Finite Group Databasebeta

← #13160052

Subgroups — #13160052

Order 1728.

1276 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 178 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×#89019C211yesmaximal, characteristic
2C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
3#11723503C211yesmaximal, characteristic
4C3×#63643C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8C3⋊D3×#103—31nomaximal
9#10590672—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6293—21no—
14C2×#6289—21no—
15C2×#6289—21no—
16#89019—22no—
17#89019C414yescomplemented
18C2×C33×C8C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C8×C3⋊(C3⋊D3)—22no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C32×C4⋊Dic3C2211yescharacteristic
23#88806—21no—
24C2×Q8×C3⋊D3—31no—
25C2×#3797—34no—
26C2×C8×C3⋊D3—31no—
27C2×C8×C3⋊D3—34no—
28C2×C8×C3⋊D3—64no—
29C4⋊C4×C3⋊D3—31no—
30#63900—34no—
31C32×#103—31no—
32#63662—31no—
33#63643D311yescharacteristic, complemented
34#63637—31no—
35C3×#816—34no—
36C3×#816D314yescomplemented
37#63565—34no—
38#63565—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4×C3⋊(C3⋊D3)C2×C412yes—
43C4×C3⋊(C3⋊D3)D412yes—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634C2×C412yescomplemented
48C2×C32×Dic3—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6C2×C412yescomplemented
52#6293—24no—
53#6289—24no—
54#6289—42no—
55C33×C8—21no—
56C3⋊(C3⋊(C3⋊C8))—21no—
57D3×#103—94no—
58D3×#103—91no—
59C2×C32×Q8—31no—
60Q8×C3⋊D3—34no—
61Q8×C3⋊D3—62no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—34no—
64C2×C4×C3⋊D3—61no—
65C2×C4×C3⋊D3—64no—
66C2×C4×C3⋊D3—61no—
67C2×#634—61no—
68C2×#634C2×D311yescharacteristic
69C2×C3×Dic6—34no—
70C2×C3×Dic6C2×D314yes—
71C2×#627—64no—
72C2×#624—64no—
73C2×#624—64no—
74#3797—316no—
75#3797—68no—
76C2×C32×C8—24no—
77C2×C32×C8C2×D314yes—
78C2×C32×C8C2×D311yescharacteristic
79C32×C4⋊C4—31no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3750—61no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85C8×C3⋊D3—62no—
86C8×C3⋊D3—68no—
87C8×C3⋊D3—616no—
88C3×C4⋊Dic3—34no—
89C3×C4⋊Dic3C2×D314yes—
90#3724—64no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)C22⋊C412yes—
93C2×C3⋊(C3⋊D3)SD1612yes—
94C2×C3⋊(C3⋊D3)Q1612yes—
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3⋊(C3⋊Dic3)C22⋊C412yes—
98C3×C3⋊Dic3—22no—
99C3×C3⋊Dic3—41no—
100C32×Dic3—22no—
101C32×Dic3—41no—
102C2×D3×Q8—94no—
103C2×D3×Q8—91no—
104C2×D3×C8—94no—
105C2×D3×C8—91no—
106C2×D3×C8—184no—
107D3×C4⋊C4—94no—
108D3×C4⋊C4—91no—
109C3×#103—34no—
110C3×#103—91no—
111#835—94no—
112#835—91no—
113#816—34no—
114#816C3⋊D311yescharacteristic, complemented
115#810—94no—
116#810—91no—
117C22×C3⋊D3—31no—
118C22×C3⋊D3—34no—
119C22×C3⋊D3—64no—
120C32×Q8—32no—
121C32×Q8—61no—
122C2×C32×C4—24no—
123C2×C32×C4—61no—
124C2×C32×C4—61no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—34no—
131C2×C3⋊Dic3—64no—
132C4×C3⋊D3—32no—
133C4×C3⋊D3—32no—
134C4×C3⋊D3—38no—
135C4×C3⋊D3—38no—
136C4×C3⋊D3—68no—
137C4×C3⋊D3—68no—
138C4×C3⋊D3—64no—
139C4×C3⋊D3—122no—
140#634—21no—
141#634—64no—
142#634C4×D312yescomplemented
143C2×C3×Dic3—24no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—64no—
147C3×Dic6—24no—
148C3×Dic6—38no—
149C3×Dic6—64no—
150C3×Dic6C4×D318yescomplemented
151#627—616no—
152#624—616no—
153#624—128no—
154C32×C8—28no—
155C32×C8—24no—
156C32×C8—21no—
157C3⋊(C3⋊C8)—68no—
158C3⋊(C3⋊C8)—64no—
159C3⋊(C3⋊C8)—61no—
160C2×#103—271noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33C2×SD1611yescharacteristic
163C2×C33C2×Q1611yescharacteristic
164C3⋊(C3⋊D3)#10314yescomplemented
165C2×C3×Q8—34no—
166C2×C3×Q8—91no—
167D3×Q8—916no—
168D3×Q8—94no—
169D3×Q8—182no—
170D3×Q8—188no—
171C2×C4×D3—94no—
172C2×C4×D3—91no—
173C2×C4×D3—184no—
174C2×C4×D3—181no—
175C2×C4×D3—184no—
176C2×C4×D3—181no—
177C2×C4×D3—184no—
178C2×Dic6—34no—
179C2×Dic6—181no—
180C2×Dic6—184no—
181C2×Dic6C2×C3⋊D311yescharacteristic
182C2×C3×C8—24no—
183C2×C3×C8D3214yes—
184C2×C3×C8C2×C3⋊D311yescharacteristic
185C3×C4⋊C4—34no—
186C3×C4⋊C4—91no—
187C4⋊Dic3—34no—
188C4⋊Dic3C2×C3⋊D311yescharacteristic
189Dic3⋊C4—181no—
190Dic3⋊C4—184no—
191C2×C3⋊C8—94no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—184no—
194D3×C8—188no—
195D3×C8—182no—
196D3×C8—1816no—
197C22×C32—24no—
198C22×C32D3×D414yes—
199C22×C32D3×D411yescharacteristic
200C2×C3⋊D3—32no—
201C2×C3⋊D3—32no—
202C2×C3⋊D3—32no—
203C2×C3⋊D3—38no—
204C2×C3⋊D3—38no—
205C2×C3⋊D3—38no—
206C2×C3⋊D3—68no—
207C2×C3⋊D3—68no—
208C2×C3⋊D3—68no—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4—62no—
212C32×C4—121no—
213C32×C4C2×C4×D314yes—
214C32×C4C2×C4×D311yescharacteristic
215C32×C4D3×D414yes—
216C32×C4D3×D411yescharacteristic
217C3⋊Dic3—22no—
218C3⋊Dic3—32no—
219C3⋊Dic3—38no—
220C3⋊Dic3—41no—
221C3⋊Dic3—68no—
222C3×Dic3—28no—
223C3×Dic3—44no—
224C3×Dic3—68no—
225C3×Dic3—124no—
226C22×Q8—271no—
227C22×C8—271no—
228C2×C4⋊C4—271no—
229#103—31no—
230#103—91no—
231#103—271no—
232#103—271no—
233C33C2×#10311yescharacteristic, complemented, Sylow
234C22×D3—94no—
235C22×D3—91no—
236C22×D3—184no—
237C3×Q8—38no—
238C3×Q8—64no—
239C3×Q8—92no—
240C3×Q8—181no—
241C2×C3×C4—24no—
242C2×C3×C4—64no—
243C2×C3×C4—64no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4C2×D3214yes—
247C2×C3×C4C22×C3⋊D311yescharacteristic
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—64no—
251C2×Dic3—64no—
252C2×Dic3—94no—
253C2×Dic3—91no—
254C2×Dic3—184no—
255C4×D3—98no—
256C4×D3—98no—
257C4×D3—92no—
258C4×D3—92no—
259C4×D3—188no—
260C4×D3—188no—
261C4×D3—184no—
262C4×D3—1816no—
263C4×D3—362no—
264C4×D3—368no—
265Dic6—21no—
266Dic6—38no—
267Dic6—64no—
268Dic6—184no—
269Dic6—1816no—
270Dic6C4×C3⋊D312yescomplemented
271C3×C8—28no—
272C3×C8—24no—
273C3×C8—21no—
274C3⋊C8—188no—
275C3⋊C8—184no—
276C3⋊C8—181no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32D3×C22⋊C414yes—
281C2×C32D3×C22⋊C411yescharacteristic
282C2×C32D3×SD1614yes—
283C2×C32D3×SD1611yescharacteristic
284C2×C32D3×Q1614yes—
285C2×C32D3×Q1611yescharacteristic
286C3⋊D3—34no—
287C3⋊D3—316no—
288C3⋊D3—616no—
289C2×Q8—31no—
290C2×Q8—91no—
291C2×Q8—274no—
292C2×Q8—541no—
293C2×Q8—542no—
294C22×C4—271no—
295C22×C4—541no—
296C22×C4—541no—
297C2×C8—271no—
298C2×C8—542no—
299C2×C8D3×C3⋊D311yescharacteristic
300C4⋊C4—31no—
301C4⋊C4—91no—
302C4⋊C4—541no—
303C22×C3—24no—
304C22×C3#380214yes—
305C22×C3D4×C3⋊D311yescharacteristic
306C2×D3—98no—
307C2×D3—98no—
308C2×D3—92no—
309C2×D3—92no—
310C2×D3—92no—
311C2×D3—98no—
312C2×D3—188no—
313C2×D3—188no—
314C2×D3—188no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—68no—
318C3×C4—124no—
319C3×C4—182no—
320C3×C4—361no—
321C3×C4C4×D3214yes—
322C3×C4#381114yes—
323C3×C4C2×C4×C3⋊D311yescharacteristic
324C3×C4D4×C3⋊D311yescharacteristic
325Dic3—22no—
326Dic3—41no—
327Dic3—68no—
328Dic3—92no—
329Dic3—98no—
330Dic3—124no—
331Dic3—188no—
332C32—24no—
333C32D3×#10314yescomplemented
334C32D3×#10311yescharacteristic, complemented
335C23—271no—
336Q8—32no—
337Q8—61no—
338Q8—92no—
339Q8—181no—
340Q8—544no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—271no—
346C2×C4—272no—
347C2×C4—272no—
348C2×C4—544no—
349C2×C4—1082no—
350C2×C4C2×D3×C3⋊D311yescharacteristic
351C8—21no—
352C8—541no—
353C2×C3—24no—
354C2×C3—24no—
355C2×C3—24no—
356C2×C3#6379614yes—
357C2×C3#6380214yes—
358C2×C3#6392114yes—
359C2×C3C22⋊C4×C3⋊D311yescharacteristic
360C2×C3SD16×C3⋊D311yescharacteristic
361C2×C3Q16×C3⋊D311yescharacteristic
362D3—916no—
363D3—94no—
364D3—1816no—
365C22—272no—
366C22—272no—
367C22—272no—
368C22#8902611yescharacteristic
369C4—62no—
370C4—121no—
371C4—182no—
372C4—272no—
373C4—361no—
374C4C4×D3×C3⋊D311yescharacteristic
375C4#8903911yescharacteristic
376C3—24no—
377C3#1059067214yescomplemented
378C3C3⋊D3×#10311yescharacteristic, complemented
379C2—274no—
380C2#1172332111yescharacteristic
381C2#1172333011yescharacteristic
382C2#1172353911yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#103
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C3×#63643C2semidirect
5. C32×#816C2semidirect
6. #11722039C2semidirect
7. #11722037C2semidirect
17. #89019C4semidirect
33. #63643D3semidirect
36. C3×#816D3semidirect
47. C3×#634C2×C4semidirect
51. C32×Dic6C2×C4semidirect
114. #816C3⋊D3semidirect
142. #634C4×D3semidirect
150. C3×Dic6C4×D3semidirect
164. C3⋊(C3⋊D3)#103semidirect
233. C33C2×#103semidirect
270. Dic6C4×C3⋊D3semidirect
333. C32D3×#103semidirect
334. C32D3×#103semidirect
377. C3#10590672semidirect
378. C3C3⋊D3×#103semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
C2×C8×C3⋊(C3⋊D3)2
#117235032
C3×#636432
C32×#8162
#117220392
#117220372
C3⋊D3×#1033
#105906723