Finite Group Databasebeta

← #10691986

Subgroups — #10691986

Order 768.

518 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3036C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic, complemented
3#71814C211yesmaximal, characteristic, complemented
4#71800C211yesmaximal, characteristic, complemented
5#71605C211yesmaximal, characteristic, complemented
6#67714C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9309—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#463—22no—
13C3×#461C412yescomplemented
14C3×#459C412yescomplemented
15C3×#423C2211yescharacteristic
16C3×#417—21no—
17C2×#832—21no—
18#5207—22no—
19#5012—22no—
20#5012—22no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4537—22no—
26D4⋊(C3⋊C8)—22no—
27#4529—21no—
28#3036D311yescharacteristic
29#2975—31no—
30#1804—31no—
31#1517—31no—
32#1480—31no—
33#1301—31no—
34#1286—31no—
35C2×C3×#41—21no—
36C22×C3×D4D411yescharacteristic
37C23×C3×C4—21no—
38C3×AGL(1,ℤ/8)—42no—
39C3×AGL(1,ℤ/8)—42no—
40C2×C3×SD16—22no—
41C2×C3×D8—22no—
42C2×C3×M16—21no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×#127C2×C411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4C2×C412yes—
53C3×C4×D4D412yes—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×C4⋊Dic3—21no—
58C2×#208—24no—
59C22×C3⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8C2×C412yes—
62C3×#103—22no—
63C3×#102—22no—
64#832—42no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67#830—24no—
68#830—21no—
69#830D412yes—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C2×AGL(1,ℤ/8)—21no—
77#532C2×D311yescharacteristic
78C2×C4×D4C2×D311yescharacteristic
79#463—22no—
80#461Dic312yescomplemented
81#459Dic312yescomplemented
82#436—62no—
83#436—62no—
84#431—31no—
85#431—31no—
86#423C2×D311yescharacteristic
87C2×C4⋊C8—31no—
88#417—21no—
89C2×#102—61no—
90#407—62no—
91#382—31no—
92#340—61no—
93#327—62no—
94D4⋊C8—62no—
95C24×C3—21no—
96C3×#41—41no—
97C3×#41—41no—
98C3×#41—41no—
99C3×#41—41no—
100C2×C3×Q8—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—41no—
110C2×C3×D4C22⋊C412yes—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—41no—
115C22×C3×C4—82no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—41no—
120C3×SD16—44no—
121C3×D8—44no—
122C3×M16—41no—
123C3×M16—41no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4C22⋊C412yes—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42C22×C411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139#208—84no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—41no—
145C2×#41—21no—
146C22×D4#6711yescharacteristic
147C23×C4—21no—
148AGL(1,ℤ/8)—42no—
149AGL(1,ℤ/8)—42no—
150C2×SD16—22no—
151C2×D8—22no—
152C2×M16—21no—
153C2×M16—64no—
154C22×C8—61no—
155C4⋊Q8C2×Dic311yescharacteristic
156#127C2×Dic311yescharacteristic
157#124—21no—
158#121—21no—
159#121—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4C2×Dic312yes—
165C4×D4#6712yes—
166C2×C4⋊C4—61no—
167C2×C4⋊C4#6711yescharacteristic
168C2×C22⋊C4—21no—
169C2×C42C22×D311yescharacteristic
170C4⋊C8—21no—
171C4⋊C8—61no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8C2×Dic312yes—
176#103—22no—
177#102—22no—
178#102—122no—
179C22⋊C8—32no—
180C22⋊C8—64no—
181C22⋊C8—61no—
182C4×C8—32no—
183C4×C8—61no—
184#95—61no—
185#95—61no—
186C23×C3—22no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—81no—
191C23×C3—81no—
192C23×C3#12111yescharacteristic
193C3×Q8—41no—
194C3×Q8—41no—
195C3×D4—22no—
196C3×D4—22no—
197C3×D4—42no—
198C3×D4—41no—
199C3×D4—41no—
200C3×D4—41no—
201C3×D4—41no—
202C3×D4—81no—
203C3×D4—81no—
204C3×D4—81no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—44no—
209C2×C3×C4—41no—
210C2×C3×C4—82no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4C2×C22⋊C411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12311yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×Dic3—81no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C3×C8—42no—
228C3⋊C8—42no—
229C3⋊C8—42no—
230C3⋊C8—81no—
231C24—21no—
232#41—41no—
233#41—41no—
234#41—41no—
235#41—41no—
236C2×Q8—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4#21712yes—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—82no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4#23711yescharacteristic
255C22×C4C2×#6711yescharacteristic
256SD16—44no—
257D8—44no—
258M16—41no—
259M16—41no—
260M16—244no—
261C2×C8—22no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—121no—
265C2×C8—121no—
266C2×C8—121no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4#21712yes—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42D3×D411yescharacteristic
278C42#23711yescharacteristic
279C42C22×Dic311yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#38611yescharacteristic
292C22×C3#46811yescharacteristic
293C22×C3#46911yescharacteristic
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—42no—
299C3×C4—81no—
300C3×C4#41711yescharacteristic
301C3×C4#41811yescharacteristic
302C3×C4#46811yescharacteristic
303C3×C4#46911yescharacteristic
304Dic3—161no—
305C23—22no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—81no—
310C23—81no—
311C23#93911yescharacteristic
312Q8—41no—
313Q8—41no—
314D4—22no—
315D4—22no—
316D4—42no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—241no—
334C2×C4—241no—
335C2×C4—241no—
336C2×C4#93211yescharacteristic
337C2×C4#93211yescharacteristic
338C2×C4D4×Dic311yescharacteristic
339C2×C4D4×Dic311yescharacteristic
340C2×C4#93511yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93911yescharacteristic
343C2×C4#94911yescharacteristic
344C2×C4#95111yescharacteristic
345C2×C4C2×#21711yescharacteristic
346C8—42no—
347C8—122no—
348C8—122no—
349C8—241no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—82no—
353C2×C3—161no—
354C2×C3#188111yescharacteristic
355C2×C3#189511yescharacteristic
356C2×C3#190411yescharacteristic
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22#503911yescharacteristic
368C22#504211yescharacteristic
369C22#522011yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—81no—
376C4—481no—
377C4#503911yescharacteristic
378C4#504211yescharacteristic
379C4#521511yescharacteristic
380C4#523211yescharacteristic
381C3#930911yescharacteristic, complemented, Sylow
382C2—21no—
383C2—21no—
384C2—82no—
385C2—161no—
386C2#7168211yescharacteristic
387C2#7208911yescharacteristic
388C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3036D3
hypocenterC3#9309
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #82304C2semidirect
3. #71814C2semidirect
4. #71800C2semidirect
5. #71605C2semidirect
6. #67714C2semidirect
7. #67713C2semidirect
13. C3×#461C4semidirect
14. C3×#459C4semidirect
80. #461Dic3semidirect
81. #459Dic3semidirect
381. C3#9309semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#823042
#718142
#718002
#716052
#677142
#677132
#93093