Finite Group Databasebeta

← #13162032

Subgroups — #13162032

Order 1728.

824 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 111 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×#89040C211yesmaximal, characteristic
2C3×#64155C211yesmaximal, characteristic
3C2×C3×#3756C212yesmaximal, complemented
4#11722076C212yesmaximal, complemented
5#11722072C211yesmaximal, characteristic
6#10591654—31nomaximal
7#10591653—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#634C2212yes—
11C2×#6296—21no—
12C2×#6294—21no—
13#89040—24no—
14C3×#3756—24no—
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C2×C3×C3⋊(C3⋊C8)C2212yes—
19C2×#3797—31no—
20C2×D3×Dic6—32no—
21#64155D311yescharacteristic
22C2×#3756D312yescomplemented
23C3×#946—32no—
24C3×#946—61no—
25C2×C3×#216—32no—
26C2×C3×#216—32no—
27C2×C3×#216—62no—
28#63567—32no—
29#63566—32no—
30#63565—32no—
31#63557—31no—
32(C2×D3)⋊(C3⋊C8)—32no—
33C22×C3×C3⋊D3D411yescharacteristic
34C33×Q8—22no—
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3D411yescharacteristic
40C3×C4×C3⋊D3—22no—
41C3×#634—22no—
42C3×#634D414yes—
43#6296—24no—
44#6294—24no—
45C3×C3⋊(C3⋊C8)—22no—
46#5186—92no—
47#5050—91no—
48C2×C32×Q8—21no—
49C2×C32×Q8—22no—
50C2×C32×Q8—41no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×Q8C2×D312yes—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634C2×D312yes—
55C2×C3×C4×D3—32no—
56C2×C3×C4×D3—61no—
57C2×C3×Dic6—32no—
58C2×C3×Dic6—32no—
59C2×C3×Dic6—62no—
60C2×#627—62no—
61C2×#627—61no—
62C2×#624—61no—
63C2×D3×Dic3—62no—
64#3797—64no—
65D3×Dic6—64no—
66D3×Dic6—64no—
67#3756—24no—
68#3752—21no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—21no—
71C2×C3⋊(C3⋊C8)C2×D312yes—
72C3×#216—64no—
73C3×#216—64no—
74C3×#216—68no—
75C3×(C2×D3)⋊C4—62no—
76C3×(C2×D3)⋊C4—62no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—61no—
79C3×Dic3⋊C4—62no—
80C3×Dic3⋊C4—62no—
81C2×C3×C3⋊C8—32no—
82C2×C3×C3⋊C8—32no—
83C2×C3×C3⋊C8—62no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3Q1612yescomplemented
88C33×C4—41no—
89C33×C4C2×D412yes—
90C3×C3⋊Dic3—24no—
91C3×C3⋊Dic3—21no—
92C3×C3⋊Dic3—41no—
93C2×D3×Q8—92no—
94C22×Dic6—91no—
95C2×C3×Q16—92no—
96C3×C22⋊Q8—91no—
97#946—32no—
98#946—61no—
99C2×#216—32no—
100C2×#216—32no—
101C2×#216—62no—
102#835—92no—
103#830—91no—
104(C2×D3)⋊C8—92no—
105#816—92no—
106#810—92no—
107C22×C3⋊D3#6711yescharacteristic
108C22×C3×D3—32no—
109C22×C3×D3—61no—
110C32×Q8—22no—
111C32×Q8—24no—
112C32×Q8—22no—
113C32×Q8—22no—
114C32×Q8—42no—
115C32×Q8—42no—
116C32×Q8—44no—
117C2×C32×C4—21no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4—42no—
124C2×C32×C4—42no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D312yes—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3#6711yescharacteristic
130C4×C3⋊D3—22no—
131#634—22no—
132#634#6714yes—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—122no—
140C3×C4×D3—62no—
141C3×C4×D3—62no—
142C3×C4×D3—122no—
143C3×Dic6—34no—
144C3×Dic6—34no—
145C3×Dic6—64no—
146C3×Dic6—62no—
147C3×Dic6—62no—
148C3×Dic6—122no—
149#627—68no—
150#627—64no—
151#624—64no—
152D3×Dic3—68no—
153C3⋊(C3⋊C8)—22no—
154C3×C3⋊C8—64no—
155C3×C3⋊C8—62no—
156C3×C3⋊C8—62no—
157#451—271noSylow
158C2×C33#12011yescharacteristic
159C2×C33C2×Q1611yescharacteristic
160C2×C33#13711yescharacteristic
161C3×C3⋊D3—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—41no—
165C2×C3×Q8—92no—
166C2×C3×Q8D3211yescharacteristic
167C2×C3×Q8D3212yes—
168C22×C3×C4—91no—
169C22×Dic3—181no—
170D3×Q8—184no—
171D3×Q8—184no—
172C2×C4×D3—32no—
173C2×C4×D3—61no—
174C2×C4×D3—182no—
175C2×Dic6—32no—
176C2×Dic6—32no—
177C2×Dic6—62no—
178C2×Dic6—184no—
179C2×Dic6—182no—
180C2×Dic6—181no—
181C3×Q16—184no—
182C2×C3×C8—92no—
183C3×C4⋊C4—91no—
184C3×C4⋊C4—181no—
185C3×C22⋊C4—181no—
186#216—64no—
187#216—64no—
188#216—68no—
189(C2×D3)⋊C4—62no—
190(C2×D3)⋊C4—62no—
191C4⋊Dic3—32no—
192C4⋊Dic3—61no—
193Dic3⋊C4—62no—
194Dic3⋊C4—62no—
195C2×C3⋊C8—32no—
196C2×C3⋊C8—32no—
197C2×C3⋊C8—62no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32D3×D412yes—
202C22×C32C2×#6711yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3#21612yescomplemented
206C2×C3×D3—34no—
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C32×C4—22no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—42no—
216C32×C4—42no—
217C32×C4—44no—
218C32×C4—44no—
219C32×C4—41no—
220C32×C4—42no—
221C32×C4D3×D412yes—
222C32×C4D3×D412yes—
223C32×C4C2×#6712yes—
224C3⋊Dic3—24no—
225C3⋊Dic3—21no—
226C3⋊Dic3—41no—
227C3×Dic3—64no—
228C3×Dic3—64no—
229C3×Dic3—62no—
230C3×Dic3—124no—
231C3×Dic3—121no—
232C3×Dic3—122no—
233C3×Dic3—122no—
234C22×Q8—271no—
235C2×Q16—92no—
236C22⋊Q8—91no—
237#103—272no—
238C22⋊C8—271no—
239C33#45111yescharacteristic, complemented, Sylow
240C23×C3—91no—
241C22×D3—32no—
242C22×D3—61no—
243C3×Q8—24no—
244C3×Q8—22no—
245C3×Q8—22no—
246C3×Q8—22no—
247C3×Q8—42no—
248C3×Q8—42no—
249C3×Q8—44no—
250C3×Q8—94no—
251C3×Q8—182no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—91no—
261C2×C3×C4—182no—
262C2×C3×C4—182no—
263C2×C3×C4—181no—
264C2×C3×C4C2×D3211yescharacteristic
265C2×C3×C4C2×D3212yes—
266C2×Dic3—32no—
267C2×Dic3—61no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—122no—
273C2×Dic3—184no—
274C4×D3—62no—
275C4×D3—62no—
276C4×D3—122no—
277C4×D3—188no—
278Dic6—34no—
279Dic6—34no—
280Dic6—64no—
281Dic6—62no—
282Dic6—62no—
283Dic6—122no—
284Dic6—184no—
285Dic6—188no—
286C3×C8—182no—
287C3⋊C8—64no—
288C3⋊C8—62no—
289C3⋊C8—62no—
290C2×C32—22no—
291C2×C32—22no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—22no—
295C2×C32—21no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32D3×Q1612yes—
300C2×C32#91812yes—
301C2×C32#93712yes—
302C2×C32#93711yescharacteristic
303C2×C32C2×#21611yescharacteristic
304C2×C32#95111yescharacteristic
305C3⋊D3—22no—
306C3×D3—64no—
307C3×D3—122no—
308C2×Q8—92no—
309C2×Q8—541no—
310C2×Q8—544no—
311C2×Q8#111311yescharacteristic
312C22×C4—91no—
313C22×C4—541no—
314Q16—184no—
315C2×C8—92no—
316C4⋊C4—91no—
317C4⋊C4—181no—
318C22⋊C4—181no—
319C22×C3—21no—
320C22×C3—22no—
321C22×C3—41no—
322C22×C3—92no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3D3×#6712yes—
326C22×C3#381111yescharacteristic
327C2×D3—34no—
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—62no—
331C2×D3—121no—
332C2×D3—121no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—42no—
337C3×C4—44no—
338C3×C4—44no—
339C3×C4—42no—
340C3×C4—41no—
341C3×C4—42no—
342C3×C4—181no—
343C3×C4—184no—
344C3×C4—361no—
345C3×C4D3×#6712yes—
346C3×C4D3×#6712yes—
347C3×C4#381112yes—
348Dic3—64no—
349Dic3—64no—
350Dic3—62no—
351Dic3—124no—
352Dic3—121no—
353Dic3—122no—
354Dic3—122no—
355C32—22no—
356C32—21no—
357C32—41no—
358C32#505011yescharacteristic, complemented
359C32#518612yescomplemented
360C23—91no—
361Q8—22no—
362Q8—94no—
363Q8—182no—
364Q8—544no—
365C2×C4—21no—
366C2×C4—91no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—181no—
370C2×C4—544no—
371C2×C4C2×#111311yescharacteristic
372C8—182no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—182no—
383C2×C3D3×#21612yes—
384C2×C3#6394312yes—
385C2×C3#6394411yescharacteristic
386C2×C3#6394511yescharacteristic
387C2×C3#6398312yes—
388C2×C3#6398311yescharacteristic
389D3—64no—
390D3—122no—
391C22—92no—
392C22—181no—
393C22—181no—
394C22#8904911yescharacteristic
395C4—41no—
396C4—181no—
397C4—184no—
398C4—361no—
399C4#8904912yes—
400C3—21no—
401C3—22no—
402C3—41no—
403C3#1059165312yescomplemented
404C3#1059165411yescharacteristic, complemented
405C2—182no—
406C2#1172371011yescharacteristic
407C2#1172371111yescharacteristic
408C2#1172372911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×Q8#1113
hypocenterC33#451
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
3. C2×C3×#3756C2semidirect
4. #11722076C2semidirect
22. C2×#3756D3semidirect
87. C2×C3×C3⋊D3Q16semidirect
205. C2×C3⋊D3#216semidirect
239. C33#451semidirect
358. C32#5050semidirect
359. C32#5186semidirect
403. C3#10591653semidirect
404. C3#10591654semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C3×#641552
C2×C3×#37562
#117220762
#117220722
#105916543
#105916533