Finite Group Databasebeta

← #13161345

Subgroups — #13161345

Order 1728.

1184 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1184 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×#89022C211yesmaximal, characteristic
2C2×#88862C211yesmaximal, characteristic, complemented
3C2×C3×#3756C211yesmaximal, characteristic, complemented
4C32×#946C211yesmaximal, characteristic, complemented
5#11722039C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592746—31nomaximal
9#10591657—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6291—21no—
15C2×#6289—21no—
16#89022—22no—
17#89022—22no—
18#88862—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3756—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×(C2×D3)⋊C4—21no—
23C32×C4⋊Dic3C2211yescharacteristic
24C32×Dic3⋊C4—21no—
25C2×#3828—31no—
26C2×#3799—34no—
27C32×C22⋊Q8—31no—
28C2×#3756D311yescharacteristic, complemented
29C2×#3755—31no—
30C2×#3755—34no—
31C2×#3755—64no—
32C3×#946D314yescomplemented
33C2×C3×#216—34no—
34#63657—31no—
35#63637—31no—
36#63636—31no—
37#63565—34no—
38#63564—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C32×D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))D412yes—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634D412yes—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6291—24no—
56#6289—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5187—91no—
60#5047—94no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D314yes—
63C2×C32×Q8C2×D311yescharacteristic
64C22×C32×C4—31no—
65C2×#638—61no—
66#3828—62no—
67#3828—62no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—34no—
70C2×C4⋊(C3⋊D3)—64no—
71C2×C4×C3⋊D3—61no—
72C2×#634C2×D311yescharacteristic
73C2×C3×C4×D3C2×D314yes—
74C2×C3×Dic6—34no—
75C2×#626—64no—
76C2×#624—64no—
77#3799—68no—
78#3799—68no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—61no—
81C32×C22⋊C4—61no—
82#3756—22no—
83#3755—62no—
84#3755—68no—
85#3755—616no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#216—68no—
91C3×(C2×D3)⋊C4—24no—
92C3×C4⋊Dic3C2×D314yes—
93C3×Dic3⋊C4—24no—
94C2×C3×C3⋊C8—34no—
95C22×C33C2×D411yescharacteristic
96C2×C3⋊(C3⋊D3)—22no—
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C32×D3—21no—
100C2×C32×D3—21no—
101C2×C32×D3—21no—
102C33×C4—41no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3×C3⋊Dic3—22no—
106C32×Dic3—21no—
107C32×Dic3—41no—
108C2×#239—91no—
109C2×#235—94no—
110C2×C3×Q16—91no—
111C3×C22⋊Q8—34no—
112#946C3⋊D311yescharacteristic, complemented
113C2×#216—34no—
114C2×#215—94no—
115C2×#215—91no—
116C2×#215—184no—
117#835—91no—
118#830—94no—
119#821—91no—
120(C2×D3)⋊C8—91no—
121#810—94no—
122#809—94no—
123C23×C32—31no—
124C22×C3⋊D3—61no—
125C22×C3⋊D3—64no—
126C22×C3⋊D3—124no—
127C22×C3×D3#6714yes—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—24no—
133C32×Q8—44no—
134C2×C32×C4—21no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—31no—
138C2×C32×C4—44no—
139C2×C32×C4—61no—
140C2×C32×C4—61no—
141C2×C32×C4—61no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144#638—64no—
145C2×C3⋊Dic3—21no—
146C4⋊(C3⋊D3)—32no—
147C4⋊(C3⋊D3)—38no—
148C4⋊(C3⋊D3)—68no—
149C4⋊(C3⋊D3)—61no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—124no—
152C4×C3⋊D3—64no—
153#634—21no—
154#634#6712yes—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—64no—
157C2×C3×Dic3#6714yes—
158C3×C4×D3—24no—
159C3×C4×D3—24no—
160C3×Dic6—38no—
161C3×Dic6—64no—
162#626—616no—
163#624—616no—
164C3⋊(C3⋊C8)—21no—
165C3⋊(C3⋊C8)—68no—
166C3⋊(C3⋊C8)—64no—
167C3⋊(C3⋊C8)—61no—
168C3×C3⋊C8—64no—
169#452—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33#13511yescharacteristic
172C2×C33#13711yescharacteristic
173C3⋊(C3⋊D3)—42no—
174C32×D3—41no—
175C2×C3×Q8—24no—
176C2×C3×Q8—91no—
177C2×C3×Q8D3214yes—
178C2×C3×Q8C2×C3⋊D311yescharacteristic
179C22×C3×C4—34no—
180C2×#67—184no—
181#239—182no—
182#239—182no—
183#235—188no—
184#235—188no—
185C2×D12—94no—
186C2×D12—91no—
187C2×D12—184no—
188C2×D12—181no—
189C2×C4×D3—184no—
190C2×C4×D3—181no—
191C2×C4×D3C2×C3⋊D311yescharacteristic
192C2×Dic6—34no—
193C3×Q16—182no—
194C2×C3×C8—91no—
195C3×C4⋊C4—34no—
196C3×C4⋊C4—64no—
197C3×C22⋊C4—64no—
198#216—68no—
199#215—188no—
200#215—182no—
201#215—1816no—
202(C2×D3)⋊C4—21no—
203C4⋊Dic3C2×C3⋊D311yescharacteristic
204Dic3⋊C4—21no—
205C2×C3⋊C8—34no—
206C2×C3⋊C8—94no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—184no—
209C22×C32—24no—
210C22×C32—61no—
211C22×C32—61no—
212C22×C32—61no—
213C22×C32D3×D411yescharacteristic
214C22×C32C2×#6714yes—
215C2×C3⋊D3—62no—
216C2×C3⋊D3—68no—
217C2×C3⋊D3—128no—
218C2×C3⋊D3—121no—
219C2×C3⋊D3—121no—
220C2×C3⋊D3—124no—
221C2×C3⋊D3—124no—
222C2×C3⋊D3—244no—
223C2×C3⋊D3—244no—
224C2×C3×D3—24no—
225C2×C3×D3—24no—
226C2×C3×D3—24no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—48no—
230C32×C4—44no—
231C32×C4—41no—
232C32×C4—61no—
233C32×C4—121no—
234C32×C4D3×D414yes—
235C32×C4D3×D414yes—
236C32×C4D3×D411yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C3⋊Dic3—22no—
239C3×Dic3—24no—
240C3×Dic3—44no—
241C3×Dic3—68no—
242C2×#41—271no—
243C2×Q16—91no—
244C2×SD16—271no—
245C22⋊Q8—31no—
246#103—271no—
247#102—271no—
248C22⋊C8—271no—
249C33#45211yescharacteristic, complemented, Sylow
250C23×C3—34no—
251C22×D3—184no—
252C22×D3—181no—
253C22×D3—364no—
254C22×D3#63811yescharacteristic
255C3×Q8—28no—
256C3×Q8—24no—
257C3×Q8—24no—
258C3×Q8—21no—
259C3×Q8—21no—
260C3×Q8—44no—
261C3×Q8—92no—
262C3×Q8—181no—
263C2×C3×C4—24no—
264C2×C3×C4—21no—
265C2×C3×C4—24no—
266C2×C3×C4—34no—
267C2×C3×C4—44no—
268C2×C3×C4—64no—
269C2×C3×C4—64no—
270C2×C3×C4—64no—
271C2×C3×C4—181no—
272C2×C3×C4C2×D3214yes—
273C2×C3×C4C22×C3⋊D311yescharacteristic
274#67—1816no—
275C2×Dic3—21no—
276C2×Dic3—64no—
277C2×Dic3#63811yescharacteristic
278D12—98no—
279D12—92no—
280D12—188no—
281D12—184no—
282D12—184no—
283D12—181no—
284D12—364no—
285C4×D3—21no—
286C4×D3—21no—
287C4×D3—184no—
288C4×D3—1816no—
289Dic6—38no—
290Dic6—64no—
291C3×C8—181no—
292C3⋊C8—64no—
293C3⋊C8—188no—
294C3⋊C8—184no—
295C3⋊C8—181no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—24no—
299C2×C32—121no—
300C2×C32#91811yescharacteristic
301C2×C32#91911yescharacteristic
302C2×C32#93714yes—
303C2×C32#93711yescharacteristic
304C2×C32#94214yes—
305C2×C32#95014yes—
306C3⋊D3—122no—
307C3⋊D3—128no—
308C3⋊D3—248no—
309C3×D3—44no—
310#41—542no—
311#41—542no—
312C2×Q8—91no—
313C2×Q8D3×C3⋊D311yescharacteristic
314C2×D4—271no—
315C2×D4—541no—
316C22×C4—31no—
317C22×C4—541no—
318Q16—182no—
319SD16—542no—
320C2×C8—91no—
321C2×C8—271no—
322C4⋊C4—31no—
323C4⋊C4—61no—
324C22⋊C4—61no—
325C22×C3—24no—
326C22×C3—64no—
327C22×C3—64no—
328C22×C3—64no—
329C22×C3D3×#6714yes—
330C22×C3C2×#63811yescharacteristic
331C2×D3—21no—
332C2×D3—21no—
333C2×D3—21no—
334C2×D3—182no—
335C2×D3—188no—
336C2×D3—368no—
337C2×D3—361no—
338C2×D3—361no—
339C2×D3—364no—
340C2×D3—364no—
341C2×D3—724no—
342C2×D3—724no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—48no—
346C3×C4—41no—
347C3×C4—44no—
348C3×C4—64no—
349C3×C4—124no—
350C3×C4—182no—
351C3×C4D3×#6714yes—
352C3×C4#381114yes—
353C3×C4D4×C3⋊D311yescharacteristic
354C3×C4D4×C3⋊D311yescharacteristic
355Dic3—21no—
356Dic3—41no—
357Dic3—68no—
358C32—24no—
359C32#504714yescomplemented
360C32#518711yescharacteristic, complemented
361C23—31no—
362C23—541no—
363Q8—21no—
364Q8—21no—
365Q8—92no—
366Q8—181no—
367D4—272no—
368D4—544no—
369D4—541no—
370C2×C4—21no—
371C2×C4—31no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—181no—
376C2×C4—544no—
377C2×C4C2×D3×C3⋊D311yescharacteristic
378C8—181no—
379C8—541no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—124no—
384C2×C3#6394814yes—
385C2×C3#6394914yes—
386C2×C3#6398314yes—
387C2×C3#6414611yescharacteristic
388C2×C3#6415111yescharacteristic
389C2×C3#6415911yescharacteristic
390D3—41no—
391D3—368no—
392D3—362no—
393D3—728no—
394C22—61no—
395C22—61no—
396C22—61no—
397C22—542no—
398C22—1081no—
399C22—1081no—
400C22D3×#63811yescharacteristic
401C4—41no—
402C4—61no—
403C4—121no—
404C4—182no—
405C4C3⋊D3×#6711yescharacteristic
406C4#8903911yescharacteristic
407C3—24no—
408C3#1059165714yescomplemented
409C3#1059274611yescharacteristic, complemented
410C2—121no—
411C2—1082no—
412C2#1172358111yescharacteristic
413C2#1172358211yescharacteristic
414C2#1172364311yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#89022C2
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×#88862C2semidirect
3. C2×C3×#3756C2semidirect
4. C32×#946C2semidirect
5. #11722039C2semidirect
6. #11722035C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
28. C2×#3756D3semidirect
32. C3×#946D3semidirect
112. #946C3⋊D3semidirect
249. C33#452semidirect
359. C32#5047semidirect
360. C32#5187semidirect
408. C3#10591657semidirect
409. C3#10592746semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
C2×#888622
C2×C3×#37562
C32×#9462
#117220392
#117220352
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927463
#105916573