Finite Group Databasebeta

← #13155472

Subgroups — #13155472

Order 1728.

834 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 128 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#11723503C212yesmaximal
2C2×#88785C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic, complemented
4C3×#63576C211yesmaximal, characteristic, complemented
5C3×#63575C212yesmaximal, complemented
6#10590687—31nomaximal
7#10590573—32nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×#6289—22no—
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14#88806—22no—
15#88785C414yescomplemented
16C2×C8×C3⋊D3—31no—
17C4⋊C4×C3⋊D3—32no—
18#63900—32no—
19#63900—32no—
20#63900—62no—
21C2×#3710—32no—
22C2×#3710—61no—
23C3×#818—32no—
24C3×#807—32no—
25C3×#807—32no—
26#63576—31no—
27#63576D311yescharacteristic, complemented
28#63575—32no—
29#63575D312yescomplemented
30C22×C3⋊(C3⋊D3)D411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C4×C3⋊(C3⋊D3)C2×C412yes—
34C4×C3⋊(C3⋊D3)Q812yes—
35C2×C3×C3⋊Dic3—22no—
36C2×C32×Dic3—22no—
37#6289—44no—
38C3×C3⋊(C3⋊C8)C2×C412yescomplemented
39C32×C3⋊C8C2×C412yescomplemented
40D3×#107—91no—
41D3×#107—92no—
42C2×C4×C3⋊D3—32no—
43C2×C4×C3⋊D3—31no—
44C2×C4×C3⋊D3—61no—
45C2×C4×C3⋊D3—62no—
46C2×C4×C3⋊D3—62no—
47C2×C4×C3⋊D3—121no—
48C2×#624—62no—
49C2×#624—62no—
50C2×#624—64no—
51C2×C32×C8—31no—
52C32×C4⋊C4—32no—
53C4⋊(C3⋊Dic3)C2×D312yes—
54#3750—62no—
55C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
56C8×C3⋊D3—34no—
57C3×C4⋊Dic3—22no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—62no—
61C3×C4⋊Dic3C2×D312yes—
62C3×C4⋊Dic3C2×D312yes—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—61no—
66C2×C3×C3⋊C8C2×D312yes—
67#3724—62no—
68#3724—62no—
69#3724—64no—
70#3710—38no—
71#3710—64no—
72C22×C33C2×D411yescharacteristic
73C2×C3⋊(C3⋊D3)C4⋊C412yes—
74C2×C3⋊(C3⋊D3)D812yes—
75C2×C3⋊(C3⋊D3)Q1612yes—
76C33×C4C22×C411yescharacteristic
77C33×C4C2×Q811yescharacteristic
78C3⋊(C3⋊Dic3)C4⋊C412yes—
79C3×C3⋊Dic3—42no—
80C32×Dic3—42no—
81C2×D3×C8—92no—
82C2×D3×C8—91no—
83C2×D3×C8—181no—
84D3×C4⋊C4—92no—
85D3×C4⋊C4—92no—
86D3×C4⋊C4—92no—
87D3×C4⋊C4—182no—
88C3×#107—91no—
89C3×#107—92no—
90#818—32no—
91#818—91no—
92#807—32no—
93#807—32no—
94#807—92no—
95C22×C3⋊D3—31no—
96C22×C3⋊D3—32no—
97C22×C3⋊D3—62no—
98C22×C3⋊D3—61no—
99C22×C3⋊D3—121no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—41no—
103C2×C32×C4—62no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—32no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—121no—
112C4×C3⋊D3—34no—
113C4×C3⋊D3—32no—
114C4×C3⋊D3—34no—
115C4×C3⋊D3—32no—
116C4×C3⋊D3—62no—
117C4×C3⋊D3—64no—
118C4×C3⋊D3—62no—
119C4×C3⋊D3—64no—
120C4×C3⋊D3—122no—
121C4×C3⋊D3—122no—
122C4×C3⋊D3—124no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129#624—124no—
130#624—124no—
131#624—128no—
132C32×C8—32no—
133C3⋊(C3⋊C8)C4×D312yescomplemented
134C3×C3⋊C8—22no—
135C3×C3⋊C8—34no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8C4×D314yescomplemented
138C2×#107—271noSylow
139C2×C33C2×C4⋊C411yescharacteristic
140C2×C33C2×D811yescharacteristic
141C2×C33C2×Q1611yescharacteristic
142C3⋊(C3⋊D3)#10714yescomplemented
143C2×C4×D3—92no—
144C2×C4×D3—91no—
145C2×C4×D3—182no—
146C2×C4×D3—181no—
147C2×C4×D3—182no—
148C2×C4×D3—182no—
149C2×C4×D3—182no—
150C2×C4×D3—184no—
151C2×C4×D3—361no—
152C2×C3×C8—32no—
153C2×C3×C8—61no—
154C2×C3×C8—91no—
155C3×C4⋊C4—32no—
156C3×C4⋊C4—32no—
157C3×C4⋊C4—62no—
158C3×C4⋊C4—92no—
159C4⋊Dic3—32no—
160C4⋊Dic3—32no—
161C4⋊Dic3—62no—
162C4⋊Dic3D3212yes—
163Dic3⋊C4—182no—
164Dic3⋊C4—182no—
165Dic3⋊C4—182no—
166Dic3⋊C4—184no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—61no—
169C2×C3⋊C8D3211yescharacteristic
170D3×C8—98no—
171D3×C8—94no—
172D3×C8—184no—
173C22×C32—21no—
174C22×C32—22no—
175C22×C32—41no—
176C22×C32D3×D412yes—
177C22×C32D3×D411yescharacteristic
178C2×C3⋊D3—32no—
179C2×C3⋊D3—34no—
180C2×C3⋊D3—32no—
181C2×C3⋊D3—34no—
182C2×C3⋊D3—34no—
183C2×C3⋊D3—32no—
184C2×C3⋊D3—64no—
185C2×C3⋊D3—62no—
186C2×C3⋊D3—64no—
187C2×C3⋊D3—62no—
188C2×C3⋊D3—62no—
189C2×C3⋊D3—64no—
190C2×C3⋊D3—122no—
191C2×C3⋊D3—122no—
192C2×C3⋊D3—122no—
193C32×C4—22no—
194C32×C4—21no—
195C32×C4—22no—
196C32×C4—21no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—122no—
200C32×C4C2×C4×D311yescharacteristic
201C32×C4C2×C4×D312yes—
202C32×C4D3×Q811yescharacteristic
203C32×C4D3×Q812yes—
204C3⋊Dic3—32no—
205C3⋊Dic3—34no—
206C3⋊Dic3—42no—
207C3⋊Dic3—64no—
208C3⋊Dic3—62no—
209C3⋊Dic3—122no—
210C3×Dic3—44no—
211C3×Dic3—42no—
212C3×Dic3—42no—
213C3×Dic3—124no—
214C3×Dic3—122no—
215C3×Dic3—122no—
216C22×C8—271no—
217C2×C4⋊C4—272no—
218#107—91no—
219#107—92no—
220#107—271no—
221C33C2×#10711yescharacteristic, complemented, Sylow
222C22×D3—91no—
223C22×D3—92no—
224C22×D3—181no—
225C22×D3—182no—
226C22×D3—361no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—64no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—182no—
234C2×C3×C4C2×D3212yes—
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—64no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—91no—
241C2×Dic3—92no—
242C2×Dic3—182no—
243C2×Dic3—181no—
244C2×Dic3—361no—
245C4×D3—94no—
246C4×D3—92no—
247C4×D3—94no—
248C4×D3—92no—
249C4×D3—182no—
250C4×D3—184no—
251C4×D3—184no—
252C4×D3—182no—
253C4×D3—362no—
254C4×D3—362no—
255C4×D3—364no—
256C4×D3—364no—
257C4×D3—364no—
258C4×D3—368no—
259C3×C8—34no—
260C3×C8—62no—
261C3×C8—92no—
262C3⋊C8—34no—
263C3⋊C8—62no—
264C3⋊C8#62412yescomplemented
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—22no—
270C2×C32—22no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32D3×C4⋊C411yescharacteristic
275C2×C32D3×C4⋊C412yes—
276C2×C32D3×D812yes—
277C2×C32D3×D811yescharacteristic
278C2×C32D3×Q1612yes—
279C2×C32D3×Q1611yescharacteristic
280C3⋊D3—34no—
281C3⋊D3—38no—
282C3⋊D3—68no—
283C3⋊D3—64no—
284C3⋊D3—124no—
285C22×C4—271no—
286C22×C4—542no—
287C2×C8—31no—
288C2×C8—91no—
289C2×C8—274no—
290C4⋊C4—32no—
291C4⋊C4—92no—
292C4⋊C4—542no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—41no—
296C22×C3#380211yescharacteristic
297C22×C3#381112yes—
298C2×D3—92no—
299C2×D3—92no—
300C2×D3—94no—
301C2×D3—92no—
302C2×D3—94no—
303C2×D3—94no—
304C2×D3—182no—
305C2×D3—184no—
306C2×D3—184no—
307C2×D3—182no—
308C2×D3—182no—
309C2×D3—184no—
310C2×D3—362no—
311C2×D3—362no—
312C2×D3—362no—
313C3×C4—21no—
314C3×C4—22no—
315C3×C4—21no—
316C3×C4—22no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—124no—
320C3×C4—122no—
321C3×C4—122no—
322C3×C4—362no—
323C3×C4#379712yes—
324C3×C4#379711yescharacteristic
325C3×C4C4×D3212yes—
326C3×C4C2×#62411yescharacteristic
327Dic3—42no—
328Dic3—94no—
329Dic3—92no—
330Dic3—124no—
331Dic3—122no—
332Dic3—122no—
333Dic3—182no—
334Dic3—184no—
335Dic3—362no—
336C32—22no—
337C32—21no—
338C32—41no—
339C32D3×#10711yescharacteristic, complemented
340C32D3×#10712yescomplemented
341C23—271no—
342C2×C4—62no—
343C2×C4—182no—
344C2×C4—271no—
345C2×C4—272no—
346C2×C4—272no—
347C2×C4—1084no—
348C2×C4D3311yescharacteristic
349C8—32no—
350C8—92no—
351C2×C3—22no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—22no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3#6379711yescharacteristic
361C2×C3#6380211yescharacteristic
362C2×C3#6388312yes—
363C2×C3#6390011yescharacteristic
364C2×C3#6392612yes—
365C2×C3#6394412yes—
366D3—98no—
367D3—94no—
368D3—188no—
369D3—184no—
370D3—364no—
371C22—272no—
372C22—272no—
373C22—272no—
374C22#8895411yescharacteristic
375C4—122no—
376C4—272no—
377C4—362no—
378C4D3×#62411yescharacteristic
379C4#8895811yescharacteristic
380C3—22no—
381C3—21no—
382C3—41no—
383C3#1059057312yescomplemented
384C3#1059068711yescharacteristic, complemented
385C2—274no—
386C2#1172283211yescharacteristic
387C2#1172284811yescharacteristic
388C2#1172291211yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypocenterC33C2×#107
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722080C2semidirect
4. C3×#63576C2semidirect
5. C3×#63575C2semidirect
15. #88785C4semidirect
27. #63576D3semidirect
29. #63575D3semidirect
38. C3×C3⋊(C3⋊C8)C2×C4semidirect
39. C32×C3⋊C8C2×C4semidirect
133. C3⋊(C3⋊C8)C4×D3semidirect
137. C3×C3⋊C8C4×D3semidirect
142. C3⋊(C3⋊D3)#107semidirect
221. C33C2×#107semidirect
264. C3⋊C8#624semidirect
339. C32D3×#107semidirect
340. C32D3×#107semidirect
383. C3#10590573semidirect
384. C3#10590687semidirect

Maximal subgroups

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

subgroupindex
#117235032
C2×#887852
#117220802
C3×#635762
C3×#635752
#105906873
#105905733