Finite Group Databasebeta

← #13160972

Subgroups — #13160972

Order 1728.

1184 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1184 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×#88861C211yesmaximal, characteristic, complemented
3#11723431C211yesmaximal, characteristic, complemented
4C2×#88800C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592482—31nomaximal
9#10591463—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#638—21no—
13C22×C3×C3⋊Dic3—21no—
14C3×#3830—22no—
15C3×#3830—22no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18#88861—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22Dic3⋊(C3⋊Dic3)—21no—
23#88803—21no—
24#88800—22no—
25C2×#3830D311yescharacteristic
26C2×C3×#237—34no—
27C2×#3754—31no—
28C2×#3754—34no—
29C2×#3754—64no—
30C2×#3746—31no—
31#64106—31no—
32#63857—34no—
33C2×#3719—34no—
34C32×#102—31no—
35#63647—31no—
36#63637—31no—
37C3×#832D314yescomplemented
38#63565—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—
46#6308—21no—
47#6308—21no—
48C3×#638—24no—
49C2×C3×C3⋊Dic3—24no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C3×#634—21no—
55C3×#634D412yes—
56C2×C32×Dic3—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5239—91no—
60#4783—94no—
61C2×C32×D4—24no—
62C2×C32×D4C2×D314yes—
63C2×C32×D4C2×D311yescharacteristic
64C2×#638—21no—
65C22×C3⋊Dic3—21no—
66#3830—22no—
67#3830—22no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634—31no—
70C2×#634—34no—
71C2×#634—64no—
72C2×#634C2×D311yescharacteristic
73C2×C3×#67—64no—
74C22×C3×Dic3—64no—
75C3×#237—68no—
76C3×#237—68no—
77C2×C3×C4×D3—34no—
78C2×C3×Dic6—34no—
79C2×C32×C8—31no—
80C32×C4⋊C4—31no—
81#3754—616no—
82#3754—68no—
83#3754—62no—
84#3752—61no—
85C4⋊(C3⋊Dic3)—61no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89#3746—62no—
90C3×C4⋊Dic3C2×D314yes—
91C2×C3×C3⋊C8C2×D314yes—
92Dic3⋊Dic3—64no—
93(C2×D3)⋊Dic3—64no—
94#3719—68no—
95C22×C33—22no—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33C2×D411yescharacteristic
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3—22no—
106C3×C3⋊Dic3—21no—
107C32×Dic3—41no—
108C2×C3×#41—91no—
109C2×#237—34no—
110C2×#216—91no—
111C2×#214—94no—
112C2×#214—91no—
113C2×#214—184no—
114#924—91no—
115C2×Dic12—94no—
116#897—94no—
117C3×#102—34no—
118#835—91no—
119#832C3⋊D311yescharacteristic, complemented
120#830—91no—
121(C2×D3)⋊C8—94no—
122#810—94no—
123C23×C32—21no—
124C23×C32—24no—
125C23×C32—44no—
126C22×C3⋊D3#6711yescharacteristic
127C22×C3×D3—34no—
128C32×D4—21no—
129C32×D4—28no—
130C32×D4—24no—
131C32×D4—44no—
132C32×D4D1218yes—
133C32×D4#6712yes—
134C2×C32×C4—24no—
135C2×C32×C4—61no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D311yescharacteristic
138#638—24no—
139C2×C3⋊Dic3—24no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144C2×C3⋊Dic3#6711yescharacteristic
145C4×C3⋊D3—21no—
146C4×C3⋊D3—21no—
147#634—21no—
148#634—68no—
149#634—64no—
150#634—61no—
151#634—61no—
152#634—64no—
153#634—124no—
154#634#6712yes—
155C3×#67—616no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—616no—
159C2×C3×Dic3—64no—
160C3×C4×D3—64no—
161C3×C4×D3—64no—
162C3×Dic6—38no—
163C3×Dic6—64no—
164C32×C8—61no—
165C3⋊(C3⋊C8)—68no—
166C3⋊(C3⋊C8)—64no—
167C3⋊(C3⋊C8)—61no—
168C3×C3⋊C8—24no—
169#452—271noSylow
170C2×C33—42no—
171C2×C33#12011yescharacteristic
172C2×C33#13511yescharacteristic
173C2×C33#13711yescharacteristic
174C3×C3⋊D3—41no—
175C3×#41—182no—
176C3×#41—182no—
177C2×C3×Q8—91no—
178C2×C3×D4—24no—
179C2×C3×D4—181no—
180C2×C3×D4D3214yes—
181C2×C3×D4C2×C3⋊D311yescharacteristic
182C22×C3×C4—91no—
183C22×C3×C4—181no—
184C2×#67—64no—
185C22×Dic3—64no—
186#237—68no—
187#237—68no—
188C2×C4×D3—34no—
189C2×Dic6—34no—
190C2×Dic6—94no—
191C2×Dic6—91no—
192C2×Dic6—184no—
193C2×C3×C8—34no—
194C3×C4⋊C4—34no—
195#217—181no—
196#216—182no—
197#214—1816no—
198#214—188no—
199#214—182no—
200(C2×D3)⋊C4—184no—
201C4⋊Dic3—184no—
202C4⋊Dic3C2×C3⋊D311yescharacteristic
203Dic3⋊C4—181no—
204C2×C3⋊C8—94no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—184no—
207C2×C3⋊C8C2×C3⋊D311yescharacteristic
208Dic12—188no—
209C22×C32—22no—
210C22×C32—24no—
211C22×C32—28no—
212C22×C32—48no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—44no—
216C22×C32—44no—
217C22×C32—84no—
218C22×C32—84no—
219C22×C32D3×D414yes—
220C22×C32C2×#6711yescharacteristic
221C2×C3⋊D3—21no—
222C2×C3⋊D3—21no—
223C2×C3⋊D3—21no—
224C2×C3×D3—64no—
225C2×C3×D3—64no—
226C2×C3×D3—64no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—121no—
230C32×C4C2×D1214yes—
231C32×C4D3×D414yes—
232C32×C4C2×#6711yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C3⋊Dic3—22no—
235C3⋊Dic3—21no—
236C3⋊Dic3—128no—
237C3⋊Dic3—124no—
238C3⋊Dic3—121no—
239C3×Dic3—44no—
240C3×Dic3—68no—
241C3×Dic3—64no—
242C2×#41—91no—
243C2×Q16—271no—
244C2×SD16—271no—
245C22⋊Q8—271no—
246#103—271no—
247#102—31no—
248C22⋊C8—271no—
249C33#45211yescharacteristic, complemented, Sylow
250C23×C3—24no—
251C23×C3—21no—
252C23×C3—44no—
253C23×C3—91no—
254C22×D3—34no—
255C3×Q8—92no—
256C3×Q8—181no—
257C3×D4—24no—
258C3×D4—21no—
259C3×D4—28no—
260C3×D4—44no—
261C3×D4—184no—
262C3×D4#62618yes—
263C3×D4C4⋊(C3⋊D3)12yes—
264C2×C3×C4—24no—
265C2×C3×C4—64no—
266C2×C3×C4—91no—
267C2×C3×C4—184no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C2×D3214yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273#67—616no—
274C2×Dic3—21no—
275C2×Dic3—34no—
276C2×Dic3—616no—
277C2×Dic3—64no—
278C2×Dic3—184no—
279C2×Dic3—181no—
280C2×Dic3—364no—
281C4×D3—64no—
282C4×D3—64no—
283Dic6—38no—
284Dic6—64no—
285Dic6—188no—
286Dic6—184no—
287Dic6—184no—
288Dic6—181no—
289Dic6—181no—
290Dic6—364no—
291C3×C8—64no—
292C3⋊C8—21no—
293C3⋊C8—188no—
294C3⋊C8—184no—
295C3⋊C8—181no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—24no—
299C2×C32—42no—
300C2×C32—48no—
301C2×C32—88no—
302C2×C32#88214yes—
303C2×C32#91214yes—
304C2×C32#91514yes—
305C2×C32#95011yescharacteristic
306C2×C32#95111yescharacteristic
307C2×C32#95311yescharacteristic
308C3⋊D3—41no—
309C3×D3—124no—
310#41—182no—
311#41—182no—
312C2×Q8—91no—
313C2×Q8—271no—
314C2×D4—181no—
315C2×D4D3×C3⋊D311yescharacteristic
316C22×C4—91no—
317C22×C4—181no—
318Q16—542no—
319SD16—542no—
320C2×C8—31no—
321C2×C8—271no—
322C4⋊C4—31no—
323C4⋊C4—541no—
324C22⋊C4—541no—
325C22×C3—22no—
326C22×C3—28no—
327C22×C3—24no—
328C22×C3—48no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—44no—
332C22×C3—44no—
333C22×C3—84no—
334C22×C3—84no—
335C22×C3—181no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3D3×#6714yes—
339C22×C3D4×C3⋊D311yescharacteristic
340C2×D3—64no—
341C2×D3—64no—
342C2×D3—64no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—124no—
346C3×C4—181no—
347C3×C4—182no—
348C3×C4C2×#62614yes—
349C3×C4D3×#6714yes—
350C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
351C3×C4D4×C3⋊D311yescharacteristic
352Dic3—41no—
353Dic3—68no—
354Dic3—64no—
355Dic3—368no—
356Dic3—361no—
357Dic3—364no—
358C32—24no—
359C32#478314yescomplemented
360C32#523911yescharacteristic, complemented
361C23—21no—
362C23—91no—
363Q8—92no—
364Q8—181no—
365Q8—541no—
366Q8—541no—
367D4—21no—
368D4—184no—
369D4#629212yes—
370C2×C4—61no—
371C2×C4—91no—
372C2×C4—184no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4—541no—
377C2×C4C2×D3×C3⋊D311yescharacteristic
378C8—61no—
379C8—541no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—42no—
384C2×C3—48no—
385C2×C3—88no—
386C2×C3—361no—
387C2×C3#6392914yes—
388C2×C3#6393314yes—
389C2×C3#6397714yes—
390C2×C3#6409111yescharacteristic
391C2×C3#6412111yescharacteristic
392C2×C3#6412411yescharacteristic
393D3—124no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—181no—
398C22—181no—
399C22—181no—
400C22C3⋊D3×#6711yescharacteristic
401C4—121no—
402C4—181no—
403C4—182no—
404C4—1081no—
405C4C2×#629211yescharacteristic
406C4C3⋊D3×#6711yescharacteristic
407C3—24no—
408C3#1059146314yescomplemented
409C3#1059248211yescharacteristic, complemented
410C2—42no—
411C2—361no—
412C2#1172355211yescharacteristic
413C2#1172355911yescharacteristic
414C2#1172363611yescharacteristic
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×#88861C2semidirect
3. #11723431C2semidirect
4. C2×#88800C2semidirect
5. C32×#832C2semidirect
6. #11722039C2semidirect
7. #11722023C2semidirect
37. C3×#832D3semidirect
119. #832C3⋊D3semidirect
249. C33#452semidirect
359. C32#4783semidirect
360. C32#5239semidirect
408. C3#10591463semidirect
409. C3#10592482semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#888612
#117234312
C2×#888002
C32×#8322
#117220392
#117220232
#105924823
#105914633