect
32. C2×C3×#215D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
54. C32×C3⋊C8D4semidirect
104. C2×#215C3⋊D3semidirect
154. C3×C3⋊C8#67semidirect
226. C33#496semidirect
268. C3⋊C8#638semidirect
324. C32#5058semidirect
325. C32#5176semidirect
368. C3#10591619semidirect
369. C3#10592875semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C3×#641562
C2×C32×#2152
#117235242
C2×#887912
C2×#887892
#117220202
#105928753
#105916193