Finite Group Databasebeta

← #13161191

Subgroups — #13161191

Order 1728.

1220 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 200 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×C32×D3×Q8C211yesmaximal, characteristic, complemented
2C2×#88862C211yesmaximal, characteristic, complemented
3#11723516C211yesmaximal, characteristic
4C2×#88795C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592926—31nomaximal
9#10591584—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C32×D3×Q8—22no—
13C32×D3×Q8—22no—
14C2×C32×C4×D3—21no—
15C2×C32×C4×D3C2211yescharacteristic
16C2×C32×Dic6—21no—
17C2×C32×Dic6C2211yescharacteristic
18C2×#6291—21no—
19#88862—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23#88804—21no—
24#88795—22no—
25C22×C32×Q8—31no—
26C2×C3×D3×Q8D314yescomplemented
27C2×#3755—31no—
28C2×#3755—34no—
29C2×#3755—31no—
30C2×#3755—64no—
31#64139—31no—
32#63909—34no—
33C2×#3717—34no—
34#63662—31no—
35#63662D311yescharacteristic, complemented
36#63657—31no—
37C3×#835—34no—
38#63565—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C32×D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8D412yescomplemented
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3D411yescharacteristic
51C32×C4×D3—24no—
52C32×C4×D3—21no—
53C32×C4×D3—21no—
54C32×Dic6—24no—
55C32×Dic6—21no—
56C32×Dic6D412yescomplemented
57#6291—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5227—94no—
61#4806—91no—
62C2×C32×Q8—24no—
63C2×C32×Q8—31no—
64C2×C32×Q8—61no—
65C2×C32×Q8—62no—
66C2×C32×Q8—62no—
67C2×C32×Q8C2×D311yescharacteristic
68C2×C32×Q8C2×D314yes—
69C22×C32×C4—31no—
70C22×C32×C4—61no—
71C2×#638—61no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—34no—
74C2×C4⋊(C3⋊D3)—64no—
75C3×D3×Q8—28no—
76C3×D3×Q8—28no—
77C2×C3×C4×D3—24no—
78C2×C3×C4×D3C2×D314yes—
79C2×C3×Dic6—24no—
80C2×C3×Dic6C2×D314yes—
81C2×#626—64no—
82#3755—62no—
83#3755—616no—
84#3755—62no—
85#3755—68no—
86#3752—61no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×C4⋊Dic3—34no—
93C2×C3×C3⋊C8—34no—
94#3722—64no—
95#3717—68no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C32×D3—21no—
101C2×C32×D3—21no—
102C2×C32×D3SD1612yescomplemented
103C33×C4—22no—
104C33×C4C2×D411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—41no—
107C32×Dic3—21no—
108C32×Dic3—22no—
109C22×C3×Q8—34no—
110C2×D3×Q8C3⋊D311yescharacteristic, complemented
111C2×#215—94no—
112C2×#215—94no—
113C2×#215—91no—
114C2×#215—184no—
115#930—94no—
116C2×#204—91no—
117#895—91no—
118C3×#103—91no—
119#835—34no—
120#835—94no—
121#830—94no—
122(C2×D3)⋊C8—91no—
123#810—91no—
124C23×C32—31no—
125C22×C3⋊D3—61no—
126C22×C3⋊D3—64no—
127C22×C3⋊D3—124no—
128C22×C3×D3#6714yes—
129C32×Q8—21no—
130C32×Q8—28no—
131C32×Q8—24no—
132C32×Q8—32no—
133C32×Q8—44no—
134C32×Q8—64no—
135C32×Q8—61no—
136C32×Q8D1212yescomplemented
137C32×Q8#6718yescomplemented
138C2×C32×C4—21no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—31no—
142C2×C32×C4—44no—
143C2×C32×C4—64no—
144C2×C32×C4—61no—
145C2×C32×C4—61no—
146C2×C32×C4—61no—
147C2×C32×C4C22×D314yes—
148C2×C32×C4C22×D311yescharacteristic
149#638—122no—
150C2×C3⋊Dic3—21no—
151C4⋊(C3⋊D3)—68no—
152C4⋊(C3⋊D3)—61no—
153C4⋊(C3⋊D3)—61no—
154C4⋊(C3⋊D3)—64no—
155C4⋊(C3⋊D3)—64no—
156C4⋊(C3⋊D3)—124no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—64no—
159C2×C3×Dic3#6714yes—
160C3×C4×D3—216no—
161C3×C4×D3—24no—
162C3×C4×D3—24no—
163C3×Dic6—216no—
164C3×Dic6—24no—
165C3×Dic6#6718yescomplemented
166#626—128no—
167C3⋊(C3⋊C8)—21no—
168C3⋊(C3⋊C8)—68no—
169C3⋊(C3⋊C8)—61no—
170C3⋊(C3⋊C8)—64no—
171C3×C3⋊C8—64no—
172#448—271noSylow
173C2×C33#12011yescharacteristic
174C2×C33C2×SD1611yescharacteristic
175C2×C33#13711yescharacteristic
176C3⋊(C3⋊D3)—81no—
177C32×D3—22no—
178C2×C3×Q8—24no—
179C2×C3×Q8—34no—
180C2×C3×Q8—64no—
181C2×C3×Q8—68no—
182C2×C3×Q8—68no—
183C2×C3×Q8D3214yes—
184C2×C3×Q8C2×C3⋊D311yescharacteristic
185C22×C3×C4—34no—
186C22×C3×C4—64no—
187C2×#67—184no—
188D3×Q8—22no—
189D3×Q8—22no—
190C2×D12—94no—
191C2×D12—91no—
192C2×D12—184no—
193C2×D12—181no—
194C2×C4×D3—21no—
195C2×C4×D3C2×C3⋊D311yescharacteristic
196C2×Dic6—21no—
197C2×Dic6C2×C3⋊D311yescharacteristic
198C2×C3×C8—91no—
199C3×C4⋊C4—91no—
200#215—188no—
201#215—1816no—
202#215—188no—
203#215—182no—
204(C2×D3)⋊C4—184no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3—34no—
207C2×C3⋊C8—34no—
208C2×C3⋊C8—94no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—184no—
211#204—182no—
212C22×C32—24no—
213C22×C32—32no—
214C22×C32—61no—
215C22×C32—61no—
216C22×C32D3×D411yescharacteristic
217C22×C32C2×#6714yes—
218C2×C3⋊D3—128no—
219C2×C3⋊D3—121no—
220C2×C3⋊D3—121no—
221C2×C3⋊D3—121no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—124no—
224C2×C3⋊D3—124no—
225C2×C3⋊D3—244no—
226C2×C3⋊D3—244no—
227C2×C3×D3—24no—
228C2×C3×D3—24no—
229C2×C3×D3#21518yescomplemented
230C32×C4—22no—
231C32×C4—24no—
232C32×C4—24no—
233C32×C4—28no—
234C32×C4—48no—
235C32×C4—61no—
236C32×C4—62no—
237C32×C4C2×D1211yescharacteristic
238C32×C4D3×D411yescharacteristic
239C32×C4C2×#6714yes—
240C32×C4C2×#6714yes—
241C3⋊Dic3—41no—
242C3×Dic3—24no—
243C3×Dic3—28no—
244C3×Dic3—124no—
245C22×Q8—31no—
246C2×SD16—271no—
247C2×SD16—271no—
248#121—271no—
249#103—91no—
250#103—271no—
251C22⋊C8—271no—
252C33#44811yescharacteristic, complemented, Sylow
253C23×C3—34no—
254C22×D3—184no—
255C22×D3—181no—
256C22×D3—364no—
257C22×D3#63811yescharacteristic
258C3×Q8—24no—
259C3×Q8—21no—
260C3×Q8—28no—
261C3×Q8—38no—
262C3×Q8—44no—
263C3×Q8—616no—
264C3×Q8—64no—
265C3×Q8#62618yescomplemented
266C3×Q8#63812yescomplemented
267C2×C3×C4—24no—
268C2×C3×C4—21no—
269C2×C3×C4—24no—
270C2×C3×C4—34no—
271C2×C3×C4—44no—
272C2×C3×C4—616no—
273C2×C3×C4—64no—
274C2×C3×C4—64no—
275C2×C3×C4—64no—
276C2×C3×C4—181no—
277C2×C3×C4C2×D3214yes—
278C2×C3×C4C22×C3⋊D311yescharacteristic
279#67—368no—
280C2×Dic3—21no—
281C2×Dic3—64no—
282C2×Dic3#63811yescharacteristic
283D12—188no—
284D12—184no—
285D12—184no—
286D12—181no—
287D12—181no—
288D12—364no—
289D12—362no—
290C4×D3—24no—
291C4×D3—21no—
292C4×D3—21no—
293Dic6—24no—
294Dic6—21no—
295Dic6#63812yescomplemented
296C3×C8—181no—
297C3⋊C8—64no—
298C3⋊C8—188no—
299C3⋊C8—184no—
300C3⋊C8—181no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—62no—
305C2×C32#88211yescharacteristic
306C2×C32D3×SD1611yescharacteristic
307C2×C32#91811yescharacteristic
308C2×C32C2×#21514yes—
309C2×C32#94214yes—
310C2×C32#95314yes—
311C3⋊D3—248no—
312C3⋊D3—244no—
313C3⋊D3—241no—
314C3×D3—28no—
315C2×Q8—31no—
316C2×Q8—61no—
317C2×Q8—62no—
318C2×Q8—62no—
319C2×Q8D3×C3⋊D311yescharacteristic
320C2×D4—271no—
321C2×D4—541no—
322C22×C4—31no—
323C22×C4—61no—
324SD16—542no—
325SD16—542no—
326C2×C8—91no—
327C2×C8—271no—
328C4⋊C4—91no—
329C22⋊C4—541no—
330C22×C3—24no—
331C22×C3—38no—
332C22×C3—64no—
333C22×C3—64no—
334C22×C3D3×#6714yes—
335C22×C3C2×#63811yescharacteristic
336C2×D3—21no—
337C2×D3—21no—
338C2×D3—368no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—361no—
342C2×D3—364no—
343C2×D3—364no—
344C2×D3—364no—
345C2×D3—724no—
346C2×D3—724no—
347C2×D3#375512yescomplemented
348C3×C4—22no—
349C3×C4—28no—
350C3×C4—24no—
351C3×C4—24no—
352C3×C4—48no—
353C3×C4—64no—
354C3×C4—68no—
355C3×C4—361no—
356C3×C4C2×#62614yes—
357C3×C4D3×#6714yes—
358C3×C4C2×#63811yescharacteristic
359C3×C4C2×#63811yescharacteristic
360Dic3—21no—
361Dic3—22no—
362Dic3—124no—
363C32—24no—
364C32#480611yescharacteristic, complemented
365C32#522714yescomplemented
366C23—31no—
367C23—541no—
368Q8—21no—
369Q8—32no—
370Q8—64no—
371Q8—61no—
372Q8#629112yescomplemented
373D4—541no—
374D4—541no—
375D4—1082no—
376C2×C4—21no—
377C2×C4—31no—
378C2×C4—64no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—181no—
383C2×C4C2×D3×C3⋊D311yescharacteristic
384C8—181no—
385C8—541no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—68no—
390C2×C3D3×#21514yes—
391C2×C3#6394814yes—
392C2×C3#6397714yes—
393C2×C3C2×#375511yescharacteristic
394C2×C3#6415111yescharacteristic
395C2×C3#6416211yescharacteristic
396D3—22no—
397D3—728no—
398D3—724no—
399D3—721no—
400C22—32no—
401C22—61no—
402C22—61no—
403C22—1081no—
404C22—1081no—
405C22—1081no—
406C22D3×#63811yescharacteristic
407C4—22no—
408C4—61no—
409C4—62no—
410C4—361no—
411C4C2×#629111yescharacteristic
412C4D3×#63811yescharacteristic
413C3—24no—
414C3#1059158414yescomplemented
415C3#1059292611yescharacteristic, complemented
416C2—62no—
417C2—2161no—
418C2D3×#375511yescharacteristic
419C2#1172358111yescharacteristic
420C2#1172363011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723516C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#448
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
1. C2×C32×D3×Q8C2semidirect
2. C2×#88862C2semidirect
4. C2×#88795C2semidirect
5. C3×#63662C2semidirect
6. #11722037C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
26. C2×C3×D3×Q8D3semidirect
35. #63662D3semidirect
43. C33×Q8D4semidirect
56. C32×Dic6D4semidirect
102. C2×C32×D3SD16semidirect
110. C2×D3×Q8C3⋊D3semidirect
136. C32×Q8D12semidirect
137. C32×Q8#67semidirect
165. C3×Dic6#67semidirect
229. C2×C3×D3#215semidirect
252. C33#448semidirect
265. C3×Q8#626semidirect
266. C3×Q8#638semidirect
295. Dic6#638semidirect
347. C2×D3#3755semidirect
364. C32#4806semidirect
365. C32#5227semidirect
372. Q8#6291semidirect
414. C3#10591584semidirect
415. C3#10592926semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
C2×#888622
#117235162
C2×#887952
C3×#636622
#117220372
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929263
#105915843