Finite Group Databasebeta

← #13160970

Subgroups — #13160970

Order 1728.

1112 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 172 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×C3×#3830C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3#11723435C211yesmaximal, characteristic, complemented
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592872—31nomaximal
9#10591461—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3830—22no—
14C3×#3830—22no—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×#214—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22#88806—21no—
23#88803—21no—
24#88798—22no—
25C2×#3830D311yescharacteristic
26C2×C3×#237—34no—
27C2×C32×SD16—31no—
28#64155—31no—
29C2×#3756—31no—
30C2×C3×#214D314yescomplemented
31#63856—34no—
32C2×#3718—34no—
33#63659—31no—
34#63659—34no—
35#63659—64no—
36#63647—31no—
37#63643—31no—
38#63566—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×D4—21no—
43C33×D4D412yes—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C3×#638—24no—
47C2×C3×C3⋊Dic3—24no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3—21no—
51C3×C4×C3⋊D3—21no—
52C3×#634—21no—
53C3×#634D412yes—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5239—91no—
60#5173—94no—
61C2×C32×Q8—31no—
62C2×C32×D4—24no—
63C2×C32×D4C2×D314yes—
64C2×C32×D4C2×D311yescharacteristic
65C2×#638—21no—
66C22×C3⋊Dic3—21no—
67#3830—22no—
68#3830—22no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×#67—64no—
72C22×C3×Dic3—64no—
73C3×#237—68no—
74C3×#237—68no—
75C2×C3×C4×D3—34no—
76C2×C3×Dic6—34no—
77C2×C3×Dic6C2×D314yes—
78C32×SD16—62no—
79C2×C32×C8—31no—
80#3756—62no—
81#3752—61no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—34no—
84C4⋊(C3⋊Dic3)—64no—
85#3750—61no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C3×#214—28no—
90C2×C3×C3⋊C8C2×D314yes—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3718—68no—
94C22×C33—22no—
95C22×C33—41no—
96C22×C33—41no—
97C22×C33C2×D411yescharacteristic
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3⋊(C3⋊Dic3)—41no—
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—21no—
106C32×Dic3—41no—
107C2×C3×#41—91no—
108C2×#237—34no—
109C2×C3×SD16—34no—
110#946—94no—
111C2×#216—94no—
112C2×#216—91no—
113C2×#214C3⋊D311yescharacteristic, complemented
114#924—91no—
115#835—91no—
116#832—94no—
117#832—91no—
118#832—184no—
119#830—91no—
120(C2×D3)⋊C8—94no—
121#816—94no—
122C23×C32—21no—
123C23×C32—24no—
124C23×C32—44no—
125C22×C3⋊D3#6711yescharacteristic
126C22×C3×D3—34no—
127C32×Q8—61no—
128C32×Q8—61no—
129C32×D4—21no—
130C32×D4—28no—
131C32×D4—24no—
132C32×D4—44no—
133C32×D4#6718yes—
134C32×D4#6712yes—
135C2×C32×C4—24no—
136C2×C32×C4—61no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139#638—24no—
140C2×C3⋊Dic3—24no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—124no—
145C2×C3⋊Dic3#6711yescharacteristic
146C4×C3⋊D3—21no—
147C4×C3⋊D3—21no—
148#634—21no—
149#634#6712yes—
150C3×#67—616no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—616no—
154C2×C3×Dic3—64no—
155C3×C4×D3—64no—
156C3×C4×D3—64no—
157C3×Dic6—24no—
158C3×Dic6—24no—
159C3×Dic6—38no—
160C3×Dic6—64no—
161C32×C8—61no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
165C3×C3⋊C8—24no—
166#452—271noSylow
167C2×C33—42no—
168C2×C33#12011yescharacteristic
169C2×C33#13511yescharacteristic
170C2×C33#13711yescharacteristic
171C3×C3⋊D3—41no—
172C3×#41—182no—
173C3×#41—182no—
174C2×C3×Q8—34no—
175C2×C3×Q8—91no—
176C2×C3×D4—24no—
177C2×C3×D4—181no—
178C2×C3×D4D3214yes—
179C2×C3×D4C2×C3⋊D311yescharacteristic
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182C2×#67—64no—
183C22×Dic3—64no—
184#237—68no—
185#237—68no—
186C2×C4×D3—34no—
187C2×Dic6—34no—
188C2×Dic6C2×C3⋊D311yescharacteristic
189C3×SD16—68no—
190C2×C3×C8—34no—
191#217—181no—
192#216—182no—
193#216—188no—
194#214—22no—
195(C2×D3)⋊C4—184no—
196C4⋊Dic3—94no—
197C4⋊Dic3—91no—
198C4⋊Dic3—184no—
199Dic3⋊C4—181no—
200Dic3⋊C4—184no—
201C2×C3⋊C8—94no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—184no—
204C2×C3⋊C8C2×C3⋊D311yescharacteristic
205C22×C32—22no—
206C22×C32—24no—
207C22×C32—28no—
208C22×C32—48no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—44no—
212C22×C32—44no—
213C22×C32—84no—
214C22×C32—84no—
215C22×C32D3×D414yes—
216C22×C32C2×#6711yescharacteristic
217C2×C3⋊D3—21no—
218C2×C3⋊D3—21no—
219C2×C3⋊D3—21no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C32×C4—24no—
224C32×C4—24no—
225C32×C4—121no—
226C32×C4D3×D414yes—
227C32×C4C2×#6714yes—
228C32×C4C2×#6711yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C3⋊Dic3—22no—
231C3⋊Dic3—21no—
232C3⋊Dic3—121no—
233C3⋊Dic3—124no—
234C3⋊Dic3—244no—
235C3×Dic3—44no—
236C3×Dic3—68no—
237C3×Dic3—64no—
238C2×#41—91no—
239C2×Q16—271no—
240C2×SD16—31no—
241C22⋊Q8—271no—
242#103—271no—
243#102—271no—
244C22⋊C8—271no—
245C33#45211yescharacteristic, complemented, Sylow
246C23×C3—24no—
247C23×C3—21no—
248C23×C3—44no—
249C23×C3—91no—
250C22×D3—34no—
251C3×Q8—64no—
252C3×Q8—64no—
253C3×Q8—92no—
254C3×Q8—181no—
255C3×D4—24no—
256C3×D4—21no—
257C3×D4—28no—
258C3×D4—44no—
259C3×D4—184no—
260C3×D4#62518yes—
261C3×D4#63812yes—
262C2×C3×C4—24no—
263C2×C3×C4—64no—
264C2×C3×C4—91no—
265C2×C3×C4—184no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4C2×D3214yes—
270C2×C3×C4C22×C3⋊D311yescharacteristic
271#67—616no—
272C2×Dic3—21no—
273C2×Dic3—34no—
274C2×Dic3—616no—
275C2×Dic3—64no—
276C2×Dic3—184no—
277C2×Dic3—181no—
278C2×Dic3—364no—
279C4×D3—64no—
280C4×D3—64no—
281Dic6—21no—
282Dic6—21no—
283Dic6—38no—
284Dic6—64no—
285C3×C8—64no—
286C3⋊C8—21no—
287C3⋊C8—184no—
288C3⋊C8—181no—
289C3⋊C8—364no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32—42no—
294C2×C32—48no—
295C2×C32—88no—
296C2×C32#91514yes—
297C2×C32#91614yes—
298C2×C32#93714yes—
299C2×C32#95011yescharacteristic
300C2×C32#95111yescharacteristic
301C2×C32#95311yescharacteristic
302C3⋊D3—41no—
303C3×D3—124no—
304#41—182no—
305#41—182no—
306C2×Q8—31no—
307C2×Q8—91no—
308C2×D4—181no—
309C2×D4D3×C3⋊D311yescharacteristic
310C22×C4—91no—
311C22×C4—181no—
312Q16—542no—
313SD16—62no—
314C2×C8—31no—
315C2×C8—271no—
316C4⋊C4—271no—
317C4⋊C4—541no—
318C22⋊C4—541no—
319C22×C3—22no—
320C22×C3—28no—
321C22×C3—24no—
322C22×C3—48no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—44no—
326C22×C3—44no—
327C22×C3—84no—
328C22×C3—84no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3D3×#6714yes—
333C22×C3D4×C3⋊D311yescharacteristic
334C2×D3—64no—
335C2×D3—64no—
336C2×D3—64no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—124no—
340C3×C4—181no—
341C3×C4—182no—
342C3×C4C2×#62514yes—
343C3×C4D3×#6714yes—
344C3×C4D4×C3⋊D311yescharacteristic
345C3×C4C2×#63811yescharacteristic
346Dic3—41no—
347Dic3—68no—
348Dic3—64no—
349Dic3—361no—
350Dic3—364no—
351Dic3—724no—
352C32—24no—
353C32#517314yescomplemented
354C32#523911yescharacteristic, complemented
355C23—21no—
356C23—91no—
357Q8—61no—
358Q8—61no—
359Q8—92no—
360Q8—181no—
361D4—21no—
362D4—184no—
363D4#629012yes—
364C2×C4—61no—
365C2×C4—91no—
366C2×C4—184no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—541no—
371C2×C4C2×D3×C3⋊D311yescharacteristic
372C8—61no—
373C8—541no—
374C2×C3—24no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—42no—
378C2×C3—48no—
379C2×C3—88no—
380C2×C3—361no—
381C2×C3#6392914yes—
382C2×C3#6393514yes—
383C2×C3#6397614yes—
384C2×C3#6412411yescharacteristic
385C2×C3#6412511yescharacteristic
386C2×C3#6414611yescharacteristic
387D3—124no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—181no—
392C22—181no—
393C22—181no—
394C22C3⋊D3×#6711yescharacteristic
395C4—121no—
396C4—181no—
397C4—182no—
398C4—1081no—
399C4C2×#629011yescharacteristic
400C4C3⋊D3×#6711yescharacteristic
401C3—24no—
402C3#1059146114yescomplemented
403C3#1059287211yescharacteristic, complemented
404C2—42no—
405C2—361no—
406C2#1172355211yescharacteristic
407C2#1172356111yescharacteristic
408C2#1172363411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C32×#214C2semidirect
3. #11723435C2semidirect
4. C2×#88798C2semidirect
5. #11722202C2semidirect
6. #11722040C2semidirect
7. #11722023C2semidirect
30. C2×C3×#214D3semidirect
113. C2×#214C3⋊D3semidirect
245. C33#452semidirect
353. C32#5173semidirect
354. C32#5239semidirect
402. C3#10591461semidirect
403. C3#10592872semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C32×#2142
#117234352
C2×#887982
#117222022
#117220402
#117220232
#105928723
#105914613