Finite Group Databasebeta

← #13156009

Subgroups — #13156009

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#11723494C211yesmaximal, characteristic
2C2×#88795C212yesmaximal
3C3×#63873C211yesmaximal, characteristic, complemented
4#11722078C211yesmaximal, characteristic, complemented
5C3×#63565C212yesmaximal, complemented
6#10591565—32nomaximal
7#10590565—31nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C2×#6289—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2211yescharacteristic
14#88804—21no—
15#88795—24no—
16C3×#3724—21no—
17C3×#3724—21no—
18C2×#3755—32no—
19#64102—31no—
20C3×#944—32no—
21#63894—32no—
22#63894—61no—
23#63873D311yescharacteristic, complemented
24C2×#3717—32no—
25C2×#3717—32no—
26C2×#3717—62no—
27C3×#835—32no—
28C3×#810—32no—
29#63572—31no—
30Dic3⋊(C3⋊C8)—32no—
31#63565D312yescomplemented
32C22×C3⋊(C3⋊D3)—21no—
33C2×C33×C4C2311yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—21no—
35C4⋊(C3⋊(C3⋊D3))D412yes—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C32×Dic3—21no—
39C2×C32×Dic3—22no—
40C32×Dic6—22no—
41C32×Dic6—22no—
42#6289—42no—
43C3×C3⋊(C3⋊C8)—22no—
44#5083—91no—
45#4818—92no—
46C2×C32×Q8—32no—
47C2×C4⋊(C3⋊D3)—31no—
48C2×C4⋊(C3⋊D3)—32no—
49C2×C4⋊(C3⋊D3)—61no—
50C2×C4⋊(C3⋊D3)—62no—
51C2×C4⋊(C3⋊D3)—121no—
52C2×C4×C3⋊D3—61no—
53C2×C3×Dic6—22no—
54C2×C3×Dic6C2×D312yes—
55C2×C3×Dic6C2×D312yes—
56C2×#624—62no—
57C2×#624—62no—
58C32×C4⋊C4—31no—
59#3755—64no—
60#3752—61no—
61C4×C3⋊Dic3C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D312yes—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3C2×D312yes—
65C3×Dic3⋊C4—62no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—62no—
72#3724—21no—
73#3724—21no—
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⋊Dic3SD1612yescomplemented
87C32×Dic3—42no—
88C32×Dic3—41no—
89C3×C4⋊Q8—91no—
90#944—32no—
91C2×#215—92no—
92C2×#215—92no—
93C2×#215—182no—
94C2×#204—92no—
95#893—92no—
96#893—181no—
97C4×D12—91no—
98C3×#103—92no—
99#835—32no—
100Dic3⋊C8—92no—
101#810—32no—
102C4⋊(C3⋊C8)—91no—
103C22×C3⋊D3—61no—
104C22×C3⋊D3—62no—
105C22×C3⋊D3—62no—
106C22×C3⋊D3—122no—
107C22×C3⋊D3—122no—
108C32×Q8—62no—
109C32×Q8—62no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—41no—
113C2×C32×C4—61no—
114C2×C32×C4—62no—
115C2×C32×C4C22×D312yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3⋊Dic3#6711yescharacteristic
119C4⋊(C3⋊D3)—32no—
120C4⋊(C3⋊D3)—34no—
121C4⋊(C3⋊D3)—62no—
122C4⋊(C3⋊D3)—64no—
123C4⋊(C3⋊D3)—61no—
124C4⋊(C3⋊D3)—62no—
125C4⋊(C3⋊D3)—62no—
126C4⋊(C3⋊D3)—122no—
127C4⋊(C3⋊D3)—122no—
128C4⋊(C3⋊D3)—122no—
129C4×C3⋊D3—122no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—42no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C3×Dic6—24no—
140C3×Dic6—22no—
141C3×Dic6—22no—
142C3×Dic6—22no—
143C3×Dic6—22no—
144C3×Dic6—42no—
145#624—124no—
146#624—124no—
147C3⋊(C3⋊C8)—22no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—62no—
151#461—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33C2×SD1611yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—42no—
156C2×C3×Q8—32no—
157C2×C3×Q8—32no—
158C2×C3×Q8—62no—
159C2×D12—91no—
160C2×D12—92no—
161C2×D12—181no—
162C2×D12—182no—
163C2×D12—361no—
164C2×C4×D3—181no—
165C2×C4×D3—182no—
166C2×C4×D3—182no—
167C2×Dic6D3212yes—
168C2×C3×C8—92no—
169C3×C4⋊C4—32no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—181no—
172C3×C4⋊C4—181no—
173C3×C42—91no—
174#215—184no—
175#215—184no—
176#215—188no—
177(C2×D3)⋊C4—181no—
178(C2×D3)⋊C4—182no—
179(C2×D3)⋊C4—182no—
180C4⋊Dic3D3211yescharacteristic
181Dic3⋊C4—62no—
182Dic3⋊C4—62no—
183C4×Dic3—32no—
184C4×Dic3—61no—
185C2×C3⋊C8—32no—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—62no—
188#204—184no—
189C22×C32—22no—
190C22×C32—21no—
191C22×C32—41no—
192C22×C32D3×D412yes—
193C22×C32C2×#6711yescharacteristic
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—21no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4—122no—
216C32×C4—121no—
217C32×C4#23512yes—
218C32×C4D3×D412yes—
219C32×C4D3×D411yescharacteristic
220C32×C4#23711yescharacteristic
221C3⋊Dic3—21no—
222C3⋊Dic3#21512yescomplemented
223C3×Dic3—34no—
224C3×Dic3—42no—
225C3×Dic3—44no—
226C3×Dic3—42no—
227C3×Dic3—42no—
228C3×Dic3—42no—
229C3×Dic3—62no—
230C3×Dic3—62no—
231C3×Dic3—121no—
232C2×SD16—272no—
233C4⋊Q8—91no—
234C4×D4—271no—
235C4⋊C8—271no—
236#103—92no—
237C33#46111yescharacteristic, complemented, Sylow
238C22×D3—182no—
239C22×D3—182no—
240C22×D3—181no—
241C22×D3—362no—
242C22×D3—362no—
243C3×Q8—64no—
244C3×Q8—62no—
245C3×Q8—62no—
246C3×Q8—62no—
247C3×Q8—62no—
248C3×Q8—122no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—91no—
257C2×C3×C4—91no—
258C2×C3×C4—122no—
259C2×C3×C4C2×D3211yescharacteristic
260C2×C3×C4C2×D3212yes—
261C2×Dic3—22no—
262C2×Dic3—21no—
263C2×Dic3—32no—
264C2×Dic3—32no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267D12—92no—
268D12—94no—
269D12—182no—
270D12—184no—
271D12—182no—
272D12—181no—
273D12—182no—
274D12—362no—
275D12—362no—
276D12—362no—
277C4×D3—364no—
278C4×D3—362no—
279C4×D3—364no—
280Dic6—22no—
281Dic6—22no—
282C3×C8—182no—
283C3⋊C8—64no—
284C3⋊C8—62no—
285C3⋊C8—62no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—22no—
291C2×C32—22no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32#88412yes—
296C2×C32D3×SD1612yes—
297C2×C32#91812yes—
298C2×C32#93811yescharacteristic
299C2×C32C2×#21511yescharacteristic
300C2×C32#94211yescharacteristic
301C3⋊D3—122no—
302C3⋊D3—124no—
303C3⋊D3—124no—
304C3⋊D3—244no—
305C3⋊D3—244no—
306C2×Q8—32no—
307C2×D4—271no—
308C22×C4—541no—
309SD16—544no—
310C2×C8—92no—
311C4⋊C4—31no—
312C4⋊C4—181no—
313C4⋊C4—181no—
314C22⋊C4—541no—
315C42—91no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—41no—
319C22×C3D3×#6712yes—
320C22×C3#381111yescharacteristic
321C2×D3—184no—
322C2×D3—182no—
323C2×D3—184no—
324C2×D3—364no—
325C2×D3—364no—
326C2×D3—362no—
327C2×D3—362no—
328C2×D3—361no—
329C2×D3—362no—
330C2×D3—361no—
331C2×D3—362no—
332C2×D3—722no—
333C2×D3—722no—
334C2×D3—722no—
335C2×D3—722no—
336C3×C4—22no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—22no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4—92no—
343C3×C4—122no—
344C3×C4—124no—
345C3×C4—122no—
346C3×C4—122no—
347C3×C4—122no—
348C3×C4—181no—
349C3×C4#379811yescharacteristic
350C3×C4#380211yescharacteristic
351C3×C4#380412yes—
352C3×C4#381112yes—
353Dic3—34no—
354Dic3—41no—
355Dic3—42no—
356Dic3—62no—
357Dic3—62no—
358Dic3—121no—
359C32—22no—
360C32—21no—
361C32—41no—
362C32#481812yescomplemented
363C32#508311yescharacteristic, complemented
364C23—541no—
365Q8—62no—
366Q8—62no—
367D4—272no—
368D4—541no—
369C2×C4—61no—
370C2×C4—62no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—1082no—
374C2×C4D3311yescharacteristic
375C8—182no—
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×C3#6391211yescharacteristic
386C2×C3#6393011yescharacteristic
387C2×C3D3×#21512yes—
388C2×C3#6394511yescharacteristic
389C2×C3#6394812yes—
390C2×C3#6395812yes—
391D3—364no—
392D3—364no—
393D3—362no—
394D3—724no—
395D3—724no—
396C22—542no—
397C22—1081no—
398C22—1081no—
399C22#8895211yescharacteristic
400C4—92no—
401C4—121no—
402C4—122no—
403C4—181no—
404C4#8895411yescharacteristic
405C4#8895911yescharacteristic
406C3—21no—
407C3—22no—
408C3—41no—
409C3#1059056511yescharacteristic, complemented
410C3#1059156512yescomplemented
411C2—1082no—
412C2#1172283811yescharacteristic
413C2#1172289011yescharacteristic
414C2#1172296411yescharacteristic
subgroupG/H
centerC22#88952
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
3. C3×#63873C2semidirect
4. #11722078C2semidirect
5. C3×#63565C2semidirect
23. #63873D3semidirect
31. #63565D3semidirect
86. C3×C3⋊Dic3SD16semidirect
222. C3⋊Dic3#215semidirect
237. C33#461semidirect
362. C32#4818semidirect
363. C32#5083semidirect
409. C3#10590565semidirect
410. C3#10591565semidirect

Maximal subgroups

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

subgroupindex
#117234942
C2×#887952
C3×#638732
#117220782
C3×#635652
#105915653
#105905653