Finite Group Databasebeta

← #13162015

Subgroups — #13162015

Order 1728.

826 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#64156C211yesmaximal, characteristic, complemented
2C2×C3×#3755C212yesmaximal, complemented
3#11723697C211yesmaximal, characteristic, complemented
4C2×#88810C212yesmaximal, complemented
5#11722071C211yesmaximal, characteristic, complemented
6#10591621—31nomaximal
7#10591619—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×#6295—22no—
12C3×#3755—24no—
13C3×#3752—22no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88810—24no—
17#64156D311yescharacteristic, complemented
18C2×#3755D312yescomplemented
19C3×#947—32no—
20C3×#947—61no—
21C2×C3×#215—32no—
22C2×C3×#215—32no—
23C2×C3×#215—62no—
24#63914—31no—
25#63911—32no—
26C2×#3714—32no—
27C2×#3714—32no—
28C2×#3713—32no—
29#63555—31no—
30#63554—32no—
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3—22no—
33C33×Q8—22no—
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3D412yes—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4⋊(C3⋊D3)—22no—
39C3×C4⋊(C3⋊D3)—22no—
40#6295—44no—
41C3×C3⋊(C3⋊C8)D414yescomplemented
42#5176—92no—
43#5058—91no—
44C2×C32×Q8—21no—
45C2×C32×Q8—22no—
46C2×C32×Q8—41no—
47C2×C32×Q8C2×D311yescharacteristic
48C2×C32×Q8C2×D312yes—
49C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
50C2×C4⋊(C3⋊D3)C2×D312yes—
51C2×C3×D12—32no—
52C2×C3×D12—32no—
53C2×C3×D12—32no—
54C2×C3×D12—62no—
55C2×C3×D12—61no—
56C2×#626—64no—
57C2×#625—62no—
58#3755—24no—
59#3752—22no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×#215—64no—
63C3×#215—64no—
64C3×#215—68no—
65C3×(C2×D3)⋊C4—64no—
66C3×(C2×D3)⋊C4—64no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—62no—
72#3714—64no—
73#3714—64no—
74#3713—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—42no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—44no—
80C33×C4—41no—
81C33×C4C2×D412yes—
82C3×C3⋊Dic3—22no—
83C2×C3×SD16—92no—
84C3×#124—91no—
85#947—32no—
86#947—61no—
87C2×#215—32no—
88C2×#215—32no—
89C2×#215—62no—
90#940—92no—
91C2×#213—92no—
92C2×#213—92no—
93C2×D24—92no—
94#874—91no—
95#815—92no—
96#803—91no—
97C22×C3⋊D3—21no—
98C22×C3⋊D3—22no—
99C22×C3×D3—62no—
100C22×C3×D3—62no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C22×C3×D3—122no—
104C32×Q8—22no—
105C32×Q8—24no—
106C32×Q8—22no—
107C32×Q8—22no—
108C32×Q8—42no—
109C32×Q8—42no—
110C32×Q8—44no—
111C2×C32×C4—21no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4—42no—
118C2×C32×C4—42no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D312yes—
121C2×C3⋊Dic3#6712yes—
122C4⋊(C3⋊D3)—22no—
123C4⋊(C3⋊D3)—22no—
124C4⋊(C3⋊D3)—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—62no—
133C3×D12—62no—
134C3×D12—122no—
135C3×D12—122no—
136#626—128no—
137#625—124no—
138C3⋊(C3⋊C8)#6714yescomplemented
139C3×C3⋊C8—34no—
140C3×C3⋊C8—34no—
141C3×C3⋊C8—64no—
142#496—271noSylow
143C2×C33#12711yescharacteristic
144C2×C33AGL(1,ℤ/8)12yes—
145C3×C3⋊D3—81no—
146C3×C3⋊D3—82no—
147C2×C3×Q8—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—41no—
150C2×C3×Q8D3211yescharacteristic
151C2×C3×Q8D3212yes—
152C2×C3×D4—91no—
153C2×C3×D4—92no—
154C2×#67—184no—
155C2×D12—32no—
156C2×D12—32no—
157C2×D12—32no—
158C2×D12—62no—
159C2×D12—61no—
160C2×D12—182no—
161C3×SD16—184no—
162C2×C3×C8—92no—
163C3×C22⋊C4—182no—
164C3×C42—91no—
165#215—64no—
166#215—64no—
167#215—68no—
168#213—184no—
169#213—184no—
170(C2×D3)⋊C4—64no—
171(C2×D3)⋊C4—64no—
172C4×Dic3—32no—
173C4×Dic3—61no—
174C2×C3⋊C8—32no—
175C2×C3⋊C8—32no—
176C2×C3⋊C8—62no—
177D24—184no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32D3×D412yes—
182C22×C32C2×#6711yescharacteristic
183C2×C3⋊D3—42no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—42no—
186C2×C3⋊D3—44no—
187C2×C3×D3—124no—
188C2×C3×D3—122no—
189C2×C3×D3—124no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194C2×C3×D3—122no—
195C2×C3×D3—124no—
196C2×C3×D3—244no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—42no—
201C32×C4—42no—
202C32×C4—44no—
203C32×C4—44no—
204C32×C4—41no—
205C32×C4—42no—
206C32×C4D3×D412yes—
207C32×C4D3×D412yes—
208C32×C4C2×#6712yes—
209C3⋊Dic3—22no—
210C3×Dic3—64no—
211C3×Dic3—122no—
212C2×SD16—92no—
213C2×D8—272no—
214#127—271no—
215#124—91no—
216#97—271no—
217C33#49611yescharacteristic, complemented, Sylow
218C23×C3—182no—
219C23×C3—181no—
220C22×D3—62no—
221C22×D3—62no—
222C22×D3—62no—
223C22×D3—62no—
224C22×D3—122no—
225C3×Q8—24no—
226C3×Q8—22no—
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—42no—
230C3×Q8—42no—
231C3×Q8—44no—
232C3×D4—182no—
233C3×D4—182no—
234C3×D4—182no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—21no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—41no—
243C2×C3×C4—92no—
244C2×C3×C4C2×D3211yescharacteristic
245C2×C3×C4C2×D3212yes—
246#67—368no—
247C2×Dic3—34no—
248C2×Dic3—62no—
249D12—64no—
250D12—62no—
251D12—62no—
252D12—62no—
253D12—62no—
254D12—62no—
255D12—62no—
256D12—122no—
257D12—122no—
258D12—364no—
259C3×C8—94no—
260C3⋊C8—34no—
261C3⋊C8—34no—
262C3⋊C8—64no—
263C2×C32—24no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—41no—
268C2×C32—42no—
269C2×C32#91414yes—
270C2×C32#94012yes—
271C2×C32#94011yescharacteristic
272C2×C32#94912yes—
273C3⋊D3—82no—
274C3⋊D3—81no—
275C3×D3—242no—
276C3×D3—244no—
277C3×D3—242no—
278C3×D3—242no—
279C3×D3—242no—
280C2×Q8#111311yescharacteristic
281C2×D4—91no—
282C2×D4—92no—
283C2×D4—542no—
284SD16—184no—
285D8—544no—
286C2×C8—92no—
287C22⋊C4—182no—
288C42—91no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—41no—
292C22×C3—364no—
293C22×C3—362no—
294C22×C3—362no—
295C22×C3—361no—
296C22×C3D3×#6712yes—
297C22×C3#381111yescharacteristic
298C2×D3—124no—
299C2×D3—122no—
300C2×D3—124no—
301C2×D3—124no—
302C2×D3—124no—
303C2×D3—124no—
304C2×D3—122no—
305C2×D3—122no—
306C2×D3—122no—
307C2×D3—244no—
308C3×C4—22no—
309C3×C4—22no—
310C3×C4—22no—
311C3×C4—42no—
312C3×C4—44no—
313C3×C4—44no—
314C3×C4—42no—
315C3×C4—41no—
316C3×C4—42no—
317C3×C4—182no—
318C3×C4D3×#6712yes—
319C3×C4D3×#6712yes—
320C3×C4#381112yes—
321Dic3—64no—
322Dic3—122no—
323C32—22no—
324C32—21no—
325C32—41no—
326C32#505811yescharacteristic, complemented
327C32#517612yescomplemented
328C23—181no—
329C23—182no—
330Q8—22no—
331D4—182no—
332D4—182no—
333D4—182no—
334D4—1084no—
335C2×C4—21no—
336C2×C4—92no—
337C2×C4C2×#111311yescharacteristic
338C8—94no—
339C2×C3—22no—
340C2×C3—24no—
341C2×C3—22no—
342C2×C3—21no—
343C2×C3—42no—
344C2×C3—41no—
345C2×C3—722no—
346C2×C3—721no—
347C2×C3#6393914yes—
348C2×C3#6394112yes—
349C2×C3#6397912yes—
350C2×C3#6397911yescharacteristic
351D3—242no—
352D3—244no—
353D3—242no—
354D3—242no—
355D3—242no—
356C22—364no—
357C22—362no—
358C22—361no—
359C22—362no—
360C22#8904911yescharacteristic
361C4—41no—
362C4—182no—
363C4#8904912yes—
364C3—22no—
365C3—21no—
366C3—41no—
367C3#1059161912yescomplemented
368C3#1059162111yescharacteristic, complemented
369C2—722no—
370C2—721no—
371C2#1172370912yes—
372C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
hypercenterC2×Q8#1113
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. C3×#64156C2semidirect
2. C2×C3×#3755C2semidirect
3. #11723697C2semidirect
4. C2×#88810C2semidirect
5. #11722071C2semidirect
17. #64156D3semidirect
18. C2×#3755D3semidirect
41. C3×C3⋊(C3⋊C8)D4semidirect
138. C3⋊(C3⋊C8)#67semidirect
217. C33#496semidirect
326. C32#5058semidirect
327. C32#5176semidirect
367. C3#10591619semidirect
368. C3#10591621semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C3×#37552
#117236972
C2×#888102
#117220712
#105916213
#105916193