Finite Group Databasebeta

← #13160102

Subgroups — #13160102

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×#89018C211yesmaximal, characteristic
2C2×#88851C211yesmaximal, characteristic, complemented
3#11723431C211yesmaximal, characteristic, complemented
4C2×#88799C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592507—31nomaximal
9#10590970—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6292—21no—
15C2×#6289—21no—
16#89018—22no—
17#89018—22no—
18C2×C33×C8C2211yescharacteristic
19#88851—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22Dic3⋊(C3⋊Dic3)—21no—
23#88803—21no—
24#88799—22no—
25C2×#3832—31no—
26C2×#3799—34no—
27C2×#3756—31no—
28C2×#3744—31no—
29C2×#3744—34no—
30C2×#3744—64no—
31#64106—31no—
32#63857—34no—
33C2×#3719—34no—
34C32×#103—31no—
35#63647D311yescharacteristic, complemented
36#63636—31no—
37C3×(C2×D3)⋊C8—34no—
38C3×#816D314yescomplemented
39#63564—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))D412yes—
46#6308—21no—
47#6308—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6292—24no—
56#6289—24no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#4808—94no—
60#4733—91no—
61C2×C32×Q8—31no—
62#3832—62no—
63#3832—62no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—34no—
66C2×C4⋊(C3⋊D3)—61no—
67C2×C4⋊(C3⋊D3)—64no—
68C2×C4×C3⋊D3—61no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634—31no—
71C2×#634—34no—
72C2×#634—64no—
73C2×C3×C4×D3—34no—
74C2×C3×Dic6C2×D314yes—
75C2×#626—64no—
76C2×#624—64no—
77#3799—68no—
78#3799—68no—
79C2×C32×C8—24no—
80C2×C32×C8C2×D314yes—
81C2×C32×C8C2×D311yescharacteristic
82C32×C4⋊C4—31no—
83#3756—62no—
84#3752—61no—
85C4⋊(C3⋊Dic3)—61no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87#3744—62no—
88#3744—68no—
89#3744—616no—
90C3×C4⋊Dic3C2×D314yes—
91C2×C3×C3⋊C8—34no—
92Dic3⋊Dic3—64no—
93(C2×D3)⋊Dic3—64no—
94#3719—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3⋊(C3⋊D3)—22no—
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
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—21no—
106C32×Dic3—22no—
107C32×Dic3—41no—
108C2×#239—94no—
109C2×#235—91no—
110C2×#216—94no—
111#924—91no—
112C2×Dic12—91no—
113C2×#204—94no—
114C2×#204—91no—
115C2×#204—184no—
116#897—94no—
117C3×#103—34no—
118C3×C22⋊C8—91no—
119#821—91no—
120(C2×D3)⋊C8—34no—
121#816C3⋊D311yescharacteristic, complemented
122#809—94no—
123C22×C3⋊D3—61no—
124C22×C3⋊D3—64no—
125C22×C3⋊D3—124no—
126C22×C3⋊D3D1211yescharacteristic
127C22×C3×D3—34no—
128C32×Q8—32no—
129C32×Q8—61no—
130C2×C32×C4—24no—
131C2×C32×C4—61no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—124no—
138C2×C3⋊Dic3D1211yescharacteristic
139C4⋊(C3⋊D3)—32no—
140C4⋊(C3⋊D3)—38no—
141C4⋊(C3⋊D3)—68no—
142C4⋊(C3⋊D3)—64no—
143C4⋊(C3⋊D3)—61no—
144C4⋊(C3⋊D3)—64no—
145C4⋊(C3⋊D3)—124no—
146C4×C3⋊D3—21no—
147C4×C3⋊D3—21no—
148C4×C3⋊D3—64no—
149#634—68no—
150#634—64no—
151#634—61no—
152#634—61no—
153#634—64no—
154#634—124no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—34no—
158C3×C4×D3—64no—
159C3×C4×D3—64no—
160C3×Dic6—24no—
161C3×Dic6D1218yes—
162#626—616no—
163#624—616no—
164C32×C8—28no—
165C32×C8—24no—
166C32×C8—21no—
167C3⋊(C3⋊C8)—21no—
168C3×C3⋊C8—64no—
169#452—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33#13511yescharacteristic
172C2×C33#13711yescharacteristic
173C3⋊(C3⋊D3)—42no—
174C3×C3⋊D3—41no—
175C2×C3×Q8—34no—
176C22×C3×C4—91no—
177C2×#67—181no—
178#239—188no—
179#239—188no—
180#235—182no—
181#235—182no—
182C2×D12—94no—
183C2×D12—91no—
184C2×D12—184no—
185C2×D12—184no—
186C2×C4×D3—34no—
187C2×C4×D3—181no—
188C2×C4×D3—184no—
189C2×Dic6—94no—
190C2×Dic6—91no—
191C2×Dic6—184no—
192C2×Dic6C2×C3⋊D311yescharacteristic
193C2×C3×C8—24no—
194C2×C3×C8—91no—
195C2×C3×C8D3214yes—
196C2×C3×C8C2×C3⋊D311yescharacteristic
197C3×C4⋊C4—34no—
198#217—181no—
199#216—188no—
200(C2×D3)⋊C4—184no—
201C4⋊Dic3—184no—
202C4⋊Dic3C2×C3⋊D311yescharacteristic
203Dic3⋊C4—181no—
204C2×C3⋊C8—34no—
205Dic12—182no—
206#204—188no—
207#204—182no—
208#204—1816no—
209C22×C32—24no—
210C22×C32C2×D1211yescharacteristic
211C22×C32D3×D414yes—
212C2×C3⋊D3—21no—
213C2×C3⋊D3—21no—
214C2×C3⋊D3—21no—
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—64no—
225C2×C3×D3—64no—
226C2×C3×D3—64no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—62no—
230C32×C4—121no—
231C32×C4C2×D1214yes—
232C32×C4D3×D414yes—
233C32×C4D3×D411yescharacteristic
234C32×C4D3×D411yescharacteristic
235C3⋊Dic3—21no—
236C3⋊Dic3—128no—
237C3⋊Dic3—124no—
238C3⋊Dic3—121no—
239C3×Dic3—28no—
240C3×Dic3—44no—
241C3×Dic3—64no—
242C2×#41—271no—
243C2×Q16—271no—
244C2×SD16—271no—
245C22⋊Q8—271no—
246#103—31no—
247#102—271no—
248C22⋊C8—91no—
249C33#45211yescharacteristic, complemented, Sylow
250C23×C3—91no—
251C22×D3—34no—
252C22×D3—184no—
253C22×D3—181no—
254C22×D3—364no—
255C3×Q8—38no—
256C3×Q8—64no—
257C2×C3×C4—24no—
258C2×C3×C4—64no—
259C2×C3×C4—64no—
260C2×C3×C4—91no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4C2×D3214yes—
264C2×C3×C4C22×C3⋊D311yescharacteristic
265#67—184no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—34no—
269C2×Dic3—184no—
270C2×Dic3—181no—
271C2×Dic3—364no—
272D12—98no—
273D12—92no—
274D12—188no—
275D12—1816no—
276D12—184no—
277D12—181no—
278D12—364no—
279C4×D3—64no—
280C4×D3—64no—
281C4×D3—184no—
282C4×D3—1816no—
283Dic6—21no—
284Dic6—188no—
285Dic6—184no—
286Dic6—184no—
287Dic6—181no—
288Dic6—181no—
289Dic6—364no—
290Dic6C4⋊(C3⋊D3)12yes—
291C3×C8—28no—
292C3×C8—24no—
293C3×C8—21no—
294C3×C8—181no—
295C3⋊C8—64no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—24no—
299C2×C32#88214yes—
300C2×C32#88211yescharacteristic
301C2×C32#90411yescharacteristic
302C2×C32#90911yescharacteristic
303C2×C32#91614yes—
304C2×C32#91814yes—
305C3⋊D3—41no—
306C3⋊D3—122no—
307C3⋊D3—128no—
308C3⋊D3—248no—
309C3×D3—124no—
310#41—542no—
311#41—542no—
312C2×Q8—31no—
313C2×Q8—271no—
314C2×D4—271no—
315C2×D4—541no—
316C22×C4—91no—
317C22×C4—541no—
318Q16—542no—
319SD16—542no—
320C2×C8—91no—
321C2×C8D3×C3⋊D311yescharacteristic
322C4⋊C4—31no—
323C4⋊C4—541no—
324C22⋊C4—541no—
325C22×C3—24no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—181no—
329C22×C3D3×D1214yes—
330C22×C3D4×C3⋊D311yescharacteristic
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—64no—
334C2×D3—182no—
335C2×D3—188no—
336C2×D3—368no—
337C2×D3—364no—
338C2×D3—364no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—724no—
342C2×D3—724no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—68no—
346C3×C4—124no—
347C3×C4—181no—
348C3×C4D3×D1214yes—
349C3×C4#381114yes—
350C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
351C3×C4D4×C3⋊D311yescharacteristic
352Dic3—22no—
353Dic3—41no—
354Dic3—64no—
355Dic3—368no—
356Dic3—361no—
357Dic3—364no—
358C32—24no—
359C32#473311yescharacteristic, complemented
360C32#480814yescomplemented
361C23—91no—
362C23—541no—
363Q8—32no—
364Q8—61no—
365Q8—541no—
366Q8—541no—
367D4—272no—
368D4—544no—
369D4—541no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—91no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—544no—
376C2×C4—541no—
377C2×C4C2×D3×C3⋊D311yescharacteristic
378C8—21no—
379C8—181no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—361no—
384C2×C3#6380114yes—
385C2×C3#6380614yes—
386C2×C3#6392314yes—
387C2×C3#6409111yescharacteristic
388C2×C3#6412511yescharacteristic
389C2×C3#6412711yescharacteristic
390D3—124no—
391D3—368no—
392D3—362no—
393D3—728no—
394C22—181no—
395C22—181no—
396C22—181no—
397C22—542no—
398C22—1081no—
399C22—1081no—
400C22C3⋊D3×D1211yescharacteristic
401C4—62no—
402C4—121no—
403C4—181no—
404C4—1081no—
405C4D3×C4⋊(C3⋊D3)11yescharacteristic
406C4#8903911yescharacteristic
407C3—24no—
408C3#1059097014yescomplemented
409C3#1059250711yescharacteristic, complemented
410C2—361no—
411C2—1082no—
412C2#1172332911yescharacteristic
413C2#1172333811yescharacteristic
414C2#1172354211yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89018C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88851C2semidirect
3. #11723431C2semidirect
4. C2×#88799C2semidirect
5. C3×#63647C2semidirect
6. C32×#816C2semidirect
7. #11722035C2semidirect
35. #63647D3semidirect
38. C3×#816D3semidirect
121. #816C3⋊D3semidirect
249. C33#452semidirect
359. C32#4733semidirect
360. C32#4808semidirect
408. C3#10590970semidirect
409. C3#10592507semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C2×#888512
#117234312
C2×#887992
C3×#636472
C32×#8162
#117220352
#105925073
#105909703