Finite Group Databasebeta

← #13159451

Subgroups — #13159451

Order 1728.

1080 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89021C211yesmaximal, characteristic
2C32×#897C211yesmaximal, characteristic, complemented
3C2×#88799C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722180C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592746—31nomaximal
9#10590614—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6290—21no—
15C2×Dic3×C3⋊D3—21no—
16#89021—22no—
17#89021—22no—
18C33×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×(C2×D3)⋊C4—21no—
22C32×C4⋊Dic3—21no—
23#88799—22no—
24#88793—22no—
25C2×#3828—31no—
26C2×#3795—34no—
27C32×C22⋊Q8—31no—
28C2×#3756—31no—
29C2×#3755—31no—
30C3×#897D314yescomplemented
31C2×#3719—34no—
32C2×#3715—34no—
33#63657—31no—
34#63637—31no—
35#63637—34no—
36#63637—64no—
37#63636D311yescharacteristic, complemented
38C3×#809—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3—21no—
41C22×C32×D3D411yescharacteristic
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308D412yes—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)D412yes—
49C2×C32×Dic3—21no—
50C2×C32×Dic3D411yescharacteristic
51C32×C4×D3—21no—
52C32×C4×D3—21no—
53C32×Dic6—21no—
54C32×Dic6—21no—
55#6290—24no—
56Dic3×C3⋊D3—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5047—94no—
60#4783—91no—
61C2×C32×Q8—31no—
62C22×C32×C4—31no—
63C2×#638—61no—
64#3828—62no—
65#3828—62no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—61no—
68C2×#634—31no—
69C2×#634—34no—
70C2×#634—64no—
71C2×C3×D12—34no—
72C2×C3×C4×D3C2×D314yes—
73C2×C3×Dic6C2×D314yes—
74C2×#625—64no—
75C2×D3×Dic3—64no—
76#3795—68no—
77#3795—68no—
78C32×C4⋊C4—24no—
79C32×C4⋊C4—61no—
80C32×C4⋊C4C2×D314yes—
81C32×C4⋊C4C2×D311yescharacteristic
82C32×C22⋊C4—61no—
83#3756—62no—
84#3755—62no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—34no—
87C2×C3⋊(C3⋊C8)—64no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×(C2×D3)⋊C4—24no—
90C3×C4⋊Dic3—24no—
91C2×C3×C3⋊C8—34no—
92#3719—68no—
93#3715—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—22no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C2×C32×D3—21no—
99C2×C32×D3—21no—
100C2×C32×D3—21no—
101C33×C4—41no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—22no—
105C32×Dic3—21no—
106C32×Dic3—41no—
107C2×#237—91no—
108C2×#235—94no—
109C3×C22⋊Q8—34no—
110C2×#216—94no—
111C2×#215—94no—
112C2×#214—91no—
113C2×Dic12—91no—
114#897C3⋊D311yescharacteristic, complemented
115C3×#102—91no—
116#830—94no—
117(C2×D3)⋊C8—91no—
118#810—94no—
119#810—91no—
120#810—184no—
121#809—34no—
122C23×C32—31no—
123C22×C3⋊D3—21no—
124C22×C3×D3—64no—
125C22×C3×D3#6714yes—
126C32×Q8—61no—
127C32×Q8—61no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—31no—
132C2×C32×C4—44no—
133C2×C32×C4—61no—
134C2×C32×C4—61no—
135C2×C32×C4—61no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D311yescharacteristic
138#638—64no—
139C2×C3⋊Dic3—61no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—124no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)D1212yes—
144C4×C3⋊D3—64no—
145#634—32no—
146#634—38no—
147#634—68no—
148#634—61no—
149#634—64no—
150#634—124no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3#6714yes—
153C3×D12—38no—
154C3×D12—64no—
155C3×C4×D3—24no—
156C3×C4×D3—24no—
157C3×Dic6—24no—
158C3×Dic6—24no—
159#625—616no—
160D3×Dic3—616no—
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#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C3×C3⋊D3—42no—
171C32×D3—41no—
172C2×C3×Q8—34no—
173C2×C3×D4—91no—
174C22×C3×C4—34no—
175C2×#67—184no—
176C2×#67—181no—
177C22×Dic3—181no—
178#237—182no—
179#237—182no—
180#235—188no—
181#235—188no—
182C2×D12—34no—
183C2×C4×D3—184no—
184C2×C4×D3C2×C3⋊D311yescharacteristic
185C2×Dic6—94no—
186C2×Dic6—91no—
187C2×Dic6—184no—
188C2×Dic6C2×C3⋊D311yescharacteristic
189C2×C3×C8—91no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—64no—
192C3×C4⋊C4D3214yes—
193C3×C4⋊C4C2×C3⋊D311yescharacteristic
194C3×C22⋊C4—64no—
195#216—188no—
196#215—188no—
197#214—182no—
198(C2×D3)⋊C4—21no—
199C4⋊Dic3—21no—
200C2×C3⋊C8—34no—
201C2×C3⋊C8—94no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—184no—
204Dic12—182no—
205C22×C32—24no—
206C22×C32—61no—
207C22×C32—61no—
208C22×C32—61no—
209C22×C32D3×D411yescharacteristic
210C22×C32C2×#6714yes—
211C2×C3⋊D3—22no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3×D3—24no—
215C2×C3×D3—24no—
216C2×C3×D3—24no—
217C2×C3×D3—68no—
218C2×C3×D3—124no—
219C2×C3×D3—124no—
220C32×C4—24no—
221C32×C4—24no—
222C32×C4—41no—
223C32×C4—44no—
224C32×C4—61no—
225C32×C4—84no—
226C32×C4—121no—
227C32×C4C2×D1211yescharacteristic
228C32×C4D3×D414yes—
229C32×C4D3×D414yes—
230C32×C4D3×D411yescharacteristic
231C3⋊Dic3—62no—
232C3⋊Dic3—68no—
233C3⋊Dic3—128no—
234C3×Dic3—24no—
235C3×Dic3—44no—
236C2×#41—271no—
237C2×Q16—271no—
238C2×SD16—271no—
239C22⋊Q8—31no—
240#103—271no—
241#102—91no—
242C22⋊C8—271no—
243C33#45211yescharacteristic, complemented, Sylow
244C23×C3—34no—
245C23×C3—181no—
246C22×D3—64no—
247C22×D3#63811yescharacteristic
248C3×Q8—64no—
249C3×Q8—64no—
250C3×D4—92no—
251C3×D4—181no—
252C2×C3×C4—24no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—34no—
256C2×C3×C4—44no—
257C2×C3×C4—64no—
258C2×C3×C4—64no—
259C2×C3×C4—64no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262#67—184no—
263#67—1816no—
264C2×Dic3—21no—
265C2×Dic3—184no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268C2×Dic3—364no—
269C2×Dic3#63811yescharacteristic
270D12—38no—
271D12—64no—
272C4×D3—21no—
273C4×D3—21no—
274C4×D3—1816no—
275Dic6—21no—
276Dic6—21no—
277Dic6—98no—
278Dic6—92no—
279Dic6—188no—
280Dic6—184no—
281Dic6—181no—
282Dic6—364no—
283C3×C8—181no—
284C3⋊C8—64no—
285C3⋊C8—184no—
286C3⋊C8—181no—
287C3⋊C8—364no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—121no—
292C2×C32#88211yescharacteristic
293C2×C32#91211yescharacteristic
294C2×C32#91511yescharacteristic
295C2×C32#93714yes—
296C2×C32#94214yes—
297C2×C32#95014yes—
298C3⋊D3—42no—
299C3×D3—44no—
300C3×D3—128no—
301#41—542no—
302#41—542no—
303C2×Q8—31no—
304C2×Q8—271no—
305C2×D4—91no—
306C2×D4—541no—
307C22×C4—31no—
308C22×C4—541no—
309Q16—542no—
310SD16—542no—
311C2×C8—91no—
312C2×C8—271no—
313C4⋊C4—61no—
314C4⋊C4D3×C3⋊D311yescharacteristic
315C22⋊C4—61no—
316C22×C3—24no—
317C22×C3—64no—
318C22×C3—64no—
319C22×C3—64no—
320C22×C3—182no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3D3×#6714yes—
324C22×C3C2×#63811yescharacteristic
325C2×D3—21no—
326C2×D3—21no—
327C2×D3—21no—
328C2×D3—68no—
329C2×D3—124no—
330C2×D3—124no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—41no—
334C3×C4—44no—
335C3×C4—64no—
336C3×C4—84no—
337C3×C4—124no—
338C3×C4D3×D1214yes—
339C3×C4#380214yes—
340C3×C4D4×C3⋊D311yescharacteristic
341C3×C4D4×C3⋊D311yescharacteristic
342Dic3—21no—
343Dic3—41no—
344Dic3—182no—
345Dic3—188no—
346Dic3—368no—
347C32—24no—
348C32#478311yescharacteristic, complemented
349C32#504714yescomplemented
350C23—31no—
351C23—181no—
352Q8—61no—
353Q8—61no—
354Q8—272no—
355Q8—541no—
356D4—92no—
357D4—181no—
358D4—544no—
359C2×C4—21no—
360C2×C4—31no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—544no—
365C2×C4—541no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—181no—
368C8—541no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—124no—
373C2×C3—362no—
374C2×C3#6392214yes—
375C2×C3#6393314yes—
376C2×C3#6394614yes—
377C2×C3#6414611yescharacteristic
378C2×C3#6415111yescharacteristic
379C2×C3#6415911yescharacteristic
380D3—41no—
381D3—128no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—182no—
386C22—361no—
387C22—361no—
388C22D3×#63811yescharacteristic
389C4—41no—
390C4—61no—
391C4—121no—
392C4—542no—
393C4C3⋊D3×D1211yescharacteristic
394C4#8902611yescharacteristic
395C3—24no—
396C3#1059061414yescomplemented
397C3#1059274611yescharacteristic, complemented
398C2—121no—
399C2—362no—
400C2#1172354011yescharacteristic
401C2#1172355611yescharacteristic
402C2#1172357911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89021C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C32×#897C2semidirect
3. C2×#88799C2semidirect
4. C2×#88793C2semidirect
5. #11722180C2semidirect
6. C3×#63636C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
30. C3×#897D3semidirect
37. #63636D3semidirect
114. #897C3⋊D3semidirect
243. C33#452semidirect
348. C32#4783semidirect
349. C32#5047semidirect
396. C3#10590614semidirect
397. C3#10592746semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C32×#8972
C2×#887992
C2×#887932
#117221802
C3×#636362
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927463
#105906143