Finite Group Databasebeta

← #13161342

Subgroups — #13161342

Order 1728.

1152 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1152 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×#89014C211yesmaximal, characteristic
2C2×#88863C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C2×C32×#215C211yesmaximal, characteristic, complemented
5#11722042C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592872—31nomaximal
9#10591655—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×D3×C3⋊Dic3—21no—
16#89014—22no—
17#89014—22no—
18#88863—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3752—21no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#3750—21no—
23C32×#215—22no—
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×#3830—31no—
26C2×#3795—34no—
27C2×C32×SD16—31no—
28#64155D311yescharacteristic, complemented
29C2×#3756—31no—
30C2×#3756—34no—
31C2×#3756—64no—
32C3×#946—34no—
33C2×C3×#215D314yescomplemented
34#63659—31no—
35#63647—31no—
36#63643—31no—
37#63567—34no—
38#63562—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C22×C32×D3—21no—
42C33×Q8—21no—
43C33×Q8—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47#6308—21no—
48#6308D412yes—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C32×D12—21no—
54C32×D12D412yes—
55#6290—24no—
56D3×C3⋊Dic3—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5173—94no—
60#5047—91no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D314yes—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×D4—31no—
65C2×#638—61no—
66C22×C3⋊Dic3—61no—
67#3830—62no—
68#3830—62no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634—31no—
71C2×#634—34no—
72C2×#634—64no—
73C2×C3×D12C2×D314yes—
74C2×C3×C4×D3—34no—
75C2×#625—64no—
76C2×D3×Dic3—64no—
77#3795—68no—
78#3795—68no—
79C32×SD16—62no—
80C2×C32×C8—31no—
81#3756—62no—
82#3756—68no—
83#3756—616no—
84#3752—21no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86#3750—21no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90C3×#215—28no—
91C3×(C2×D3)⋊C4—64no—
92C3×C4⋊Dic3—34no—
93C3×Dic3⋊C4—64no—
94C2×C3×C3⋊C8C2×D314yes—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3—21no—
99C2×C32×D3—22no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102C33×C4—41no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)—22no—
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—41no—
108C2×#237—94no—
109C2×#235—91no—
110C2×C3×SD16—34no—
111C3×C22⋊Q8—91no—
112#946—34no—
113C2×#216—94no—
114C2×#216—91no—
115C2×#216—184no—
116C2×#215C3⋊D311yescharacteristic, complemented
117#832—94no—
118#830—91no—
119(C2×D3)⋊C8—94no—
120#816—94no—
121#810—91no—
122#809—91no—
123C23×C32—61no—
124C22×C3⋊D3#6711yescharacteristic
125C22×C3×D3—24no—
126C22×C3×D3—34no—
127C32×Q8—28no—
128C32×Q8—24no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—24no—
132C32×Q8—44no—
133C32×D4—32no—
134C32×D4—61no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—44no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141#638—64no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—64no—
146C2×C3⋊Dic3—124no—
147C2×C3⋊Dic3#6711yescharacteristic
148C4×C3⋊D3—21no—
149C4×C3⋊D3—21no—
150#634—32no—
151#634—38no—
152#634—68no—
153#634—61no—
154#634—64no—
155#634—124no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—64no—
158C3×D12—24no—
159C3×D12#6718yes—
160C3×C4×D3—64no—
161C3×C4×D3—64no—
162#625—616no—
163D3×Dic3—616no—
164C32×C8—61no—
165C3⋊(C3⋊C8)—68no—
166C3⋊(C3⋊C8)—64no—
167C3⋊(C3⋊C8)—61no—
168C3×C3⋊C8—24no—
169#452—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33#13511yescharacteristic
172C2×C33#13711yescharacteristic
173C3×C3⋊D3—41no—
174C32×D3—42no—
175C2×C3×Q8—24no—
176C2×C3×Q8D3214yes—
177C2×C3×Q8C2×C3⋊D311yescharacteristic
178C2×C3×D4—34no—
179C22×C3×C4—91no—
180C2×#67—181no—
181C2×#67—184no—
182C22×Dic3—184no—
183#237—188no—
184#237—188no—
185#235—182no—
186#235—182no—
187C2×D12C2×C3⋊D311yescharacteristic
188C2×C4×D3—34no—
189C2×C4×D3—181no—
190C2×Dic6—94no—
191C2×Dic6—91no—
192C2×Dic6—184no—
193C3×SD16—68no—
194C2×C3×C8—34no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4—181no—
197C3×C22⋊C4—181no—
198#216—188no—
199#216—182no—
200#216—1816no—
201#215—22no—
202(C2×D3)⋊C4—64no—
203C4⋊Dic3—34no—
204Dic3⋊C4—64no—
205C2×C3⋊C8—94no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—184no—
208C2×C3⋊C8C2×C3⋊D311yescharacteristic
209C22×C32—24no—
210C22×C32—62no—
211C22×C32—121no—
212C22×C32—121no—
213C22×C32D3×D414yes—
214C22×C32C2×#6711yescharacteristic
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3×D3—28no—
219C2×C3×D3—44no—
220C2×C3×D3—44no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C2×C3×D3—64no—
224C32×C4—24no—
225C32×C4—24no—
226C32×C4—48no—
227C32×C4—44no—
228C32×C4—41no—
229C32×C4D3×D414yes—
230C32×C4D3×D411yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4C2×#6714yes—
233C3⋊Dic3—21no—
234C3⋊Dic3—41no—
235C3⋊Dic3—62no—
236C3⋊Dic3—68no—
237C3⋊Dic3—128no—
238C3×Dic3—64no—
239C3×Dic3—124no—
240C2×#41—271no—
241C2×Q16—271no—
242C2×SD16—31no—
243C22⋊Q8—91no—
244#103—271no—
245#102—271no—
246C22⋊C8—271no—
247C33#45211yescharacteristic, complemented, Sylow
248C23×C3—64no—
249C23×C3—91no—
250C22×D3—21no—
251C22×D3—34no—
252C3×Q8—28no—
253C3×Q8—24no—
254C3×Q8—24no—
255C3×Q8—21no—
256C3×Q8—21no—
257C3×Q8—44no—
258C3×D4—38no—
259C3×D4—64no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—44no—
264C2×C3×C4—91no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3214yes—
269C2×C3×C4C22×C3⋊D311yescharacteristic
270#67—184no—
271#67—1816no—
272C2×Dic3—34no—
273C2×Dic3—64no—
274C2×Dic3—1816no—
275C2×Dic3—184no—
276C2×Dic3—181no—
277C2×Dic3—364no—
278D12—21no—
279D12#63812yes—
280C4×D3—64no—
281C4×D3—64no—
282C4×D3—184no—
283Dic6—98no—
284Dic6—92no—
285Dic6—188no—
286Dic6—184no—
287Dic6—181no—
288Dic6—364no—
289C3×C8—64no—
290C3⋊C8—21no—
291C3⋊C8—188no—
292C3⋊C8—184no—
293C3⋊C8—181no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—122no—
298C2×C32#91514yes—
299C2×C32#91614yes—
300C2×C32#93714yes—
301C2×C32#93711yescharacteristic
302C2×C32#94211yescharacteristic
303C2×C32#95011yescharacteristic
304C3⋊D3—41no—
305C3×D3—48no—
306C3×D3—124no—
307#41—542no—
308#41—542no—
309C2×Q8—271no—
310C2×Q8D3×C3⋊D311yescharacteristic
311C2×D4—31no—
312C2×D4—541no—
313C22×C4—91no—
314C22×C4—541no—
315Q16—542no—
316SD16—62no—
317C2×C8—31no—
318C2×C8—271no—
319C4⋊C4—91no—
320C4⋊C4—181no—
321C22⋊C4—181no—
322C22×C3—24no—
323C22×C3—68no—
324C22×C3—124no—
325C22×C3—124no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—181no—
329C22×C3D3×#6714yes—
330C22×C3D4×C3⋊D311yescharacteristic
331C2×D3—22no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—64no—
335C2×D3—64no—
336C2×D3—64no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—48no—
340C3×C4—41no—
341C3×C4—44no—
342C3×C4—181no—
343C3×C4—361no—
344C3×C4D3×#6714yes—
345C3×C4#381114yes—
346C3×C4D4×C3⋊D311yescharacteristic
347C3×C4C2×#63811yescharacteristic
348Dic3—64no—
349Dic3—124no—
350Dic3—182no—
351Dic3—188no—
352Dic3—368no—
353C32—24no—
354C32#504711yescharacteristic, complemented
355C32#517314yescomplemented
356C23—61no—
357C23—91no—
358Q8—21no—
359Q8—21no—
360Q8—272no—
361Q8—541no—
362D4—32no—
363D4—61no—
364D4—544no—
365C2×C4—21no—
366C2×C4—91no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—544no—
371C2×C4—541no—
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—61no—
374C8—541no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—128no—
379C2×C3—361no—
380C2×C3#6394614yes—
381C2×C3#6394714yes—
382C2×C3#6398314yes—
383C2×C3#6412411yescharacteristic
384C2×C3#6412511yescharacteristic
385C2×C3#6414611yescharacteristic
386D3—42no—
387D3—124no—
388C22—62no—
389C22—121no—
390C22—121no—
391C22—181no—
392C22—181no—
393C22—181no—
394C22C3⋊D3×#6711yescharacteristic
395C4—41no—
396C4—181no—
397C4—361no—
398C4—542no—
399C4D3×#63811yescharacteristic
400C4#8903911yescharacteristic
401C3—24no—
402C3#1059165514yescomplemented
403C3#1059287211yescharacteristic, complemented
404C2—122no—
405C2—361no—
406C2#1172358311yescharacteristic
407C2#1172358411yescharacteristic
408C2#1172364311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89014C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#452
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
2. C2×#88863C2semidirect
3. C3×#64155C2semidirect
4. C2×C32×#215C2semidirect
5. #11722042C2semidirect
6. #11722031C2semidirect
7. #11722023C2semidirect
28. #64155D3semidirect
33. C2×C3×#215D3semidirect
116. C2×#215C3⋊D3semidirect
247. C33#452semidirect
354. C32#5047semidirect
355. C32#5173semidirect
402. C3#10591655semidirect
403. C3#10592872semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C2×#888632
C3×#641552
C2×C32×#2152
#117220422
#117220312
#117220232
#105928723
#105916553