Finite Group Databasebeta

← #13161987

Subgroups — #13161987

Order 1728.

612 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 103 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×#64155C211yesmaximal, characteristic
2#11723689C211yesmaximal, characteristic
3C3×#63662C212yesmaximal, complemented
4#11722079C212yesmaximal, complemented
5#11722072C211yesmaximal, characteristic
6#10591553—31nomaximal
7#10591549—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6294—21no—
11C3×#3752—21no—
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×#3750—21no—
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88815—21no—
17#64155D311yescharacteristic
18C3×#946—32no—
19C3×#946—61no—
20#63900—31no—
21D3×C4⋊Dic3—32no—
22#63662D312yescomplemented
23C3×#835—32no—
24C3×#835—32no—
25C3×#835—62no—
26#63574—32no—
27#63573—32no—
28#63573—32no—
29#63557—31no—
30(C2×D3)⋊(C3⋊C8)—32no—
31C22×C3×C3⋊D3D411yescharacteristic
32C33×Q8—22no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3—22no—
39#6294—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5043—91no—
42#4805—92no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D312yes—
47C2×C32×Q8C2×D311yescharacteristic
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C2×C3×C4×D3—32no—
50C2×C3×C4×D3—61no—
51C2×#624—61no—
52C2×D3×Dic3—62no—
53#3752—21no—
54C4⋊(C3⋊Dic3)C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)C2×D312yes—
56#3750—21no—
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×(C2×D3)⋊C4—62no—
59C3×(C2×D3)⋊C4—62no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3—62no—
65C3×Dic3⋊C4—62no—
66C3×Dic3⋊C4—62no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3724—61no—
71Dic3⋊Dic3—62no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3SD1612yes—
76C33×C4—41no—
77C33×C4#4112yes—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3—42no—
80C3×C3⋊Dic3—41no—
81C3×C22⋊Q8—91no—
82#946—32no—
83#946—61no—
84C2×C4⋊Dic3—91no—
85D3×C4⋊C4—92no—
86C3×#103—92no—
87#835—32no—
88#835—32no—
89#835—62no—
90#830—91no—
91(C2×D3)⋊C8—92no—
92#817—92no—
93#808—92no—
94#808—92no—
95C22×C3⋊D3#6711yescharacteristic
96C22×C3×D3—32no—
97C22×C3×D3—61no—
98C32×Q8—22no—
99C32×Q8—24no—
100C32×Q8—22no—
101C32×Q8—22no—
102C32×Q8—42no—
103C32×Q8—42no—
104C32×Q8—44no—
105C2×C32×C4—21no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—42no—
112C2×C32×C4—42no—
113C2×C32×C4C22×D312yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3#6711yescharacteristic
118C4×C3⋊D3—22no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—122no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—122no—
129#624—122no—
130D3×Dic3—124no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—122no—
135#483—271noSylow
136C2×C33#12311yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33#13711yescharacteristic
139C3×C3⋊D3—22no—
140C2×C3×Q8—21no—
141C2×C3×Q8—22no—
142C2×C3×Q8—41no—
143C2×C3×Q8D3211yescharacteristic
144C2×C3×Q8D3212yes—
145C22×C3×C4—91no—
146C22×Dic3—181no—
147C2×C4×D3—32no—
148C2×C4×D3—61no—
149C2×C4×D3—182no—
150C2×C3×C8—92no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—92no—
153C3×C4⋊C4—181no—
154C3×C22⋊C4—181no—
155(C2×D3)⋊C4—62no—
156(C2×D3)⋊C4—62no—
157C4⋊Dic3—32no—
158C4⋊Dic3—32no—
159C4⋊Dic3—32no—
160C4⋊Dic3—61no—
161C4⋊Dic3—62no—
162C4⋊Dic3—181no—
163Dic3⋊C4—62no—
164Dic3⋊C4—62no—
165Dic3⋊C4—182no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—62no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32D3×D412yes—
173C22×C32C2×#6711yescharacteristic
174C2×C3⋊D3—21no—
175C2×C3⋊D3—21no—
176C2×C3⋊D3#21512yes—
177C2×C3×D3—34no—
178C2×C3×D3—62no—
179C2×C3×D3—62no—
180C2×C3×D3—62no—
181C2×C3×D3—121no—
182C2×C3×D3—121no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—42no—
188C32×C4—44no—
189C32×C4—44no—
190C32×C4—42no—
191C32×C4—41no—
192C32×C4#23512yes—
193C32×C4#23712yes—
194C32×C4#23712yes—
195C3⋊Dic3—21no—
196C3⋊Dic3—41no—
197C3⋊Dic3—42no—
198C3×Dic3—62no—
199C3×Dic3—121no—
200C3×Dic3—122no—
201C3×Dic3—122no—
202C3×Dic3—122no—
203C3×Dic3—122no—
204C3×Dic3—242no—
205C22⋊Q8—91no—
206C2×C4⋊C4—271no—
207#106—272no—
208#103—92no—
209C22⋊C8—271no—
210C33#48311yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C22×D3—32no—
213C22×D3—61no—
214C3×Q8—24no—
215C3×Q8—22no—
216C3×Q8—22no—
217C3×Q8—22no—
218C3×Q8—42no—
219C3×Q8—42no—
220C3×Q8—44no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—91no—
230C2×C3×C4—182no—
231C2×C3×C4—182no—
232C2×C3×C4—181no—
233C2×C3×C4C2×D3211yescharacteristic
234C2×C3×C4C2×D3212yes—
235C2×Dic3—32no—
236C2×Dic3—61no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—122no—
242C2×Dic3—362no—
243C4×D3—62no—
244C4×D3—62no—
245C4×D3—122no—
246C4×D3—364no—
247C3×C8—182no—
248C3⋊C8—62no—
249C3⋊C8—62no—
250C3⋊C8—122no—
251C2×C32—22no—
252C2×C32—21no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32#88312yes—
261C2×C32D3×SD1612yes—
262C2×C32#91812yes—
263C2×C32#93511yescharacteristic
264C2×C32C2×#21511yescharacteristic
265C2×C32#95111yescharacteristic
266C3⋊D3—22no—
267C3×D3—64no—
268C3×D3—122no—
269C2×Q8#111311yescharacteristic
270C22×C4—91no—
271C22×C4—541no—
272C2×C8—92no—
273C4⋊C4—92no—
274C4⋊C4—91no—
275C4⋊C4—181no—
276C4⋊C4—541no—
277C22⋊C4—181no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3—92no—
282C22×C3—181no—
283C22×C3—181no—
284C22×C3D3×#6712yes—
285C22×C3#381111yescharacteristic
286C2×D3—34no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—62no—
290C2×D3—121no—
291C2×D3—121no—
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—42no—
296C3×C4—44no—
297C3×C4—44no—
298C3×C4—42no—
299C3×C4—41no—
300C3×C4—42no—
301C3×C4—181no—
302C3×C4—362no—
303C3×C4—361no—
304C3×C4#380412yes—
305C3×C4#380412yes—
306C3×C4#380512yes—
307Dic3—62no—
308Dic3—121no—
309Dic3—122no—
310Dic3—122no—
311Dic3—122no—
312Dic3—122no—
313Dic3—242no—
314C32—22no—
315C32—21no—
316C32—41no—
317C32#480512yescomplemented
318C32#504311yescharacteristic, complemented
319C23—91no—
320Q8—22no—
321C2×C4—21no—
322C2×C4—91no—
323C2×C4—182no—
324C2×C4—181no—
325C2×C4—182no—
326C2×C4—1082no—
327C2×C4C2×#111311yescharacteristic
328C8—182no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—21no—
334C2×C3—22no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—182no—
339C2×C3D3×#21512yes—
340C2×C3#6394011yescharacteristic
341C2×C3#6394312yes—
342C2×C3#6394511yescharacteristic
343C2×C3#6396912yes—
344C2×C3#6397511yescharacteristic
345D3—64no—
346D3—122no—
347C22—92no—
348C22—181no—
349C22—181no—
350C22#8904911yescharacteristic
351C4—41no—
352C4—181no—
353C4—362no—
354C4—361no—
355C4#8904612yes—
356C3—21no—
357C3—22no—
358C3—41no—
359C3#1059154912yescomplemented
360C3#1059155311yescharacteristic, complemented
361C2—182no—
362C2#1172370811yescharacteristic
363C2#1172371111yescharacteristic
364C2#1172372411yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×Q8#1113
hypocenterC33#483
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
3. C3×#63662C2semidirect
4. #11722079C2semidirect
22. #63662D3semidirect
210. C33#483semidirect
317. C32#4805semidirect
318. C32#5043semidirect
359. C3#10591549semidirect
360. C3#10591553semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
#117236892
C3×#636622
#117220792
#117220722
#105915533
#105915493