Finite Group Databasebeta

← #10694327

Subgroups — #10694327

Order 768.

764 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2856C211yesmaximal, characteristic
2C2×#4537C214yesmaximal, complemented
3#71534C211yesmaximal, characteristic, complemented
4#71530C211yesmaximal, characteristic, complemented
5#9715—31nomaximal, Sylow
6C2×C3×#127C411yescharacteristic, complemented
7C2×C3×#121—21no—
8C2×C3×#121—21no—
9C2×C3×#121C2212yes—
10C22×C3×C42C2211yescharacteristic
11C3×#390—21no—
12C3×#390—21no—
13C3×#390C412yescomplemented
14C3×#384C411yescharacteristic, complemented
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19C2×C4⋊(C3⋊C8)—21no—
20#4537—28no—
21#4524—21no—
22#2856D311yescharacteristic
23#1797—31no—
24C2×#327—34no—
25#1434—31no—
26C22×C3×D4—22no—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4C2×C412yes—
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C23×Dic3—22no—
34C3×#127—42no—
35C3×#121—24no—
36C3×#121—24no—
37C3×#121—44no—
38C3×#121—42no—
39C3×#121—42no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C412yes—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C42—42no—
47C2×C3×C42C2×C411yescharacteristic
48C2×C3×C42C2×C411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C22×C3⋊C8—22no—
52C3⋊#95—22no—
53C3⋊#95—42no—
54C3⋊#95—42no—
55C3⋊#95D414yescomplemented
56#830—24no—
57C4⋊(C3⋊C8)—42no—
58C2×#127Dic311yescharacteristic, complemented
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×D312yes—
62C22×C42C2×D311yescharacteristic
63C2×C4⋊C8—61no—
64C2×C22⋊C8—32no—
65#390—21no—
66#390—21no—
67#390Dic312yescomplemented
68#384Dic311yescharacteristic, complemented
69C2×#95—32no—
70C2×#95—61no—
71C2×#95—61no—
72#336—61no—
73#327—68no—
74C24×C3—22no—
75C24×C3C2×D411yescharacteristic
76C2×C3×D4—24no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—44no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—44no—
87C2×C3×D4—42no—
88C2×C3×D4—42no—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4D812yes—
102C22×C3×C4SD1612yes—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D412yes—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4112yes—
107C22×Dic3—24no—
108C22×Dic3—42no—
109C22×Dic3—42no—
110C22×Dic3—42no—
111C22×Dic3—42no—
112C22×Dic3—42no—
113C22×Dic3—42no—
114C3×C4⋊C4—24no—
115C3×C4⋊C4—42no—
116C3×C4⋊C4—42no—
117C3×C22⋊C4—44no—
118C3×C22⋊C4—44no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—42no—
121C3×C42—22no—
122C3×C42—22no—
123C3×C42—41no—
124C3×C42—41no—
125C2×C3⋊C8—42no—
126C2×C3⋊C8—42no—
127C2×C3⋊C8—42no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C2×Dic312yes—
132C23×C4—62no—
133C23×C4#6711yescharacteristic
134C23×C4#6711yescharacteristic
135C23×C4C22×D311yescharacteristic
136C22×C8—62no—
137#127—42no—
138#121—24no—
139#121—24no—
140#121—44no—
141#121—42no—
142#121—42no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4C2×Dic312yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C42—42no—
150C2×C42C2×Dic311yescharacteristic
151C2×C42C2×Dic311yescharacteristic
152C2×C42#6711yescharacteristic
153C2×C42#6711yescharacteristic
154C4⋊C8—122no—
155C22⋊C8—64no—
156#95—34no—
157#95—62no—
158#95—122no—
159#95—122no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—42no—
165C23×C3—42no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—42no—
169C23×C3—42no—
170C23×C3—42no—
171C23×C3C2×C22⋊C411yescharacteristic
172C23×C3#12112yes—
173C23×C3C2×D811yescharacteristic
174C23×C3C2×SD1611yescharacteristic
175C23×C3AGL(1,ℤ/8)12yes—
176C3×D4—84no—
177C3×D4—84no—
178C3×D4—84no—
179C3×D4—84no—
180C3×D4—84no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—44no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4#9912yes—
203C2×C3×C4#10212yes—
204C2×C3×C4#10212yes—
205C2×C3×C4C4×D412yes—
206C2×C3×C4#12411yescharacteristic
207C2×C3×C4#12711yescharacteristic
208C2×Dic3—44no—
209C2×Dic3—44no—
210C2×Dic3—44no—
211C2×Dic3—82no—
212C2×Dic3—82no—
213C2×Dic3—82no—
214C2×Dic3—82no—
215C3⋊C8—82no—
216C24—22no—
217C24C2×#6711yescharacteristic
218C2×D4—24no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—44no—
229C2×D4—42no—
230C2×D4—42no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—64no—
240C22×C4—122no—
241C22×C4—122no—
242C22×C4—122no—
243C22×C4—122no—
244C22×C4—122no—
245C22×C4—122no—
246C22×C4#21312yes—
247C22×C4#21412yes—
248C22×C4#21711yescharacteristic
249C22×C4#21711yescharacteristic
250C22×C4#21711yescharacteristic
251C22×C4#21711yescharacteristic
252C22×C4D3×D412yes—
253C22×C4#23712yes—
254C22×C4C22×Dic311yescharacteristic
255C22×C4C2×#6711yescharacteristic
256C2×C8—122no—
257C2×C8—122no—
258C2×C8—122no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42—41no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3—82no—
281C22×C3—82no—
282C22×C3—82no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3#39011yescharacteristic
286C22×C3C2×#9911yescharacteristic
287C22×C3#41011yescharacteristic
288C22×C3C2×#10211yescharacteristic
289C22×C3#41811yescharacteristic
290C22×C3#42111yescharacteristic
291C22×C3#42111yescharacteristic
292C22×C3#45912yes—
293C22×C3#46012yes—
294C3×C4—21no—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—41no—
301C3×C4—82no—
302Dic3—84no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23C2×#21311yescharacteristic
315C23#93212yes—
316C23C2×#21411yescharacteristic
317C23#93812yes—
318C23C2×#21711yescharacteristic
319D4—84no—
320D4—84no—
321D4—84no—
322D4—84no—
323D4—84no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—124no—
346C2×C4—124no—
347C2×C4—124no—
348C2×C4—242no—
349C2×C4—242no—
350C2×C4—242no—
351C2×C4—242no—
352C2×C4#83212yes—
353C2×C4#83212yes—
354C2×C4#83412yes—
355C2×C4D4×Dic312yes—
356C2×C4#93611yescharacteristic
357C2×C4#94011yescharacteristic
358C8—242no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—162no—
362C2×C3#151011yescharacteristic
363C2×C3#151411yescharacteristic
364C2×C3#151711yescharacteristic
365C2×C3#151911yescharacteristic
366C2×C3#196311yescharacteristic
367C2×C3#196411yescharacteristic
368C2×C3#198211yescharacteristic
369C22—22no—
370C22—22no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22#507212yes—
385C22#507512yes—
386C22C2×#83211yescharacteristic
387C22#521511yescharacteristic
388C22C2×#83411yescharacteristic
389C22#522111yescharacteristic
390C22#523611yescharacteristic
391C22#523611yescharacteristic
392C22#524011yescharacteristic
393C4—21no—
394C4—22no—
395C4—21no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—82no—
401C4—244no—
402C3#971511yescharacteristic, complemented, Sylow
403C2—22no—
404C2—22no—
405C2—162no—
406C2#7167411yescharacteristic
407C2#7167511yescharacteristic
408C2#7168611yescharacteristic
409C2#7176111yescharacteristic
410C2#7181311yescharacteristic
411C2#7181411yescharacteristic
412C2#7181611yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2856C2
socleC23×C3C2×C22⋊C4
cosocleC23×C3×C4C23
hypercenter#2856D3
hypocenterC3#9715
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2856

Complements

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

NH ≅ G/Ntype
2. C2×#4537C2semidirect
3. #71534C2semidirect
4. #71530C2semidirect
6. C2×C3×#127C4semidirect
13. C3×#390C4semidirect
14. C3×#384C4semidirect
55. C3⋊#95D4semidirect
58. C2×#127Dic3semidirect
67. #390Dic3semidirect
68. #384Dic3semidirect
402. C3#9715semidirect

Maximal subgroups

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

subgroupindex
C3×#28562
C2×#45372
#715342
#715302
#97153