Finite Group Databasebeta

← #13161498

Subgroups — #13161498

Order 1728.

554 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 119 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×#64079C211yesmaximal, characteristic
2C3×#64078C211yesmaximal, characteristic
3C3×#63630C211yesmaximal, characteristic
4#11722080C212yesmaximal
5#11722079C212yesmaximal
6#10590315—31nomaximal
7#10590306—31nomaximal
8#10590305—31nomaximal
9C2×C3×#634—21no—
10C33×C42C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#3750—21no—
16C3×#3750—21no—
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18#64079D311yescharacteristic
19#64078D311yescharacteristic
20C3×#870—31no—
21C3×#870—31no—
22C3×#870—61no—
23C3×#869—31no—
24C3×#869—31no—
25C3×#869—61no—
26#63630D311yescharacteristic
27C3×#803—31no—
28C3×#803—31no—
29C3×#803—61no—
30#63576—32no—
31#63575—32no—
32#63575—32no—
33#63574—32no—
34#63573—32no—
35#63573—32no—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—22no—
42C3×#634—42no—
43C3×C3⋊(C3⋊C8)Q814yescomplemented
44#5081—91no—
45#5064—91no—
46#4702—91no—
47C2×#634—21no—
48C2×C3×Dic6—62no—
49C2×C3×Dic6—61no—
50C2×C3×Dic6—61no—
51C32×C42—21no—
52C32×C42—21no—
53C32×C42—21no—
54C32×C42—41no—
55C32×C42C2×D311yescharacteristic
56C32×C42C2×D311yescharacteristic
57C32×C42C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C2×D312yes—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63#3750—21no—
64C2×C3⋊(C3⋊C8)C2×D312yes—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—32no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—62no—
72C3×C4⋊Dic3—61no—
73C3×C4⋊Dic3—61no—
74C3×C4⋊Dic3—62no—
75C3×C4⋊Dic3—61no—
76C3×C4⋊Dic3—61no—
77C3×Dic3⋊C4—62no—
78C3×Dic3⋊C4—62no—
79C3×Dic3⋊C4—61no—
80C3×Dic3⋊C4—61no—
81C3×Dic3⋊C4—61no—
82C3×Dic3⋊C4—61no—
83C2×C3×C3⋊C8—32no—
84C2×C3×C3⋊C8—32no—
85C2×C3×C3⋊C8—62no—
86C22×C33C2×D411yescharacteristic
87C33×C4—21no—
88C33×C4—21no—
89C33×C4C2×Q812yes—
90C3×C3⋊Dic3—42no—
91C3×C3⋊Dic3—41no—
92C3×C3⋊Dic3—41no—
93C3×C4⋊Q8—91no—
94C3×#125—91no—
95#870—31no—
96#870—31no—
97#870—61no—
98#869—31no—
99#869—31no—
100#869—61no—
101C3×#97—91no—
102#818—92no—
103#817—92no—
104#808—92no—
105#808—92no—
106#807—92no—
107#807—92no—
108#803—31no—
109#803—31no—
110#803—61no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C2×C32×C4—41no—
123C2×C32×C4D1211yescharacteristic
124C2×C32×C4D1211yescharacteristic
125C2×C32×C4#6711yescharacteristic
126C2×C32×C4#6711yescharacteristic
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—22no—
135#634—42no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—62no—
145C3×Dic6—124no—
146C3×Dic6—122no—
147C3×Dic6—122no—
148C3⋊(C3⋊C8)Dic614yescomplemented
149C3×C3⋊C8—34no—
150C3×C3⋊C8—34no—
151C3×C3⋊C8—64no—
152#501—271noSylow
153C2×C33C4⋊Q811yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C2×C33#13711yescharacteristic
156C2×C3×Q8—181no—
157C2×Dic6—62no—
158C2×Dic6—61no—
159C2×Dic6—61no—
160C2×C3×C8—92no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—92no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—181no—
165C3×C4⋊C4—181no—
166C3×C4⋊C4—181no—
167C3×C42—21no—
168C3×C42—21no—
169C3×C42—21no—
170C3×C42—41no—
171C3×C42D3211yescharacteristic
172C3×C42D3211yescharacteristic
173C3×C42D3211yescharacteristic
174C4⋊Dic3—31no—
175C4⋊Dic3—32no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—32no—
179C4⋊Dic3—31no—
180C4⋊Dic3—62no—
181C4⋊Dic3—61no—
182C4⋊Dic3—61no—
183C4⋊Dic3—62no—
184C4⋊Dic3—61no—
185C4⋊Dic3—61no—
186Dic3⋊C4—62no—
187Dic3⋊C4—62no—
188Dic3⋊C4—61no—
189Dic3⋊C4—61no—
190Dic3⋊C4—61no—
191Dic3⋊C4—61no—
192C2×C3⋊C8—32no—
193C2×C3⋊C8—32no—
194C2×C3⋊C8—62no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—41no—
199C22×C32C2×D1211yescharacteristic
200C22×C32C2×#6711yescharacteristic
201C22×C32C2×#6711yescharacteristic
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—22no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—42no—
214C32×C4—41no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—41no—
218C32×C4—42no—
219C32×C4—42no—
220C32×C4C2×Dic612yes—
221C32×C4D3×Q812yes—
222C32×C4D3×Q812yes—
223C3⋊Dic3—42no—
224C3⋊Dic3—41no—
225C3⋊Dic3—41no—
226C3×Dic3—124no—
227C3×Dic3—122no—
228C3×Dic3—122no—
229C3×Dic3—122no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234C3×Dic3—122no—
235C4⋊Q8—91no—
236#125—91no—
237#107—272no—
238#106—272no—
239#97—91no—
240C33#50111yescharacteristic, complemented, Sylow
241C3×Q8—362no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4—182no—
257C2×C3×C4#62511yescharacteristic
258C2×C3×C4#62511yescharacteristic
259C2×C3×C4#62611yescharacteristic
260C2×C3×C4#62611yescharacteristic
261C2×C3×C4#62611yescharacteristic
262C2×C3×C4#62611yescharacteristic
263C2×C3×C4C2×D3211yescharacteristic
264C2×C3×C4C2×D3211yescharacteristic
265C2×C3×C4C2×D3211yescharacteristic
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—64no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—62no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—62no—
275Dic6—124no—
276Dic6—122no—
277Dic6—122no—
278C3×C8—94no—
279C3⋊C8—34no—
280C3⋊C8—34no—
281C3⋊C8—64no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32#86911yescharacteristic
295C2×C32#90811yescharacteristic
296C2×C32#90911yescharacteristic
297C2×C32#93211yescharacteristic
298C2×C32#94211yescharacteristic
299C2×C32#94411yescharacteristic
300C2×C32#94411yescharacteristic
301C2×C32#94911yescharacteristic
302C2×C32#95111yescharacteristic
303C2×Q8—181no—
304C2×C8—92no—
305C4⋊C4—91no—
306C4⋊C4—92no—
307C4⋊C4—91no—
308C4⋊C4—181no—
309C4⋊C4—181no—
310C4⋊C4—181no—
311C42#111311yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—41no—
316C22×C3C2×#62511yescharacteristic
317C22×C3C2×#62611yescharacteristic
318C22×C3C2×#62611yescharacteristic
319C3×C4—22no—
320C3×C4—22no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—362no—
338C3×C4—361no—
339C3×C4—361no—
340C3×C4D3×Dic612yes—
341C3×C4D3×Dic612yes—
342C3×C4#379712yes—
343Dic3—124no—
344Dic3—122no—
345Dic3—122no—
346Dic3—122no—
347Dic3—121no—
348Dic3—121no—
349Dic3—121no—
350Dic3—122no—
351Dic3—121no—
352C32—21no—
353C32—21no—
354C32—21no—
355C32—41no—
356C32#470211yescharacteristic, complemented
357C32#506411yescharacteristic, complemented
358C32#508111yescharacteristic, complemented
359Q8—362no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—182no—
363C2×C4#629511yescharacteristic
364C2×C4#629511yescharacteristic
365C2×C4C2×#111311yescharacteristic
366C8—94no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3#6382311yescharacteristic
380C2×C3#6382511yescharacteristic
381C2×C3#6382711yescharacteristic
382C2×C3#6383011yescharacteristic
383C2×C3#6383111yescharacteristic
384C2×C3#6383311yescharacteristic
385C2×C3#6386611yescharacteristic
386C2×C3#6386611yescharacteristic
387C2×C3#6387311yescharacteristic
388C22C2×#629511yescharacteristic
389C4—21no—
390C4—21no—
391C4—362no—
392C4—361no—
393C4—361no—
394C4#8904012yes—
395C3—21no—
396C3—21no—
397C3—21no—
398C3—41no—
399C3#1059030511yescharacteristic, complemented
400C3#1059030611yescharacteristic, complemented
401C3#1059031511yescharacteristic, complemented
402C2#1172366111yescharacteristic
403C2#1172366311yescharacteristic
404C2#1172367811yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42#1113
hypocenterC33#501
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
43. C3×C3⋊(C3⋊C8)Q8semidirect
148. C3⋊(C3⋊C8)Dic6semidirect
240. C33#501semidirect
356. C32#4702semidirect
357. C32#5064semidirect
358. C32#5081semidirect
399. C3#10590305semidirect
400. C3#10590306semidirect
401. C3#10590315semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C3×#640782
C3×#636302
#117220802
#117220792
#105903153
#105903063
#105903053