Finite Group Databasebeta

← #13161132

Subgroups — #13161132

Order 1728.

1596 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1596 conjugacy classes of subgroups — too many to draw. Only its 188 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×Q8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8C3⋊D3×#103—31nomaximal
9#10591552—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11Q8×C3⋊(C3⋊D3)—22no—
12Q8×C3⋊(C3⋊D3)C414yescomplemented
13C2×C4×C3⋊(C3⋊D3)—21no—
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×#6308—21no—
16C2×#6308C2211yescharacteristic
17C2×#6289—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88806—21no—
23#88785—22no—
24C2×Q8×C3⋊D3—31no—
25C2×Q8×C3⋊D3—34no—
26C2×Q8×C3⋊D3—64no—
27C2×C8×C3⋊D3—31no—
28C4⋊C4×C3⋊D3—31no—
29#63900—34no—
30C2×#3710—34no—
31C32×#103—31no—
32#63662D311yescharacteristic, complemented
33#63643—31no—
34#63637—31no—
35C3×#835—34no—
36C3×#835D314yescomplemented
37#63567—34no—
38#63567—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C33×Q8—21no—
41C33×Q8C2×C412yescomplemented
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C4×C3⋊(C3⋊D3)—24no—
47C4×C3⋊(C3⋊D3)C2×C412yes—
48C4×C3⋊(C3⋊D3)D412yes—
49#6308—21no—
50#6308—24no—
51#6308C2×C412yescomplemented
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54#6289—42no—
55C3×C3⋊(C3⋊C8)—21no—
56C32×C3⋊C8—21no—
57D3×#103—94no—
58D3×#103—91no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C2×D311yescharacteristic
62Q8×C3⋊D3—34no—
63Q8×C3⋊D3—316no—
64Q8×C3⋊D3—62no—
65Q8×C3⋊D3—68no—
66Q8×C3⋊D3—616no—
67Q8×C3⋊D3—616no—
68C2×C4×C3⋊D3—31no—
69C2×C4×C3⋊D3—34no—
70C2×C4×C3⋊D3—61no—
71C2×C4×C3⋊D3—64no—
72C2×C4×C3⋊D3—64no—
73C2×C4×C3⋊D3—68no—
74C2×C4×C3⋊D3—61no—
75C2×#634—31no—
76C2×#634—34no—
77C2×#634—61no—
78C2×#634—64no—
79C2×#634—64no—
80C2×#634—68no—
81C2×#624—64no—
82C2×C32×C8—31no—
83C32×C4⋊C4—31no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85#3750—61no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C8×C3⋊D3—62no—
88C3×C4⋊Dic3—34no—
89C3×C4⋊Dic3C2×D314yes—
90C2×C3×C3⋊C8—34no—
91C2×C3×C3⋊C8C2×D314yes—
92#3724—64no—
93#3710—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3⋊(C3⋊D3)C22⋊C412yes—
96C2×C3⋊(C3⋊D3)SD1612yes—
97C2×C3⋊(C3⋊D3)Q1612yes—
98C33×C4—22no—
99C33×C4C22×C411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—22no—
102C3⋊(C3⋊Dic3)C22⋊C412yes—
103C3×C3⋊Dic3—41no—
104C32×Dic3—41no—
105C2×D3×Q8—94no—
106C2×D3×Q8—91no—
107C2×D3×Q8—184no—
108C2×D3×C8—94no—
109C2×D3×C8—91no—
110D3×C4⋊C4—94no—
111D3×C4⋊C4—91no—
112C3×#103—34no—
113C3×#103—91no—
114#835—34no—
115#835C3⋊D311yescharacteristic, complemented
116#816—94no—
117#816—91no—
118#810—94no—
119#810—91no—
120C22×C3⋊D3—31no—
121C22×C3⋊D3—34no—
122C22×C3⋊D3—64no—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—21no—
126C32×Q8—28no—
127C32×Q8C4×D318yescomplemented
128C32×Q8C4×D312yescomplemented
129C2×C32×C4—28no—
130C2×C32×C4—24no—
131C2×C32×C4—21no—
132C2×C32×C4—24no—
133C2×C32×C4—61no—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—31no—
138C2×C3⋊Dic3—34no—
139C2×C3⋊Dic3—64no—
140C2×C3⋊Dic3—68no—
141C2×C3⋊Dic3—64no—
142C2×C3⋊Dic3—61no—
143C4×C3⋊D3—32no—
144C4×C3⋊D3—32no—
145C4×C3⋊D3—38no—
146C4×C3⋊D3—38no—
147C4×C3⋊D3—68no—
148C4×C3⋊D3—68no—
149C4×C3⋊D3—64no—
150C4×C3⋊D3—616no—
151C4×C3⋊D3—632no—
152C4×C3⋊D3—122no—
153#634—32no—
154#634—38no—
155#634—68no—
156#634—64no—
157#634—616no—
158#634—632no—
159#634—68no—
160#634—64no—
161#634—61no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—64no—
164#624—128no—
165C32×C8—61no—
166C3⋊(C3⋊C8)—21no—
167C3×C3⋊C8—24no—
168C3×C3⋊C8—64no—
169C2×#103—271noSylow
170C2×C33C2×C22⋊C411yescharacteristic
171C2×C33C2×SD1611yescharacteristic
172C2×C33C2×Q1611yescharacteristic
173C3⋊(C3⋊D3)#10314yescomplemented
174C2×C3×Q8—24no—
175C2×C3×Q8D3214yes—
176C2×C3×Q8C2×C3⋊D311yescharacteristic
177D3×Q8—916no—
178D3×Q8—94no—
179D3×Q8—1816no—
180D3×Q8—188no—
181D3×Q8—182no—
182D3×Q8—1816no—
183C2×C4×D3—94no—
184C2×C4×D3—91no—
185C2×C4×D3—184no—
186C2×C4×D3—184no—
187C2×C4×D3—181no—
188C2×C4×D3—188no—
189C2×C4×D3—181no—
190C2×C4×D3—184no—
191C2×Dic6—94no—
192C2×Dic6—91no—
193C2×Dic6—184no—
194C2×Dic6—184no—
195C2×Dic6—181no—
196C2×Dic6—188no—
197C2×C3×C8—34no—
198C2×C3×C8—91no—
199C3×C4⋊C4—34no—
200C3×C4⋊C4—91no—
201C4⋊Dic3—34no—
202C4⋊Dic3C2×C3⋊D311yescharacteristic
203Dic3⋊C4—181no—
204Dic3⋊C4—184no—
205C2×C3⋊C8—34no—
206C2×C3⋊C8C2×C3⋊D311yescharacteristic
207D3×C8—182no—
208D3×C8—188no—
209C22×C32—24no—
210C22×C32D3×D414yes—
211C22×C32D3×D411yescharacteristic
212C2×C3⋊D3—32no—
213C2×C3⋊D3—32no—
214C2×C3⋊D3—32no—
215C2×C3⋊D3—38no—
216C2×C3⋊D3—38no—
217C2×C3⋊D3—38no—
218C2×C3⋊D3—68no—
219C2×C3⋊D3—68no—
220C2×C3⋊D3—68no—
221C32×C4—216no—
222C32×C4—28no—
223C32×C4—22no—
224C32×C4—24no—
225C32×C4—24no—
226C32×C4—121no—
227C32×C4C2×C4×D314yes—
228C32×C4C2×C4×D311yescharacteristic
229C32×C4D3×D414yes—
230C32×C4D3×D411yescharacteristic
231C3⋊Dic3—32no—
232C3⋊Dic3—38no—
233C3⋊Dic3—41no—
234C3⋊Dic3—68no—
235C3⋊Dic3—616no—
236C3⋊Dic3—68no—
237C3⋊Dic3—62no—
238C3×Dic3—44no—
239C3×Dic3—124no—
240C22×Q8—271no—
241C22×C8—271no—
242C2×C4⋊C4—271no—
243#103—31no—
244#103—91no—
245#103—271no—
246#103—271no—
247C33C2×#10311yescharacteristic, complemented, Sylow
248C22×D3—94no—
249C22×D3—91no—
250C22×D3—184no—
251C3×Q8—28no—
252C3×Q8—24no—
253C3×Q8—21no—
254C3×Q8—28no—
255C3×Q8#62418yescomplemented
256C3×Q8C4×C3⋊D312yescomplemented
257C2×C3×C4—28no—
258C2×C3×C4—24no—
259C2×C3×C4—21no—
260C2×C3×C4—24no—
261C2×C3×C4—64no—
262C2×C3×C4—181no—
263C2×C3×C4C2×D3214yes—
264C2×C3×C4C22×C3⋊D311yescharacteristic
265C2×Dic3—21no—
266C2×Dic3—64no—
267C2×Dic3—94no—
268C2×Dic3—91no—
269C2×Dic3—184no—
270C2×Dic3—188no—
271C2×Dic3—184no—
272C2×Dic3—181no—
273C4×D3—98no—
274C4×D3—98no—
275C4×D3—92no—
276C4×D3—92no—
277C4×D3—188no—
278C4×D3—188no—
279C4×D3—184no—
280C4×D3—1816no—
281C4×D3—1832no—
282C4×D3—362no—
283C4×D3—368no—
284Dic6—98no—
285Dic6—92no—
286Dic6—188no—
287Dic6—184no—
288Dic6—1816no—
289Dic6—1832no—
290Dic6—188no—
291Dic6—184no—
292Dic6—181no—
293C3×C8—64no—
294C3×C8—181no—
295C3⋊C8—21no—
296C3⋊C8—64no—
297C2×C32—24no—
298C2×C32—24no—
299C2×C32—24no—
300C2×C32D3×C22⋊C414yes—
301C2×C32D3×C22⋊C411yescharacteristic
302C2×C32D3×SD1614yes—
303C2×C32D3×SD1611yescharacteristic
304C2×C32D3×Q1614yes—
305C2×C32D3×Q1611yescharacteristic
306C3⋊D3—34no—
307C3⋊D3—316no—
308C3⋊D3—616no—
309C2×Q8—274no—
310C2×Q8—271no—
311C2×Q8—541no—
312C2×Q8—542no—
313C2×Q8D3×C3⋊D311yescharacteristic
314C22×C4—271no—
315C22×C4—541no—
316C22×C4—541no—
317C2×C8—31no—
318C2×C8—91no—
319C2×C8—542no—
320C4⋊C4—31no—
321C4⋊C4—91no—
322C4⋊C4—541no—
323C22×C3—24no—
324C22×C3#381114yes—
325C22×C3D4×C3⋊D311yescharacteristic
326C2×D3—92no—
327C2×D3—92no—
328C2×D3—92no—
329C2×D3—98no—
330C2×D3—98no—
331C2×D3—98no—
332C2×D3—188no—
333C2×D3—188no—
334C2×D3—188no—
335C3×C4—216no—
336C3×C4—22no—
337C3×C4—28no—
338C3×C4—24no—
339C3×C4—24no—
340C3×C4—124no—
341C3×C4—361no—
342C3×C4C2×#62414yes—
343C3×C4#381114yes—
344C3×C4C2×C4×C3⋊D311yescharacteristic
345C3×C4D4×C3⋊D311yescharacteristic
346Dic3—41no—
347Dic3—92no—
348Dic3—98no—
349Dic3—124no—
350Dic3—188no—
351Dic3—1816no—
352Dic3—182no—
353Dic3—188no—
354C32—24no—
355C32D3×#10314yescomplemented
356C32D3×#10311yescharacteristic, complemented
357C23—271no—
358Q8—21no—
359Q8—272no—
360Q8—544no—
361Q8—541no—
362Q8#628912yescomplemented
363C2×C4—21no—
364C2×C4—61no—
365C2×C4—181no—
366C2×C4—271no—
367C2×C4—272no—
368C2×C4—272no—
369C2×C4—544no—
370C2×C4—541no—
371C2×C4—1082no—
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—61no—
374C8—181no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3#6394014yes—
379C2×C3#6394414yes—
380C2×C3#6397414yes—
381C2×C3C22⋊C4×C3⋊D311yescharacteristic
382C2×C3SD16×C3⋊D311yescharacteristic
383C2×C3Q16×C3⋊D311yescharacteristic
384D3—916no—
385D3—94no—
386D3—1816no—
387C22—272no—
388C22—272no—
389C22—272no—
390C22#8903911yescharacteristic
391C4—22no—
392C4—121no—
393C4—272no—
394C4—361no—
395C4—542no—
396C4C2×#628911yescharacteristic
397C4#8903911yescharacteristic
398C3—24no—
399C3#1059155214yescomplemented
400C3C3⋊D3×#10311yescharacteristic, complemented
401C2—274no—
402C2#1172357111yescharacteristic
403C2#1172357711yescharacteristic
404C2#1172362711yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C2×#103
LCSG ▷ 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
4. C3×#63662C2semidirect
5. C32×#835C2semidirect
6. #11722042C2semidirect
7. #11722041C2semidirect
12. Q8×C3⋊(C3⋊D3)C4semidirect
32. #63662D3semidirect
36. C3×#835D3semidirect
41. C33×Q8C2×C4semidirect
51. #6308C2×C4semidirect
115. #835C3⋊D3semidirect
127. C32×Q8C4×D3semidirect
128. C32×Q8C4×D3semidirect
173. C3⋊(C3⋊D3)#103semidirect
247. C33C2×#103semidirect
255. C3×Q8#624semidirect
256. C3×Q8C4×C3⋊D3semidirect
355. C32D3×#103semidirect
356. C32D3×#103semidirect
362. Q8#6289semidirect
399. C3#10591552semidirect
400. C3C3⋊D3×#103semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊(C3⋊D3)2
#117235032
C2×#887852
C3×#636622
C32×#8352
#117220422
#117220412
C3⋊D3×#1033
#105915523