Finite Group Databasebeta

← #13161156

Subgroups — #13161156

Order 1728.

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

normal subgroups only

This group has 992 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
1C2×#88862C211yesmaximal, characteristic
2C32×#944C211yesmaximal, characteristic, complemented
3#11723484C211yesmaximal, characteristic
4C2×#88795C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592782—31nomaximal
9#10591565—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6289—21no—
14#88862—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3C2211yescharacteristic
21#88804—21no—
22#88795—22no—
23C32×C4⋊Q8—31no—
24C2×#3755—31no—
25C2×#3755—34no—
26C2×#3755—31no—
27C2×#3755—64no—
28C4×C4⋊(C3⋊D3)—31no—
29C3×#944D314yescomplemented
30#63894—34no—
31C2×#3717—34no—
32#63662—31no—
33#63662D311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))—31no—
35C3×#835—34no—
36Dic3⋊(C3⋊C8)—34no—
37#63565—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))D412yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×Dic6—21no—
50C32×Dic6—21no—
51#6289—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)—21no—
54#5083—94no—
55#4818—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8—31no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4×C3⋊D3—61no—
64C2×C3×Dic6C2×D314yes—
65C2×#624—64no—
66C32×C4⋊C4—61no—
67C32×C4⋊C4—61no—
68C32×C42—31no—
69#3755—62no—
70#3755—62no—
71#3755—68no—
72#3755—616no—
73#3752—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×C4⋊Dic3—34no—
80C3×Dic3⋊C4—24no—
81C3×Dic3⋊C4—24no—
82C3×C4×Dic3C2×D314yes—
83C2×C3×C3⋊C8—34no—
84#3722—64no—
85#3717—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—22no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C33×C4—41no—
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3×C3⋊Dic3—41no—
94C32×Dic3—21no—
95C32×Dic3—41no—
96C32×Dic3SD1612yescomplemented
97C3×C4⋊Q8—34no—
98#944C3⋊D311yescharacteristic, complemented
99C2×#215—94no—
100C2×#215—94no—
101C2×#215—91no—
102C2×#215—184no—
103C2×#204—91no—
104#893—91no—
105C4×D12—94no—
106C3×#103—91no—
107#835—34no—
108#835—94no—
109Dic3⋊C8—91no—
110#810—91no—
111C4⋊(C3⋊C8)—94no—
112C22×C3⋊D3—61no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—124no—
115C32×Q8—28no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—21no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×Q8—61no—
122C32×Q8—61no—
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—24no—
126C2×C32×C4—31no—
127C2×C32×C4—31no—
128C2×C32×C4—44no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C4⋊(C3⋊D3)—32no—
134C4⋊(C3⋊D3)—38no—
135C4⋊(C3⋊D3)—68no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—124no—
139C4×C3⋊D3—122no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3#6714yes—
143C2×C3×Dic3#6714yes—
144C3×Dic6—24no—
145C3×Dic6—24no—
146#624—128no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)—68no—
149C3⋊(C3⋊C8)—64no—
150C3⋊(C3⋊C8)—61no—
151C3×C3⋊C8—64no—
152#461—271noSylow
153C2×C33#12111yescharacteristic
154C2×C33C2×SD1611yescharacteristic
155C2×C33#13711yescharacteristic
156C3⋊(C3⋊D3)—42no—
157C2×C3×Q8—24no—
158C2×C3×Q8—34no—
159C2×C3×Q8D3214yes—
160C2×C3×Q8C2×C3⋊D311yescharacteristic
161C2×D12—94no—
162C2×D12—91no—
163C2×D12—184no—
164C2×C4×D3—184no—
165C2×C4×D3—181no—
166C2×Dic6C2×C3⋊D311yescharacteristic
167C2×C3×C8—91no—
168C3×C4⋊C4—64no—
169C3×C4⋊C4—64no—
170C3×C4⋊C4—91no—
171C3×C42—34no—
172#215—188no—
173#215—188no—
174#215—182no—
175#215—1816no—
176(C2×D3)⋊C4—184no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3—34no—
179Dic3⋊C4—21no—
180Dic3⋊C4—21no—
181C4×Dic3C2×C3⋊D311yescharacteristic
182C2×C3⋊C8—34no—
183C2×C3⋊C8—94no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—184no—
186#204—182no—
187C22×C32—24no—
188C22×C32D3×D411yescharacteristic
189C22×C32C2×#6714yes—
190C2×C3⋊D3—62no—
191C2×C3⋊D3—68no—
192C2×C3⋊D3—128no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—124no—
197C2×C3⋊D3—244no—
198C2×C3⋊D3—244no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—32no—
202C32×C4—48no—
203C32×C4—41no—
204C32×C4—44no—
205C32×C4—61no—
206C32×C4—121no—
207C32×C4#23511yescharacteristic
208C32×C4D3×D414yes—
209C32×C4D3×D411yescharacteristic
210C32×C4#23714yes—
211C3⋊Dic3—41no—
212C3×Dic3—24no—
213C3×Dic3—44no—
214C3×Dic3—124no—
215C3×Dic3#21518yescomplemented
216C2×SD16—271no—
217C2×SD16—271no—
218C4⋊Q8—31no—
219C4×D4—271no—
220C4⋊C8—271no—
221#103—91no—
222#103—271no—
223C33#46111yescharacteristic, complemented, Sylow
224C22×D3—184no—
225C22×D3—181no—
226C22×D3—364no—
227C3×Q8—28no—
228C3×Q8—24no—
229C3×Q8—24no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—44no—
233C3×Q8—64no—
234C3×Q8—64no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—34no—
239C2×C3×C4—34no—
240C2×C3×C4—44no—
241C2×C3×C4—64no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—64no—
247C2×Dic3#63811yescharacteristic
248C2×Dic3#63811yescharacteristic
249D12—98no—
250D12—92no—
251D12—188no—
252D12—184no—
253D12—181no—
254D12—364no—
255C4×D3—368no—
256C4×D3—362no—
257Dic6—21no—
258Dic6—21no—
259C3×C8—181no—
260C3⋊C8—64no—
261C3⋊C8—188no—
262C3⋊C8—184no—
263C3⋊C8—181no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32#88411yescharacteristic
268C2×C32D3×SD1611yescharacteristic
269C2×C32#91811yescharacteristic
270C2×C32#93814yes—
271C2×C32C2×#21514yes—
272C2×C32#94214yes—
273C3⋊D3—122no—
274C3⋊D3—128no—
275C3⋊D3—248no—
276C2×Q8—31no—
277C2×Q8D3×C3⋊D311yescharacteristic
278C2×D4—271no—
279C22×C4—541no—
280SD16—542no—
281SD16—542no—
282C2×C8—91no—
283C2×C8—271no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—91no—
287C22⋊C4—541no—
288C42—31no—
289C22×C3—24no—
290C22×C3D3×#6714yes—
291C22×C3C2×#63811yescharacteristic
292C2×D3—182no—
293C2×D3—188no—
294C2×D3—368no—
295C2×D3—361no—
296C2×D3—361no—
297C2×D3—364no—
298C2×D3—364no—
299C2×D3—724no—
300C2×D3—724no—
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—38no—
304C3×C4—48no—
305C3×C4—41no—
306C3×C4—44no—
307C3×C4—64no—
308C3×C4—124no—
309C3×C4—361no—
310C3×C4#380414yes—
311C3×C4#381114yes—
312C3×C4D4×C3⋊D311yescharacteristic
313C3×C4#383011yescharacteristic
314Dic3—21no—
315Dic3—41no—
316Dic3—124no—
317Dic3#375512yescomplemented
318C32—24no—
319C32#481811yescharacteristic, complemented
320C32#508314yescomplemented
321C23—541no—
322Q8—21no—
323Q8—21no—
324Q8—61no—
325Q8—61no—
326D4—272no—
327D4—541no—
328C2×C4—21no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—61no—
332C2×C4—181no—
333C2×C4—1082no—
334C2×C4C2×D3×C3⋊D311yescharacteristic
335C8—181no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3D3×#21514yes—
341C2×C3#6394814yes—
342C2×C3#6395814yes—
343C2×C3#6414711yescharacteristic
344C2×C3C2×#375511yescharacteristic
345C2×C3#6415111yescharacteristic
346D3—362no—
347D3—368no—
348D3—728no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22D3×#63811yescharacteristic
353C4—32no—
354C4—41no—
355C4—61no—
356C4—121no—
357C4—361no—
358C4#8902811yescharacteristic
359C4#8903911yescharacteristic
360C3—24no—
361C3#1059156514yescomplemented
362C3#1059278211yescharacteristic, complemented
363C2—1082no—
364C2D3×#375511yescharacteristic
365C2#1172358111yescharacteristic
366C2#1172359911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#461
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
2. C32×#944C2semidirect
5. C3×#63662C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722037C2semidirect
29. C3×#944D3semidirect
33. #63662D3semidirect
96. C32×Dic3SD16semidirect
98. #944C3⋊D3semidirect
215. C3×Dic3#215semidirect
223. C33#461semidirect
317. Dic3#3755semidirect
319. C32#4818semidirect
320. C32#5083semidirect
361. C3#10591565semidirect
362. C3#10592782semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C32×#9442
#117234842
C2×#887952
C3×#636622
Dic3⋊(C3⋊(C3⋊C8))2
#117220372
#105927823
#105915653