Finite Group Databasebeta

← #13161184

Subgroups — #13161184

Order 1728.

1220 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 176 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×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#88862C211yesmaximal, characteristic, complemented
3#11723526C211yesmaximal, characteristic
4C2×#88797C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592505—31nomaximal
9#10591584—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C3×Q8×C3⋊D3—22no—
13C3×Q8×C3⋊D3—22no—
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634—21no—
17C2×C3×#634C2211yescharacteristic
18C2×#6292—21no—
19#88862—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88804—21no—
24#88797—22no—
25C2×Q8×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×Q8—34no—
27C2×#3755—31no—
28C2×#3755—34no—
29C2×#3755—64no—
30C2×#3744—31no—
31#64104—31no—
32#63909—34no—
33C2×#3717—34no—
34C32×#103—31no—
35#63647—31no—
36#63637—31no—
37C3×#835D314yescomplemented
38#63565—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊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×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3—24no—
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C3×#634—24no—
54C3×#634—21no—
55C3×#634D412yescomplemented
56C2×C32×Dic3—21no—
57#6292—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5227—91no—
61#4806—94no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65Q8×C3⋊D3—22no—
66Q8×C3⋊D3—22no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—64no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634—21no—
74C2×#634C2×D311yescharacteristic
75C3×D3×Q8—68no—
76C3×D3×Q8—68no—
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3—64no—
79C2×C3×Dic6—34no—
80C2×C3×Dic6—64no—
81C2×#626—64no—
82C2×C32×C8—31no—
83C32×C4⋊C4—31no—
84#3755—616no—
85#3755—68no—
86#3755—62no—
87#3752—61no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91#3744—62no—
92C3×C4⋊Dic3C2×D314yes—
93C2×C3×C3⋊C8C2×D314yes—
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×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3SD1612yescomplemented
103C33×C4—22no—
104C33×C4C2×D411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—22no—
108C32×Dic3—41no—
109C22×C3×Q8—91no—
110C2×D3×Q8—34no—
111C2×#215—94no—
112C2×#215—91no—
113C2×#215—91no—
114C2×#215—184no—
115#930—91no—
116C2×#204—94no—
117#895—94no—
118C3×#103—34no—
119#835—91no—
120#835C3⋊D311yescharacteristic, complemented
121#830—91no—
122(C2×D3)⋊C8—94no—
123#810—94no—
124C22×C3⋊D3—61no—
125C22×C3⋊D3—64no—
126C22×C3⋊D3—124no—
127C22×C3⋊D3#6711yescharacteristic
128C22×C3×D3—34no—
129C32×Q8—21no—
130C32×Q8—28no—
131C32×Q8—24no—
132C32×Q8—44no—
133C32×Q8D1218yescomplemented
134C32×Q8#6712yescomplemented
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—44no—
139C2×C32×C4—61no—
140C2×C32×C4C22×D314yes—
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3#6711yescharacteristic
144C4⋊(C3⋊D3)—68no—
145C4⋊(C3⋊D3)—64no—
146C4⋊(C3⋊D3)—61no—
147C4⋊(C3⋊D3)—61no—
148C4⋊(C3⋊D3)—64no—
149C4⋊(C3⋊D3)—124no—
150C4⋊(C3⋊D3)—122no—
151C4×C3⋊D3—24no—
152C4×C3⋊D3—21no—
153C4×C3⋊D3—21no—
154#634—24no—
155#634—21no—
156#634#6712yescomplemented
157C2×C3×Dic3—24no—
158C2×C3×Dic3—34no—
159C2×C3×Dic3—64no—
160C3×C4×D3—616no—
161C3×C4×D3—64no—
162C3×C4×D3—64no—
163C3×Dic6—38no—
164C3×Dic6—616no—
165C3×Dic6—64no—
166#626—128no—
167C32×C8—61no—
168C3⋊(C3⋊C8)—68no—
169C3⋊(C3⋊C8)—64no—
170C3⋊(C3⋊C8)—61no—
171C3×C3⋊C8—24no—
172#448—271noSylow
173C2×C33#12011yescharacteristic
174C2×C33C2×SD1611yescharacteristic
175C2×C33#13711yescharacteristic
176C3⋊(C3⋊D3)—81no—
177C3×C3⋊D3—22no—
178C2×C3×Q8—24no—
179C2×C3×Q8—91no—
180C2×C3×Q8—181no—
181C2×C3×Q8—182no—
182C2×C3×Q8—182no—
183C2×C3×Q8D3214yes—
184C2×C3×Q8C2×C3⋊D311yescharacteristic
185C22×C3×C4—91no—
186C22×C3×C4—181no—
187C2×#67—181no—
188D3×Q8—68no—
189D3×Q8—68no—
190C2×D12—94no—
191C2×D12—91no—
192C2×D12—184no—
193C2×D12—184no—
194C2×C4×D3—34no—
195C2×C4×D3—64no—
196C2×Dic6—34no—
197C2×Dic6—64no—
198C2×C3×C8—34no—
199C3×C4⋊C4—34no—
200#215—182no—
201#215—1816no—
202#215—188no—
203#215—182no—
204(C2×D3)⋊C4—181no—
205(C2×D3)⋊C4—184no—
206C4⋊Dic3C2×C3⋊D311yescharacteristic
207C2×C3⋊C8—94no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—184no—
210C2×C3⋊C8C2×C3⋊D311yescharacteristic
211#204—188no—
212C22×C32—24no—
213C22×C32D3×D414yes—
214C22×C32C2×#6711yescharacteristic
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3⋊D3—128no—
218C2×C3⋊D3—124no—
219C2×C3⋊D3—121no—
220C2×C3⋊D3—121no—
221C2×C3⋊D3—121no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—124no—
224C2×C3⋊D3—244no—
225C2×C3⋊D3—244no—
226C2×C3⋊D3#21512yescomplemented
227C2×C3×D3—38no—
228C2×C3×D3—64no—
229C2×C3×D3—64no—
230C32×C4—22no—
231C32×C4—24no—
232C32×C4—24no—
233C32×C4—28no—
234C32×C4—48no—
235C32×C4—121no—
236C32×C4C2×D1214yes—
237C32×C4D3×D414yes—
238C32×C4C2×#6711yescharacteristic
239C32×C4C2×#6711yescharacteristic
240C3⋊Dic3—21no—
241C3⋊Dic3—22no—
242C3×Dic3—44no—
243C3×Dic3—68no—
244C3×Dic3—64no—
245C22×Q8—91no—
246C2×SD16—271no—
247C2×SD16—271no—
248#121—271no—
249#103—31no—
250#103—271no—
251C22⋊C8—271no—
252C33#44811yescharacteristic, complemented, Sylow
253C23×C3—91no—
254C22×D3—34no—
255C22×D3—184no—
256C22×D3—181no—
257C22×D3—364no—
258C3×Q8—24no—
259C3×Q8—21no—
260C3×Q8—28no—
261C3×Q8—44no—
262C3×Q8—92no—
263C3×Q8—184no—
264C3×Q8—181no—
265C3×Q8#62618yescomplemented
266C3×Q8C4⋊(C3⋊D3)12yescomplemented
267C2×C3×C4—24no—
268C2×C3×C4—21no—
269C2×C3×C4—24no—
270C2×C3×C4—44no—
271C2×C3×C4—64no—
272C2×C3×C4—91no—
273C2×C3×C4—184no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4C2×D3214yes—
278C2×C3×C4C22×C3⋊D311yescharacteristic
279#67—362no—
280C2×Dic3—21no—
281C2×Dic3—34no—
282C2×Dic3—64no—
283D12—188no—
284D12—184no—
285D12—184no—
286D12—181no—
287D12—181no—
288D12—364no—
289D12—368no—
290C4×D3—616no—
291C4×D3—64no—
292C4×D3—64no—
293Dic6—38no—
294Dic6—616no—
295Dic6—64no—
296C3×C8—64no—
297C3⋊C8—21no—
298C3⋊C8—188no—
299C3⋊C8—184no—
300C3⋊C8—181no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32#88214yes—
305C2×C32D3×SD1614yes—
306C2×C32#91814yes—
307C2×C32C2×#21511yescharacteristic
308C2×C32#94211yescharacteristic
309C2×C32#95311yescharacteristic
310C3⋊D3—22no—
311C3⋊D3—248no—
312C3⋊D3—241no—
313C3⋊D3—244no—
314C3×D3—68no—
315C2×Q8—91no—
316C2×Q8—181no—
317C2×Q8—182no—
318C2×Q8—182no—
319C2×Q8D3×C3⋊D311yescharacteristic
320C2×D4—271no—
321C2×D4—541no—
322C22×C4—91no—
323C22×C4—181no—
324SD16—542no—
325SD16—542no—
326C2×C8—31no—
327C2×C8—271no—
328C4⋊C4—31no—
329C22⋊C4—541no—
330C22×C3—24no—
331C22×C3—92no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3D3×#6714yes—
335C22×C3D4×C3⋊D311yescharacteristic
336C2×D3—38no—
337C2×D3—64no—
338C2×D3—64no—
339C2×D3—368no—
340C2×D3—361no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—364no—
344C2×D3—364no—
345C2×D3—364no—
346C2×D3—724no—
347C2×D3—724no—
348C3×C4—22no—
349C3×C4—28no—
350C3×C4—24no—
351C3×C4—24no—
352C3×C4—48no—
353C3×C4—124no—
354C3×C4—181no—
355C3×C4—182no—
356C3×C4C2×#62614yes—
357C3×C4D3×#6714yes—
358C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
359C3×C4D4×C3⋊D311yescharacteristic
360Dic3—41no—
361Dic3—68no—
362Dic3—64no—
363C32—24no—
364C32#480614yescomplemented
365C32#522711yescharacteristic, complemented
366C23—91no—
367C23—541no—
368Q8—21no—
369Q8—92no—
370Q8—184no—
371Q8—181no—
372Q8#629212yescomplemented
373D4—541no—
374D4—541no—
375D4—1082no—
376C2×C4—21no—
377C2×C4—61no—
378C2×C4—91no—
379C2×C4—184no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4C2×D3×C3⋊D311yescharacteristic
384C8—61no—
385C8—541no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—182no—
390C2×C3D3×#21514yes—
391C2×C3#6394814yes—
392C2×C3#6397714yes—
393C2×C3#6409111yescharacteristic
394C2×C3SD16×C3⋊D311yescharacteristic
395C2×C3#6412711yescharacteristic
396D3—68no—
397D3—728no—
398D3—724no—
399D3—721no—
400C22—92no—
401C22—181no—
402C22—181no—
403C22—1081no—
404C22—1081no—
405C22—1081no—
406C22C3⋊D3×#6711yescharacteristic
407C4—22no—
408C4—121no—
409C4—181no—
410C4—182no—
411C4C2×#629211yescharacteristic
412C4C3⋊D3×#6711yescharacteristic
413C3—24no—
414C3#1059158414yescomplemented
415C3#1059250511yescharacteristic, complemented
416C2—182no—
417C2—2161no—
418C2C3⋊D3×#21511yescharacteristic
419C2#1172358611yescharacteristic
420C2#1172363611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723526C2
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×C3×Q8×C3⋊D3C2semidirect
2. C2×#88862C2semidirect
4. C2×#88797C2semidirect
5. C32×#835C2semidirect
6. #11722039C2semidirect
7. #11722023C2semidirect
25. C2×Q8×C3⋊D3D3semidirect
37. C3×#835D3semidirect
43. C33×Q8D4semidirect
55. C3×#634D4semidirect
102. C2×C3×C3⋊D3SD16semidirect
120. #835C3⋊D3semidirect
133. C32×Q8D12semidirect
134. C32×Q8#67semidirect
156. #634#67semidirect
226. C2×C3⋊D3#215semidirect
252. C33#448semidirect
265. C3×Q8#626semidirect
266. C3×Q8C4⋊(C3⋊D3)semidirect
364. C32#4806semidirect
365. C32#5227semidirect
372. Q8#6292semidirect
414. C3#10591584semidirect
415. C3#10592505semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×#888622
#117235262
C2×#887972
C32×#8352
#117220392
#117220232
#105925053
#105915843