Finite Group Databasebeta

← #13160979

Subgroups — #13160979

Order 1728.

1112 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 188 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×C32×#237C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3#11723425C211yesmaximal, characteristic, complemented
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592938—31nomaximal
9#10591461—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×#67—21no—
13C22×C32×Dic3—21no—
14C32×#237—22no—
15C32×#237—22no—
16C2×C32×C4×D3C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3754—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88806—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88798—22no—
25C2×C32×#41—31no—
26C2×C3×#237D314yes—
27C2×#3756—31no—
28C2×#3754D311yescharacteristic, complemented
29#64133—31no—
30C2×C3×#214—34no—
31#63856—34no—
32C2×#3718—34no—
33#63662—31no—
34#63659—31no—
35#63659—34no—
36#63659—64no—
37#63657—31no—
38#63566—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C32×D3D411yescharacteristic
42C33×D4—21no—
43C33×D4D412yes—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634—21no—
49C32×#67—24no—
50C2×C32×Dic3—24no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55C32×Dic6—21no—
56C32×Dic6D412yes—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5239—94no—
60#5173—91no—
61C32×#41—62no—
62C32×#41—62no—
63C2×C32×Q8—31no—
64C2×C32×D4—24no—
65C2×C32×D4—61no—
66C2×C32×D4C2×D314yes—
67C2×C32×D4C2×D311yescharacteristic
68C22×C32×C4—31no—
69C22×C32×C4—61no—
70C2×#634C2×D311yescharacteristic
71C2×C3×#67—24no—
72C22×C3×Dic3—24no—
73C3×#237—28no—
74C3×#237—28no—
75C2×C3×C4×D3C2×D314yes—
76C2×C3×Dic6—34no—
77C2×C3×Dic6C2×D314yes—
78#3757—61no—
79#3756—62no—
80#3754—22no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—34no—
83C4⋊(C3⋊Dic3)—64no—
84#3750—61no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—34no—
87C2×C3⋊(C3⋊C8)—64no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×#214—68no—
90C2×C3×C3⋊C8—34no—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3718—68no—
94C22×C33—22no—
95C22×C33—41no—
96C22×C33—41no—
97C22×C33C2×D411yescharacteristic
98C2×C32×D3—21no—
99C2×C32×D3—21no—
100C2×C32×D3—21no—
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3⋊(C3⋊Dic3)—41no—
104C3×C3⋊Dic3—41no—
105C32×Dic3—22no—
106C32×Dic3—21no—
107C2×C3×#41—34no—
108C2×#237C3⋊D311yescharacteristic
109C2×C3×SD16—91no—
110#946—91no—
111C2×#216—94no—
112C2×#216—91no—
113C2×#214—34no—
114#924—94no—
115#835—94no—
116#832—94no—
117#832—91no—
118#832—184no—
119#830—94no—
120(C2×D3)⋊C8—91no—
121#816—91no—
122C23×C32—21no—
123C23×C32—24no—
124C23×C32—31no—
125C23×C32—44no—
126C22×C3×D3#6714yes—
127C32×Q8—32no—
128C32×Q8—61no—
129C32×D4—21no—
130C32×D4—28no—
131C32×D4—24no—
132C32×D4—44no—
133C32×D4—64no—
134C32×D4#6718yes—
135C32×D4#6712yes—
136C2×C32×C4—24no—
137C2×C32×C4—31no—
138C2×C32×C4—64no—
139C2×C32×C4—61no—
140C2×C32×C4—61no—
141C2×C32×C4—61no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—64no—
147C2×C3⋊Dic3—124no—
148#634—21no—
149#634—21no—
150C3×#67—216no—
151C2×C3×Dic3—216no—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3#6714yes—
155C3×C4×D3—24no—
156C3×C4×D3—24no—
157C3×Dic6—24no—
158C3×Dic6—64no—
159C3×Dic6—64no—
160C3×Dic6#6718yes—
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
165C3×C3⋊C8—64no—
166#452—271noSylow
167C2×C33—42no—
168C2×C33#12011yescharacteristic
169C2×C33#13511yescharacteristic
170C2×C33#13711yescharacteristic
171C32×D3—41no—
172C3×#41—68no—
173C3×#41—68no—
174C2×C3×Q8—34no—
175C2×C3×Q8—91no—
176C2×C3×D4—24no—
177C2×C3×D4—64no—
178C2×C3×D4D3214yes—
179C2×C3×D4C2×C3⋊D311yescharacteristic
180C22×C3×C4—34no—
181C22×C3×C4—64no—
182C2×#67—21no—
183C22×Dic3—21no—
184#237—22no—
185#237—22no—
186C2×C4×D3C2×C3⋊D311yescharacteristic
187C2×Dic6—34no—
188C2×Dic6C2×C3⋊D311yescharacteristic
189C3×SD16—182no—
190C2×C3×C8—91no—
191#217—184no—
192#216—188no—
193#216—182no—
194#214—68no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—94no—
197C4⋊Dic3—91no—
198C4⋊Dic3—184no—
199Dic3⋊C4—184no—
200Dic3⋊C4—181no—
201C2×C3⋊C8—34no—
202C2×C3⋊C8—94no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—184no—
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—61no—
214C22×C32—61no—
215C22×C32—61no—
216C22×C32—84no—
217C22×C32—84no—
218C22×C32D3×D411yescharacteristic
219C22×C32C2×#6714yes—
220C2×C3×D3—24no—
221C2×C3×D3—24no—
222C2×C3×D3—24no—
223C32×C4—24no—
224C32×C4—24no—
225C32×C4—61no—
226C32×C4—62no—
227C32×C4D3×D411yescharacteristic
228C32×C4C2×#6714yes—
229C32×C4C2×#6714yes—
230C32×C4C2×#6711yescharacteristic
231C3⋊Dic3—41no—
232C3⋊Dic3—121no—
233C3⋊Dic3—124no—
234C3⋊Dic3—244no—
235C3×Dic3—28no—
236C3×Dic3—24no—
237C3×Dic3—124no—
238C2×#41—31no—
239C2×Q16—271no—
240C2×SD16—91no—
241C22⋊Q8—271no—
242#103—271no—
243#102—271no—
244C22⋊C8—271no—
245C33#45211yescharacteristic, complemented, Sylow
246C23×C3—24no—
247C23×C3—21no—
248C23×C3—34no—
249C23×C3—44no—
250C22×D3#63811yescharacteristic
251C3×Q8—38no—
252C3×Q8—64no—
253C3×Q8—181no—
254C3×Q8—181no—
255C3×D4—24no—
256C3×D4—21no—
257C3×D4—28no—
258C3×D4—44no—
259C3×D4—616no—
260C3×D4#62518yes—
261C3×D4#63812yes—
262C2×C3×C4—24no—
263C2×C3×C4—34no—
264C2×C3×C4—616no—
265C2×C3×C4—64no—
266C2×C3×C4—64no—
267C2×C3×C4—64no—
268C2×C3×C4—181no—
269C2×C3×C4C2×D3214yes—
270C2×C3×C4C22×C3⋊D311yescharacteristic
271#67—24no—
272C2×Dic3—24no—
273C2×Dic3—21no—
274C2×Dic3—64no—
275C2×Dic3—184no—
276C2×Dic3—181no—
277C2×Dic3—364no—
278C2×Dic3#63811yescharacteristic
279C4×D3—21no—
280C4×D3—21no—
281Dic6—21no—
282Dic6—64no—
283Dic6—64no—
284Dic6#63812yes—
285C3×C8—181no—
286C3⋊C8—64no—
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—121no—
297C2×C32#91511yescharacteristic
298C2×C32#91611yescharacteristic
299C2×C32#93711yescharacteristic
300C2×C32#95014yes—
301C2×C32#95114yes—
302C2×C32#95314yes—
303C3×D3—44no—
304#41—62no—
305#41—62no—
306C2×Q8—31no—
307C2×Q8—91no—
308C2×D4—61no—
309C2×D4D3×C3⋊D311yescharacteristic
310C22×C4—31no—
311C22×C4—61no—
312Q16—542no—
313SD16—182no—
314C2×C8—91no—
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—64no—
328C22×C3—64no—
329C22×C3—64no—
330C22×C3—84no—
331C22×C3—84no—
332C22×C3D3×#6714yes—
333C22×C3C2×#63811yescharacteristic
334C2×D3—21no—
335C2×D3—21no—
336C2×D3—21no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—64no—
340C3×C4—68no—
341C3×C4—361no—
342C3×C4C2×#62514yes—
343C3×C4D3×#6714yes—
344C3×C4C2×#63811yescharacteristic
345C3×C4C2×#63811yescharacteristic
346Dic3—22no—
347Dic3—21no—
348Dic3—124no—
349Dic3—361no—
350Dic3—364no—
351Dic3—724no—
352C32—24no—
353C32#517311yescharacteristic, complemented
354C32#523914yescomplemented
355C23—21no—
356C23—31no—
357Q8—32no—
358Q8—61no—
359Q8—181no—
360Q8—181no—
361D4—21no—
362D4—64no—
363D4#629012yes—
364C2×C4—31no—
365C2×C4—64no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—181no—
370C2×C4—541no—
371C2×C4C2×D3×C3⋊D311yescharacteristic
372C8—181no—
373C8—541no—
374C2×C3—24no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—42no—
378C2×C3—48no—
379C2×C3—88no—
380C2×C3—124no—
381C2×C3#6392914yes—
382C2×C3#6393514yes—
383C2×C3#6397614yes—
384C2×C3#6415911yescharacteristic
385C2×C3#6416011yescharacteristic
386C2×C3#6416211yescharacteristic
387D3—41no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—61no—
392C22—61no—
393C22—61no—
394C22D3×#63811yescharacteristic
395C4—61no—
396C4—62no—
397C4—361no—
398C4—1081no—
399C4C2×#629011yescharacteristic
400C4D3×#63811yescharacteristic
401C3—24no—
402C3#1059146114yescomplemented
403C3#1059293811yescharacteristic, complemented
404C2—42no—
405C2—121no—
406C2#1172355011yescharacteristic
407C2#1172355811yescharacteristic
408C2#1172362911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×#237C2
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×C3×#3754C2semidirect
3. #11723425C2semidirect
4. C2×#88798C2semidirect
5. #11722202C2semidirect
6. #11722038C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
28. C2×#3754D3semidirect
245. C33#452semidirect
353. C32#5173semidirect
354. C32#5239semidirect
402. C3#10591461semidirect
403. C3#10592938semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C2×C3×#37542
#117234252
C2×#887982
#117222022
#117220382
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929383
#105914613