Finite Group Databasebeta

← #13161225

Subgroups — #13161225

Order 1728.

952 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 166 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
1Q8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic
4#11723434C211yesmaximal, characteristic
5#11722050C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic
8#10592865—31nomaximal
9#10591612—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3756—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#215—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22(C2×D3)⋊(C3⋊Dic3)—21no—
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3—31no—
25Q8×C3⋊Dic3—34no—
26Q8×C3⋊Dic3—64no—
27C2×#3756D311yescharacteristic, complemented
28#64145—31no—
29C2×C3×#216—34no—
30C2×C3×#215D314yes—
31#63863—34no—
32#63659—31no—
33#63643—31no—
34#63641—31no—
35#63572—34no—
36#63566—34no—
37#63563—34no—
38C22×C32×D3—21no—
39C33×Q8—21no—
40C33×Q8D412yes—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634—21no—
49C32×D12—21no—
50C32×D12—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C32×C3⋊C8—21no—
53#5185—91no—
54#5166—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×D4—31no—
59C2×#634C2×D311yescharacteristic
60C2×C3×D12C2×D314yes—
61C2×C3×Dic6—34no—
62C32×SD16—62no—
63C2×C32×C8—31no—
64#3757—61no—
65#3757—61no—
66#3756—22no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—61no—
70C4⋊(C3⋊Dic3)—64no—
71C4⋊(C3⋊Dic3)—64no—
72C4⋊(C3⋊Dic3)—68no—
73C4×C3⋊Dic3—31no—
74C4×C3⋊Dic3—34no—
75C4×C3⋊Dic3—61no—
76C4×C3⋊Dic3—64no—
77C4×C3⋊Dic3—64no—
78C4×C3⋊Dic3—68no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×#216—68no—
81C3×#215—28no—
82C2×C3×C3⋊C8—34no—
83C2×C3×C3⋊C8C2×D314yes—
84(C2×D3)⋊Dic3—64no—
85(C2×D3)⋊Dic3—64no—
86C22×C33C2×D411yescharacteristic
87C2×C32×D3—41no—
88C2×C32×D3—41no—
89C2×C32×D3—41no—
90C33×C4—22no—
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3⋊(C3⋊Dic3)—21no—
94C3⋊(C3⋊Dic3)—21no—
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—34no—
99#947—91no—
100Q8×Dic3—94no—
101Q8×Dic3—91no—
102Q8×Dic3—184no—
103C2×#216—34no—
104C2×#215C3⋊D311yescharacteristic
105#936—94no—
106#835—91no—
107#832—94no—
108#821—91no—
109#816—94no—
110Dic3⋊C8—94no—
111Dic3⋊C8—91no—
112C23×C32—61no—
113C22×C3×D3—24no—
114C32×Q8—28no—
115C32×Q8—24no—
116C32×Q8—21no—
117C32×Q8—28no—
118C32×Q8#6718yes—
119C32×Q8#6712yes—
120C32×D4—61no—
121C32×D4—61no—
122C2×C32×C4—28no—
123C2×C32×C4—24no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—68no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—61no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—64no—
141C3×D12—24no—
142C3×D12—24no—
143C3×Dic6—64no—
144C3×Dic6—64no—
145C32×C8—61no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—24no—
148C3×C3⋊C8—64no—
149#464—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33#13511yescharacteristic
152C2×C33#13711yescharacteristic
153C32×D3—81no—
154C2×C3×Q8—24no—
155C2×C3×Q8—91no—
156C2×C3×Q8D3214yes—
157C2×C3×Q8C2×C3⋊D311yescharacteristic
158C2×C3×D4—34no—
159C2×D12C2×C3⋊D311yescharacteristic
160C2×Dic6—34no—
161C3×Q16—182no—
162C3×SD16—68no—
163C2×C3×C8—34no—
164C2×C3×C8—91no—
165#217—184no—
166#217—184no—
167#216—68no—
168#215—22no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171C4⋊Dic3—94no—
172C4⋊Dic3—91no—
173C4⋊Dic3—184no—
174C4⋊Dic3—184no—
175C4⋊Dic3—181no—
176C4⋊Dic3—188no—
177C4×Dic3—94no—
178C4×Dic3—91no—
179C4×Dic3—184no—
180C4×Dic3—184no—
181C4×Dic3—181no—
182C4×Dic3—188no—
183C2×C3⋊C8—34no—
184C2×C3⋊C8C2×C3⋊D311yescharacteristic
185C22×C32—24no—
186C22×C32—121no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32D3×D414yes—
190C22×C32D3×D411yescharacteristic
191C2×C3×D3—44no—
192C2×C3×D3—44no—
193C2×C3×D3—44no—
194C32×C4—216no—
195C32×C4—28no—
196C32×C4—22no—
197C32×C4—24no—
198C32×C4—24no—
199C32×C4#23714yes—
200C32×C4#23711yescharacteristic
201C32×C4C2×#6714yes—
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—41no—
204C3⋊Dic3—68no—
205C3⋊Dic3—64no—
206C3⋊Dic3—61no—
207C3⋊Dic3—61no—
208C3⋊Dic3—64no—
209C3⋊Dic3—124no—
210C3⋊Dic3—128no—
211C3⋊Dic3—124no—
212C3⋊Dic3—121no—
213C3×Dic3—124no—
214C2×Q16—91no—
215C2×SD16—31no—
216#124—271no—
217C4×Q8—271no—
218C4⋊C8—271no—
219#103—271no—
220#102—271no—
221C33#46411yescharacteristic, complemented, Sylow
222C23×C3—64no—
223C22×D3—21no—
224C3×Q8—28no—
225C3×Q8—24no—
226C3×Q8—21no—
227C3×Q8—28no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×Q8#62518yes—
231C3×Q8#63812yes—
232C3×D4—64no—
233C3×D4—64no—
234C2×C3×C4—28no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—181no—
239C2×C3×C4C2×D3214yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241C2×Dic3—64no—
242C2×Dic3—94no—
243C2×Dic3—94no—
244C2×Dic3—91no—
245C2×Dic3—91no—
246C2×Dic3—184no—
247C2×Dic3—184no—
248C2×Dic3—188no—
249C2×Dic3—184no—
250C2×Dic3—181no—
251D12—21no—
252D12—21no—
253Dic6—64no—
254Dic6—64no—
255C3×C8—64no—
256C3×C8—181no—
257C3⋊C8—21no—
258C3⋊C8—64no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—24no—
262C2×C32—241no—
263C2×C32#91514yes—
264C2×C32#91614yes—
265C2×C32#91811yescharacteristic
266C2×C32#91911yescharacteristic
267C2×C32#93914yes—
268C2×C32#93911yescharacteristic
269C3×D3—84no—
270C2×Q8—91no—
271C2×Q8D3×C3⋊D311yescharacteristic
272C2×D4—31no—
273Q16—182no—
274SD16—62no—
275C2×C8—31no—
276C2×C8—91no—
277C4⋊C4—271no—
278C4⋊C4—541no—
279C22⋊C4—541no—
280C22⋊C4—541no—
281C42—271no—
282C42—541no—
283C22×C3—24no—
284C22×C3—124no—
285C22×C3—124no—
286C22×C3—124no—
287C22×C3#381114yes—
288C22×C3D4×C3⋊D311yescharacteristic
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C3×C4—216no—
293C3×C4—22no—
294C3×C4—28no—
295C3×C4—24no—
296C3×C4—24no—
297C3×C4—361no—
298C3×C4#380514yes—
299C3×C4C2×#62514yes—
300C3×C4#383011yescharacteristic
301C3×C4C2×#63811yescharacteristic
302Dic3—124no—
303Dic3—188no—
304Dic3—181no—
305Dic3—181no—
306Dic3—184no—
307Dic3—184no—
308Dic3—364no—
309Dic3—368no—
310Dic3—361no—
311Dic3—364no—
312C32—24no—
313C32#516614yescomplemented
314C32#518511yescharacteristic, complemented
315C23—61no—
316Q8—21no—
317Q8—181no—
318Q8—181no—
319Q8#629012yes—
320D4—61no—
321D4—61no—
322C2×C4—21no—
323C2×C4—181no—
324C2×C4—271no—
325C2×C4—271no—
326C2×C4—541no—
327C2×C4C2×D3×C3⋊D311yescharacteristic
328C8—61no—
329C8—181no—
330C2×C3—24no—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—244no—
334C2×C3#6395014yes—
335C2×C3#6395114yes—
336C2×C3#6398014yes—
337C2×C3#6412411yescharacteristic
338C2×C3#6412511yescharacteristic
339C2×C3#6414811yescharacteristic
340D3—81no—
341C22—121no—
342C22—121no—
343C22—121no—
344C22#8903911yescharacteristic
345C4—22no—
346C4—361no—
347C4—541no—
348C4—541no—
349C4—1081no—
350C4#8902911yescharacteristic
351C4C2×#629011yescharacteristic
352C3—24no—
353C3#1059161214yescomplemented
354C3#1059286511yescharacteristic, complemented
355C2—241no—
356C2#1172358711yescharacteristic
357C2#1172358811yescharacteristic
358C2#1172363711yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#464
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. Q8×C3⋊(C3⋊Dic3)C2semidirect
2. C2×C3×#3756C2semidirect
5. #11722050C2semidirect
6. #11722040C2semidirect
27. C2×#3756D3semidirect
221. C33#464semidirect
313. C32#5166semidirect
314. C32#5185semidirect
353. C3#10591612semidirect
354. C3#10592865semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊Dic3)2
C2×C3×#37562
C2×C32×#2152
#117234342
#117220502
#117220402
#117220322
#105928653
#105916123