Finite Group Databasebeta

← #13161306

Subgroups — #13161306

Order 1728.

992 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 156 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
1#11723985C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic
3#11723499C211yesmaximal, characteristic
4C2×#88794C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592861—31nomaximal
9#10591569—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14#88857—21no—
15#88857—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#215—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88794—22no—
23C2×C32×SD16—31no—
24#64153—31no—
25#64153—34no—
26#64153—64no—
27D4×C3⋊Dic3—31no—
28C2×#3754—31no—
29C2×C3×#215D314yes—
30#63898—34no—
31C2×#3716—34no—
32#63662D311yescharacteristic, complemented
33#63643—31no—
34#63641—31no—
35C3×#835—34no—
36#63572—34no—
37#63567—34no—
38C22×C32×D3—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46#6308—21no—
47#6308—21no—
48C2×C3×C3⋊Dic3—21no—
49C32×D12—21no—
50C32×D12D412yes—
51D3×C3⋊Dic3—42no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8—21no—
54#5162—94no—
55#4818—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×D4—31no—
60C22×C3⋊Dic3—61no—
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C2×C3×D12C2×D314yes—
65C2×D3×Dic3—64no—
66C32×SD16—62no—
67C2×C32×C8—31no—
68#3757—61no—
69#3754—62no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71#3750—61no—
72#3750—61no—
73#3750—64no—
74#3750—64no—
75#3750—68no—
76#3750—68no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—34no—
79C4×C3⋊Dic3—64no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#215—28no—
82C3×C4⋊Dic3—34no—
83C2×C3×C3⋊C8—34no—
84C2×C3×C3⋊C8C2×D314yes—
85(C2×D3)⋊Dic3—64no—
86#3716—68no—
87C22×C33C2×D411yescharacteristic
88C2×C32×D3—22no—
89C2×C32×D3—41no—
90C2×C32×D3—41no—
91C33×C4—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—21no—
95C3⋊(C3⋊Dic3)—41no—
96C3⋊(C3⋊Dic3)SD1612yescomplemented
97C3×C3⋊Dic3—41no—
98C2×C3×SD16—34no—
99#944—94no—
100#944—91no—
101#944—184no—
102C2×#215C3⋊D311yescharacteristic
103D4×Dic3—94no—
104C2×#214—94no—
105C2×#204—91no—
106#893—91no—
107C3×#103—91no—
108#835—34no—
109#816—94no—
110Dic3⋊C8—94no—
111Dic3⋊C8—91no—
112#810—91no—
113C23×C32—61no—
114C22×C3×D3—24no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—24no—
119C32×Q8—44no—
120C32×Q8—44no—
121C32×D4—32no—
122C32×D4—61no—
123C2×C32×C4—28no—
124C2×C32×C4—24no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—31no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—34no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—68no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—61no—
139C2×C3⋊Dic3—122no—
140#634—61no—
141#634—61no—
142#634—64no—
143#634—64no—
144#634—124no—
145#634—124no—
146C2×C3×Dic3—64no—
147C3×D12—24no—
148C3×D12#6718yes—
149D3×Dic3—128no—
150C32×C8—61no—
151C3⋊(C3⋊C8)—21no—
152C3×C3⋊C8—24no—
153C3×C3⋊C8—64no—
154#461—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33C2×SD1611yescharacteristic
157C2×C33#13711yescharacteristic
158C32×D3—42no—
159C2×C3×Q8—24no—
160C2×C3×Q8D3214yes—
161C2×C3×Q8C2×C3⋊D311yescharacteristic
162C2×C3×D4—34no—
163C22×Dic3—184no—
164C2×D12C2×C3⋊D311yescharacteristic
165C2×C4×D3—181no—
166C2×Dic6—94no—
167C2×Dic6—91no—
168C2×Dic6—184no—
169C3×SD16—68no—
170C2×C3×C8—34no—
171C2×C3×C8—91no—
172C3×C4⋊C4—91no—
173#217—184no—
174#215—22no—
175#214—188no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—34no—
178Dic3⋊C4—184no—
179Dic3⋊C4—181no—
180Dic3⋊C4—184no—
181Dic3⋊C4—181no—
182Dic3⋊C4—188no—
183Dic3⋊C4—188no—
184C4×Dic3—94no—
185C4×Dic3—91no—
186C4×Dic3—184no—
187C2×C3⋊C8—34no—
188C2×C3⋊C8C2×C3⋊D311yescharacteristic
189#204—182no—
190C22×C32—24no—
191C22×C32—62no—
192C22×C32—121no—
193C22×C32—121no—
194C22×C32D3×D414yes—
195C22×C32D3×D411yescharacteristic
196C2×C3×D3—28no—
197C2×C3×D3—44no—
198C2×C3×D3—44no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—41no—
203C32×C4—44no—
204C32×C4#23511yescharacteristic
205C32×C4D3×D411yescharacteristic
206C32×C4#23714yes—
207C32×C4C2×#6714yes—
208C3⋊Dic3—32no—
209C3⋊Dic3—38no—
210C3⋊Dic3—41no—
211C3⋊Dic3—68no—
212C3⋊Dic3—68no—
213C3⋊Dic3—64no—
214C3⋊Dic3—61no—
215C3⋊Dic3—128no—
216C3⋊Dic3—121no—
217C3⋊Dic3—124no—
218C3×Dic3—124no—
219C2×SD16—31no—
220C2×SD16—271no—
221C4⋊Q8—271no—
222C4×D4—271no—
223C4⋊C8—271no—
224#103—91no—
225#103—271no—
226C33#46111yescharacteristic, complemented, Sylow
227C23×C3—64no—
228C22×D3—21no—
229C3×Q8—24no—
230C3×Q8—24no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—44no—
234C3×Q8—44no—
235C3×D4—38no—
236C3×D4—64no—
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3214yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244C2×Dic3—64no—
245C2×Dic3—94no—
246C2×Dic3—91no—
247C2×Dic3—94no—
248C2×Dic3—91no—
249C2×Dic3—184no—
250C2×Dic3—184no—
251C2×Dic3—188no—
252C2×Dic3—184no—
253C2×Dic3—181no—
254C2×Dic3—368no—
255D12—21no—
256D12#63812yes—
257C4×D3—362no—
258Dic6—184no—
259Dic6—184no—
260Dic6—181no—
261Dic6—181no—
262Dic6—364no—
263Dic6—364no—
264C3×C8—64no—
265C3×C8—181no—
266C3⋊C8—21no—
267C3⋊C8—64no—
268C2×C32—24no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—122no—
272C2×C32#88411yescharacteristic
273C2×C32D3×SD1614yes—
274C2×C32D3×SD1611yescharacteristic
275C2×C32#91514yes—
276C2×C32#91811yescharacteristic
277C2×C32#93914yes—
278C3×D3—48no—
279C2×Q8—271no—
280C2×Q8D3×C3⋊D311yescharacteristic
281C2×D4—31no—
282C22×C4—541no—
283SD16—62no—
284SD16—542no—
285C2×C8—31no—
286C2×C8—91no—
287C4⋊C4—91no—
288C4⋊C4—541no—
289C4⋊C4—541no—
290C22⋊C4—541no—
291C42—271no—
292C22×C3—24no—
293C22×C3—68no—
294C22×C3—124no—
295C22×C3—124no—
296C22×C3#381114yes—
297C22×C3D4×C3⋊D311yescharacteristic
298C2×D3—22no—
299C2×D3—41no—
300C2×D3—41no—
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—48no—
304C3×C4—41no—
305C3×C4—44no—
306C3×C4—361no—
307C3×C4#380414yes—
308C3×C4D3×#6714yes—
309C3×C4#383011yescharacteristic
310C3×C4C2×#63811yescharacteristic
311Dic3—92no—
312Dic3—98no—
313Dic3—124no—
314Dic3—188no—
315Dic3—188no—
316Dic3—181no—
317Dic3—184no—
318Dic3—368no—
319Dic3—361no—
320Dic3—364no—
321C32—24no—
322C32#481811yescharacteristic, complemented
323C32#516214yescomplemented
324C23—61no—
325Q8—21no—
326Q8—21no—
327Q8—541no—
328Q8—541no—
329D4—32no—
330D4—61no—
331C2×C4—21no—
332C2×C4—181no—
333C2×C4—271no—
334C2×C4—271no—
335C2×C4—541no—
336C2×C4—1082no—
337C2×C4C2×D3×C3⋊D311yescharacteristic
338C8—61no—
339C8—181no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—24no—
343C2×C3—128no—
344C2×C3#6394014yes—
345C2×C3#6395114yes—
346C2×C3#6397114yes—
347C2×C3SD16×C3⋊D311yescharacteristic
348C2×C3#6412411yescharacteristic
349C2×C3#6414811yescharacteristic
350D3—42no—
351C22—62no—
352C22—121no—
353C22—121no—
354C22#8903911yescharacteristic
355C4—41no—
356C4—272no—
357C4—361no—
358C4—541no—
359C4—1081no—
360C4#8902811yescharacteristic
361C4D3×#63811yescharacteristic
362C3—24no—
363C3#1059156914yescomplemented
364C3#1059286111yescharacteristic, complemented
365C2—122no—
366C2#1172357111yescharacteristic
367C2#1172358811yescharacteristic
368C2#1172361611yescharacteristic
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#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
1. #11723985C2semidirect
5. C3×#63662C2semidirect
6. #11722050C2semidirect
7. #11722042C2semidirect
32. #63662D3semidirect
96. C3⋊(C3⋊Dic3)SD16semidirect
226. C33#461semidirect
322. C32#4818semidirect
323. C32#5162semidirect
363. C3#10591569semidirect
364. C3#10592861semidirect

Maximal subgroups

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

subgroupindex
#117239852
C2×C32×#2152
#117234992
C2×#887942
C3×#636622
#117220502
#117220422
#105928613
#105915693