Finite Group Databasebeta

← #13161968

Subgroups — #13161968

Order 1728.

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

normal subgroups only

This group has 680 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×D4×C3⋊Dic3C211yesmaximal, characteristic
2C4⋊#6296C211yesmaximal, characteristic, complemented
3C3×#63659C212yesmaximal
4#11722079C212yesmaximal, complemented
5#11722078C211yesmaximal, characteristic, complemented
6#10591495—31nomaximal
7#10591408—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×#6296—21no—
11C3×#3757—21no—
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88815—21no—
17D4×C3⋊Dic3D311yescharacteristic
18C3×D4×Dic3—32no—
19C3×D4×Dic3—61no—
20#63919—31no—
21#63918—32no—
22#63659D312yes—
23C3×#832—32no—
24C3×#832—32no—
25C3×#832—62no—
26#63574—32no—
27#63573—32no—
28#63573—32no—
29#63572—31no—
30Dic3⋊(C3⋊C8)—32no—
31C23×C33—21no—
32C33×D4—21no—
33C33×D4Q812yescomplemented
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—42no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C3×C3⋊Dic3D411yescharacteristic
40#6296—42no—
41C3×C3⋊(C3⋊C8)—22no—
42#5008—91no—
43#4761—92no—
44C2×C32×D4—21no—
45C2×C32×D4—22no—
46C2×C32×D4—41no—
47C2×C32×D4C2×D312yes—
48C2×C32×D4C2×D311yescharacteristic
49C22×C3⋊Dic3—21no—
50C22×C3×Dic3—62no—
51C22×C3×Dic3—62no—
52C2×#627—62no—
53C2×#627—61no—
54#3757—21no—
55C4⋊(C3⋊Dic3)C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)C2×D312yes—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D312yes—
59C3×#217—62no—
60C3×#217—62no—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—32no—
64C3×C4⋊Dic3—61no—
65C3×C4⋊Dic3—62no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—62no—
71#3724—61no—
72Dic3⋊Dic3—62no—
73C22×C33—22no—
74C22×C33—41no—
75C22×C33—41no—
76C22×C33C2×D411yescharacteristic
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3—42no—
82C3×C3⋊Dic3SD1612yes—
83C3×C4×D4—91no—
84D4×Dic3—32no—
85D4×Dic3—61no—
86#888—92no—
87#869—91no—
88C3×#102—92no—
89#832—32no—
90#832—32no—
91#832—62no—
92#817—92no—
93Dic3⋊C8—92no—
94#808—92no—
95#808—92no—
96C4⋊(C3⋊C8)—91no—
97C23×C32—21no—
98C23×C32—22no—
99C23×C32—22no—
100C23×C32—42no—
101C23×C32—42no—
102C32×D4—21no—
103C32×D4—22no—
104C32×D4—22no—
105C32×D4—22no—
106C32×D4—24no—
107C32×D4—42no—
108C32×D4—42no—
109C32×D4—42no—
110C32×D4Dic612yescomplemented
111C32×D4Dic614yescomplemented
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D312yes—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—42no—
120C2×C3⋊Dic3#6711yescharacteristic
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3×Dic3—32no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—122no—
131C2×C3×Dic3—124no—
132C2×C3×Dic3—124no—
133#627—124no—
134#627—122no—
135C3⋊(C3⋊C8)—22no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—122no—
139#476—271noSylow
140C2×C33—42no—
141C2×C33C22⋊Q811yescharacteristic
142C2×C33C2×SD1611yescharacteristic
143C2×C33AGL(1,ℤ/8)11yescharacteristic
144C2×C3×D4—21no—
145C2×C3×D4—22no—
146C2×C3×D4—41no—
147C2×C3×D4D3211yescharacteristic
148C2×C3×D4D3212yes—
149C22×C3×C4—181no—
150C22×Dic3—62no—
151C22×Dic3—62no—
152C2×Dic6—182no—
153C2×Dic6—181no—
154C2×C3×C8—92no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—92no—
157C3×C22⋊C4—181no—
158C3×C42—91no—
159#217—62no—
160#217—62no—
161C4⋊Dic3—32no—
162C4⋊Dic3—32no—
163C4⋊Dic3—32no—
164C4⋊Dic3—61no—
165C4⋊Dic3—62no—
166C4⋊Dic3—181no—
167Dic3⋊C4—182no—
168C4×Dic3—32no—
169C4×Dic3—61no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—32no—
172C2×C3⋊C8—62no—
173C22×C32—22no—
174C22×C32—24no—
175C22×C32—24no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—44no—
180C22×C32—44no—
181C22×C32—41no—
182C22×C32—41no—
183C22×C32—42no—
184C22×C32—42no—
185C22×C32—42no—
186C22×C32—42no—
187C22×C32—82no—
188C22×C32—82no—
189C22×C32—82no—
190C22×C32—82no—
191C22×C32D3×D412yes—
192C22×C32C2×#6711yescharacteristic
193C32×C4—22no—
194C32×C4—21no—
195C32×C4—22no—
196C32×C4—21no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4C2×Dic612yes—
200C32×C4C2×Dic611yescharacteristic
201C32×C4#23511yescharacteristic
202C32×C4#23712yes—
203C3⋊Dic3—21no—
204C3⋊Dic3—41no—
205C3⋊Dic3—42no—
206C3⋊Dic3#21412yes—
207C3×Dic3—34no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C3×Dic3—121no—
211C3×Dic3—122no—
212C3×Dic3—122no—
213C3×Dic3—122no—
214C3×Dic3—122no—
215C3×Dic3—242no—
216C4⋊Q8—271no—
217C4×D4—91no—
218#106—272no—
219C4⋊C8—271no—
220#102—92no—
221C33#47611yescharacteristic, complemented, Sylow
222C23×C3—22no—
223C23×C3—22no—
224C23×C3—21no—
225C23×C3—42no—
226C23×C3—42no—
227C3×D4—22no—
228C3×D4—22no—
229C3×D4—21no—
230C3×D4—22no—
231C3×D4—24no—
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4#62714yescomplemented
236C3×D4#62712yescomplemented
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—182no—
243C2×C3×C4—181no—
244C2×C3×C4—362no—
245C2×C3×C4C2×D3212yes—
246C2×C3×C4C2×D3211yescharacteristic
247C2×Dic3—32no—
248C2×Dic3—32no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—122no—
256C2×Dic3—124no—
257C2×Dic3—124no—
258Dic6—364no—
259Dic6—362no—
260C3×C8—182no—
261C3⋊C8—62no—
262C3⋊C8—62no—
263C3⋊C8—122no—
264C2×C32—22no—
265C2×C32—21no—
266C2×C32—22no—
267C2×C32—21no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—42no—
274C2×C32—44no—
275C2×C32—44no—
276C2×C32—84no—
277C2×C32—84no—
278C2×C32#87812yes—
279C2×C32#91112yes—
280C2×C32D3×SD1612yes—
281C2×C32#92411yescharacteristic
282C2×C32C2×#21411yescharacteristic
283C2×C32#94911yescharacteristic
284C2×Q8—541no—
285C2×D4#111311yescharacteristic
286C22×C4—181no—
287C2×C8—92no—
288C4⋊C4—92no—
289C4⋊C4—91no—
290C4⋊C4—541no—
291C22⋊C4—181no—
292C42—91no—
293C22×C3—24no—
294C22×C3—24no—
295C22×C3—22no—
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3—42no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—82no—
311C22×C3D3×#6712yes—
312C22×C3#381111yescharacteristic
313C3×C4—22no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—22no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—92no—
320C3×C4—181no—
321C3×C4—362no—
322C3×C4—361no—
323C3×C4#380412yes—
324C3×C4#380511yescharacteristic
325C3×C4C2×#62712yes—
326C3×C4C2×#62711yescharacteristic
327Dic3—34no—
328Dic3—62no—
329Dic3—62no—
330Dic3—121no—
331Dic3—122no—
332Dic3—122no—
333Dic3—122no—
334Dic3—122no—
335Dic3—242no—
336C32—21no—
337C32—22no—
338C32—41no—
339C32#476112yescomplemented
340C32#500811yescharacteristic, complemented
341C23—21no—
342Q8—1082no—
343D4—21no—
344D4#629612yescomplemented
345C2×C4—91no—
346C2×C4—91no—
347C2×C4—181no—
348C2×C4—182no—
349C2×C4—362no—
350C2×C4C2×#111311yescharacteristic
351C8—182no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—44no—
359C2×C3—42no—
360C2×C3—44no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—84no—
365C2×C3—84no—
366C2×C3#6392512yes—
367C2×C3#6392711yescharacteristic
368C2×C3D3×#21412yes—
369C2×C3#6393011yescharacteristic
370C2×C3#6396412yes—
371C2×C3#6398211yescharacteristic
372C22—22no—
373C22—41no—
374C22—41no—
375C22#8904911yescharacteristic
376C4—92no—
377C4—181no—
378C4—362no—
379C4—361no—
380C4#8904611yescharacteristic
381C4C2×#629611yescharacteristic
382C3—21no—
383C3—22no—
384C3—41no—
385C3#1059140812yescomplemented
386C3#1059149511yescharacteristic, complemented
387C2—42no—
388C2#1172370311yescharacteristic
389C2#1172370411yescharacteristic
390C2#1172372111yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#476
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C4⋊#6296C2semidirect
4. #11722079C2semidirect
5. #11722078C2semidirect
33. C33×D4Q8semidirect
110. C32×D4Dic6semidirect
111. C32×D4Dic6semidirect
221. C33#476semidirect
235. C3×D4#627semidirect
236. C3×D4#627semidirect
339. C32#4761semidirect
340. C32#5008semidirect
344. D4#6296semidirect
385. C3#10591408semidirect
386. C3#10591495semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C4⋊#62962
C3×#636592
#117220792
#117220782
#105914953
#105914083