Finite Group Databasebeta

← #13156013

Subgroups — #13156013

Order 1728.

764 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 104 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#11723678C211yesmaximal, characteristic, complemented
2#11723494C211yesmaximal, characteristic
3C2×#88797C212yesmaximal
4C3×#63572C211yesmaximal, characteristic, complemented
5C3×#63567C212yesmaximal, complemented
6#10590760—32nomaximal
7#10590565—31nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×#6289—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15#88815—21no—
16#88804—21no—
17#88797—24no—
18C2×#3744—32no—
19#64102—31no—
20#63894—32no—
21#63894—61no—
22#63873—31no—
23#63866—32no—
24C2×#3717—32no—
25C2×#3717—32no—
26C2×#3717—62no—
27C3×#816—32no—
28C3×Dic3⋊C8—32no—
29C3×#810—32no—
30#63572D311yescharacteristic, complemented
31#63567D312yescomplemented
32C22×C3⋊(C3⋊D3)—21no—
33C2×C33×C4C2311yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—21no—
35C4⋊(C3⋊(C3⋊D3))D412yes—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3D411yescharacteristic
39C3×#634—22no—
40C3×#634—22no—
41C2×C32×Dic3—21no—
42#6289—42no—
43C32×C3⋊C8—22no—
44#4841—91no—
45#4818—92no—
46C2×C4⋊(C3⋊D3)—31no—
47C2×C4⋊(C3⋊D3)—32no—
48C2×C4⋊(C3⋊D3)—61no—
49C2×C4⋊(C3⋊D3)—62no—
50C2×C4⋊(C3⋊D3)—121no—
51C2×C4×C3⋊D3—61no—
52C2×#634C2×D312yes—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—62no—
56C2×#624—62no—
57C2×#624—62no—
58C2×C32×C8—32no—
59C32×C4⋊C4—31no—
60#3752—61no—
61C4×C3⋊Dic3C2×D311yescharacteristic
62#3744—64no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—22no—
68C2×C3×C3⋊C8C2×D312yes—
69C2×C3×C3⋊C8C2×D312yes—
70#3724—61no—
71#3724—61no—
72Dic3⋊Dic3—62no—
73Dic3⋊Dic3—62no—
74#3722—62no—
75#3722—62no—
76#3717—64no—
77#3717—64no—
78#3717—68no—
79C22×C33C2×D411yescharacteristic
80C2×C3⋊(C3⋊D3)—22no—
81C2×C3⋊(C3⋊D3)—41no—
82C2×C3⋊(C3⋊D3)—41no—
83C33×C4C2×D411yescharacteristic
84C33×C4#4111yescharacteristic
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—42no—
87C3×C3⋊Dic3SD1612yescomplemented
88C32×Dic3—41no—
89#944—92no—
90C2×#215—92no—
91C2×#204—92no—
92C2×#204—92no—
93C2×#204—182no—
94#893—92no—
95#893—181no—
96C4×D12—91no—
97#869—91no—
98C3×C4⋊C8—91no—
99C3×#103—92no—
100#816—32no—
101Dic3⋊C8—32no—
102#810—32no—
103C22×C3⋊D3—61no—
104C22×C3⋊D3—62no—
105C22×C3⋊D3—62no—
106C22×C3⋊D3—122no—
107C22×C3⋊D3—122no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D312yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3D1211yescharacteristic
116C2×C3⋊Dic3D1211yescharacteristic
117C4⋊(C3⋊D3)—32no—
118C4⋊(C3⋊D3)—34no—
119C4⋊(C3⋊D3)—62no—
120C4⋊(C3⋊D3)—64no—
121C4⋊(C3⋊D3)—61no—
122C4⋊(C3⋊D3)—62no—
123C4⋊(C3⋊D3)—62no—
124C4⋊(C3⋊D3)—122no—
125C4⋊(C3⋊D3)—122no—
126C4⋊(C3⋊D3)—122no—
127C4×C3⋊D3—122no—
128#634—22no—
129#634—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—122no—
139C3×Dic6—64no—
140C3×Dic6—62no—
141C3×Dic6—62no—
142C3×Dic6—62no—
143C3×Dic6—62no—
144C3×Dic6—122no—
145#624—124no—
146#624—124no—
147C32×C8—62no—
148C3×C3⋊C8—24no—
149C3×C3⋊C8—22no—
150C3×C3⋊C8—22no—
151#461—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33C2×SD1611yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—42no—
156C2×C3×Q8—92no—
157C2×D12—91no—
158C2×D12—92no—
159C2×D12—181no—
160C2×D12—182no—
161C2×D12—361no—
162C2×C4×D3—181no—
163C2×C4×D3—182no—
164C2×C4×D3—182no—
165C2×Dic6—32no—
166C2×Dic6—32no—
167C2×Dic6—62no—
168C2×C3×C8—32no—
169C2×C3×C8—32no—
170C2×C3×C8—62no—
171C3×C4⋊C4—32no—
172C3×C4⋊C4—61no—
173C3×C42—91no—
174#215—184no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—182no—
177(C2×D3)⋊C4—182no—
178C4⋊Dic3—181no—
179C4⋊Dic3—181no—
180C4⋊Dic3D3211yescharacteristic
181Dic3⋊C4—182no—
182Dic3⋊C4—182no—
183C4×Dic3—32no—
184C4×Dic3—61no—
185C2×C3⋊C8D3212yes—
186#204—184no—
187#204—184no—
188#204—188no—
189C22×C32—22no—
190C22×C32—21no—
191C22×C32—41no—
192C22×C32C2×D1211yescharacteristic
193C22×C32D3×D412yes—
194C2×C3⋊D3—62no—
195C2×C3⋊D3—64no—
196C2×C3⋊D3—64no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C2×C3⋊D3—122no—
202C2×C3⋊D3—122no—
203C2×C3⋊D3—122no—
204C2×C3⋊D3—122no—
205C2×C3⋊D3—242no—
206C2×C3⋊D3—242no—
207C2×C3⋊D3—242no—
208C2×C3⋊D3—242no—
209C32×C4—21no—
210C32×C4—22no—
211C32×C4—21no—
212C32×C4—22no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4—121no—
216C32×C4#23512yes—
217C32×C4D3×D412yes—
218C32×C4D3×D411yescharacteristic
219C32×C4#23911yescharacteristic
220C3⋊Dic3—21no—
221C3⋊Dic3—42no—
222C3⋊Dic3#20412yescomplemented
223C3×Dic3—34no—
224C3×Dic3—42no—
225C3×Dic3—42no—
226C3×Dic3—62no—
227C3×Dic3—62no—
228C3×Dic3—121no—
229C3×Dic3—124no—
230C3×Dic3—122no—
231C3×Dic3—122no—
232C2×SD16—272no—
233C4⋊Q8—271no—
234C4×D4—271no—
235C4⋊C8—91no—
236#103—92no—
237C33#46111yescharacteristic, complemented, Sylow
238C22×D3—182no—
239C22×D3—182no—
240C22×D3—181no—
241C22×D3—362no—
242C22×D3—362no—
243C3×Q8—182no—
244C3×Q8—182no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—62no—
249C2×C3×C4—62no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—182no—
253C2×C3×C4C2×D3211yescharacteristic
254C2×C3×C4C2×D3212yes—
255C2×Dic3—21no—
256C2×Dic3—32no—
257C2×Dic3—32no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—122no—
263D12—92no—
264D12—94no—
265D12—182no—
266D12—184no—
267D12—182no—
268D12—181no—
269D12—182no—
270D12—362no—
271D12—362no—
272D12—362no—
273C4×D3—364no—
274C4×D3—362no—
275C4×D3—364no—
276Dic6—64no—
277Dic6—62no—
278Dic6—62no—
279Dic6—62no—
280Dic6—62no—
281Dic6—122no—
282C3×C8—64no—
283C3×C8—62no—
284C3×C8—62no—
285C3⋊C8—22no—
286C2×C32—22no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32#88412yes—
296C2×C32#89511yescharacteristic
297C2×C32C2×#20411yescharacteristic
298C2×C32#90911yescharacteristic
299C2×C32D3×SD1612yes—
300C2×C32#91812yes—
301C3⋊D3—122no—
302C3⋊D3—124no—
303C3⋊D3—124no—
304C3⋊D3—244no—
305C3⋊D3—244no—
306C2×Q8—92no—
307C2×D4—271no—
308C22×C4—541no—
309SD16—544no—
310C2×C8—32no—
311C4⋊C4—31no—
312C4⋊C4—541no—
313C4⋊C4—541no—
314C22⋊C4—541no—
315C42—91no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—41no—
319C22×C3D3×D1212yes—
320C22×C3#381111yescharacteristic
321C2×D3—184no—
322C2×D3—184no—
323C2×D3—182no—
324C2×D3—364no—
325C2×D3—364no—
326C2×D3—362no—
327C2×D3—362no—
328C2×D3—362no—
329C2×D3—362no—
330C2×D3—361no—
331C2×D3—361no—
332C2×D3—722no—
333C2×D3—722no—
334C2×D3—722no—
335C2×D3—722no—
336C3×C4—22no—
337C3×C4—21no—
338C3×C4—22no—
339C3×C4—21no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4—92no—
343C3×C4—122no—
344C3×C4—122no—
345C3×C4—181no—
346C3×C4—362no—
347C3×C4#379811yescharacteristic
348C3×C4#379912yes—
349C3×C4#380211yescharacteristic
350C3×C4#381112yes—
351Dic3—34no—
352Dic3—41no—
353Dic3—62no—
354Dic3—62no—
355Dic3—121no—
356Dic3—124no—
357Dic3—122no—
358Dic3—122no—
359C32—22no—
360C32—21no—
361C32—41no—
362C32#481812yescomplemented
363C32#484111yescharacteristic, complemented
364C23—541no—
365Q8—182no—
366Q8—182no—
367D4—272no—
368D4—541no—
369C2×C4—61no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—182no—
373C2×C4—1082no—
374C2×C4D3311yescharacteristic
375C8—62no—
376C2×C3—21no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—22no—
381C2×C3—21no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3D3×#20412yes—
386C2×C3#6380112yes—
387C2×C3#6391012yes—
388C2×C3#6391211yescharacteristic
389C2×C3#6393011yescharacteristic
390C2×C3#6394511yescharacteristic
391D3—364no—
392D3—364no—
393D3—362no—
394D3—724no—
395D3—724no—
396C22—542no—
397C22—1081no—
398C22—1081no—
399C22#8895511yescharacteristic
400C4—92no—
401C4—121no—
402C4—181no—
403C4—362no—
404C4#8895411yescharacteristic
405C4#8896511yescharacteristic
406C3—21no—
407C3—22no—
408C3—41no—
409C3#1059056511yescharacteristic, complemented
410C3#1059076012yescomplemented
411C2—1082no—
412C2#1172284211yescharacteristic
413C2#1172289711yescharacteristic
414C2#1172296811yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11723678C2semidirect
4. C3×#63572C2semidirect
5. C3×#63567C2semidirect
30. #63572D3semidirect
31. #63567D3semidirect
87. C3×C3⋊Dic3SD16semidirect
222. C3⋊Dic3#204semidirect
237. C33#461semidirect
362. C32#4818semidirect
363. C32#4841semidirect
409. C3#10590565semidirect
410. C3#10590760semidirect

Maximal subgroups

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

subgroupindex
#117236782
#117234942
C2×#887972
C3×#635722
C3×#635672
#105907603
#105905653