Finite Group Databasebeta

← #10690775

Subgroups — #10690775

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82080C211yesmaximal, characteristic
2C2×#4926C211yesmaximal, characteristic, complemented
3C3×#1434C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal
5#67612C212yesmaximal, complemented
6#9118—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#929—21no—
10C2×C3×#95—21no—
11C2×C3×#95—21no—
12C2×C3×#95C2212yes—
13#5213—21no—
14#5212C2211yescharacteristic
15C2×#830C2212yes—
16#4940—21no—
17#4940—21no—
18#4939C411yescharacteristic, complemented
19#4939C411yescharacteristic, complemented
20#4928C411yescharacteristic, complemented
21#4928C411yescharacteristic, complemented
22#4926—24no—
23C2×C4⋊(C3⋊C8)—21no—
24C2×#803—21no—
25#4451—24no—
26C2×#432—31no—
27#2854—31no—
28C2×#327—32no—
29#1434D311yescharacteristic, complemented
30#1294—32no—
31C23×C3×C4—22no—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C22×#67—21no—
36C22×D12C2×C411yescharacteristic
37C2×C3×C42—22no—
38C2×C3×C42—22no—
39C2×C3×C42C2×C411yescharacteristic
40C2×C3×C42C2×C411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×#217—21no—
44#930—22no—
45#930—22no—
46#929—44no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×C4⋊Dic3C2×C411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C22×C3⋊C8—22no—
58C3×#95—24no—
59C3×#95—22no—
60C3×#95—22no—
61C3×#95—42no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—22no—
66#830D414yes—
67C4⋊(C3⋊C8)—24no—
68#803—24no—
69C2×#123—61no—
70C2×#121—31no—
71C22×C42C2×D311yescharacteristic
72#432—64no—
73C2×C4⋊C8—61no—
74C2×C22⋊C8—32no—
75C2×#97—61no—
76#390—31no—
77#390—31no—
78#389—31no—
79#389—31no—
80#388—61no—
81#388—61no—
82#386—61no—
83#380—31no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95C2×D312yes—
87#327—64no—
88C24×C3C2×D411yescharacteristic
89C23×D3—21no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—21no—
95C22×C3×C4—22no—
96C22×C3×C4—24no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4—42no—
103C22×C3×C4—42no—
104C22×C3×C4—42no—
105C22×C3×C4—42no—
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D412yes—
113C22×C3×C4#4112yes—
114C2×#67—42no—
115C2×#67—42no—
116C2×#67—41no—
117C2×#67—41no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C2×D12—22no—
122C2×D12—41no—
123C2×D12—41no—
124C3×C42—22no—
125C3×C42—22no—
126C3×C42—21no—
127C3×C42—21no—
128C3×C42—21no—
129C3×C42—21no—
130#217—42no—
131(C2×D3)⋊C4—42no—
132(C2×D3)⋊C4—42no—
133(C2×D3)⋊C4—42no—
134(C2×D3)⋊C4—42no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137C4⋊Dic3—22no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140Dic3⋊C4—42no—
141Dic3⋊C4—42no—
142C2×C3⋊C8—24no—
143C2×C3⋊C8—42no—
144C2×C3⋊C8—42no—
145C22×D4—31no—
146C22×D4—61no—
147C23×C4—22no—
148C23×C4#6711yescharacteristic
149C23×C4#6711yescharacteristic
150C23×C4C22×D311yescharacteristic
151C22×C8—62no—
152#123—124no—
153#121—62no—
154#121—62no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42C4×D311yescharacteristic
169C2×C42C4×D311yescharacteristic
170C2×C42D1211yescharacteristic
171C2×C42D1211yescharacteristic
172C4⋊C8—64no—
173C22⋊C8—34no—
174C22⋊C8—62no—
175#97—64no—
176#95—24no—
177#95—22no—
178#95—22no—
179#95—42no—
180#95—61no—
181#95—61no—
182#95—61no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3C2×C22⋊C411yescharacteristic
188C23×C3#12112yes—
189C23×C3AGL(1,ℤ/8)11yescharacteristic
190C23×C3AGL(1,ℤ/8)12yes—
191C23×C3#13711yescharacteristic
192C22×D3—41no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—44no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—44no—
220C2×C3×C4—44no—
221C2×C3×C4—44no—
222C2×C3×C4—82no—
223C2×C3×C4—82no—
224C2×C3×C4—82no—
225C2×C3×C4—82no—
226C2×C3×C4#10412yes—
227C2×C3×C4C4×D412yes—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12311yescharacteristic
230#67—82no—
231#67—82no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241D12—82no—
242C3⋊C8—44no—
243C24—61no—
244C24C2×#6711yescharacteristic
245C2×D4—62no—
246C2×D4—121no—
247C2×D4—121no—
248C2×D4—121no—
249C2×D4—122no—
250C2×D4—121no—
251C2×D4—122no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4(C2×D3)⋊C411yescharacteristic
272C22×C4(C2×D3)⋊C411yescharacteristic
273C22×C4(C2×D3)⋊C411yescharacteristic
274C22×C4(C2×D3)⋊C411yescharacteristic
275C22×C4C2×C4×D311yescharacteristic
276C22×C4C2×D1211yescharacteristic
277C22×C4D3×D412yes—
278C22×C4#23712yes—
279C2×C8—64no—
280C2×C8—122no—
281C2×C8—122no—
282C4⋊C4—62no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—122no—
286C4⋊C4—122no—
287C22⋊C4—122no—
288C22⋊C4—122no—
289C22⋊C4—122no—
290C22⋊C4—122no—
291C22⋊C4—122no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—22no—
295C42—22no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3#38611yescharacteristic
309C22×C3#41011yescharacteristic
310C22×C3#41011yescharacteristic
311C22×C3#41711yescharacteristic
312C22×C3#41811yescharacteristic
313C22×C3C2×#10411yescharacteristic
314C22×C3#42111yescharacteristic
315C22×C3#46812yes—
316C22×C3#46912yes—
317C2×D3—81no—
318C2×D3—81no—
319C2×D3—81no—
320C2×D3—81no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—81no—
324C3×C4—22no—
325C3×C4—21no—
326C3×C4—22no—
327C3×C4—21no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—84no—
332Dic3—81no—
333Dic3—81no—
334Dic3—81no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—121no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23C2×(C2×D3)⋊C411yescharacteristic
347C23#93211yescharacteristic
348C23#93912yes—
349C23#94211yescharacteristic
350C23#94912yes—
351D4—242no—
352D4—242no—
353D4—242no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4#80512yes—
391C2×C4#67⋊C412yes—
392C2×C4#88211yescharacteristic
393C2×C4#88611yescharacteristic
394C8—124no—
395C2×C3—22no—
396C2×C3—22no—
397C2×C3#150511yescharacteristic
398C2×C3#150911yescharacteristic
399C2×C3#151211yescharacteristic
400C2×C3#151411yescharacteristic
401C2×C3#187111yescharacteristic
402C2×C3#189511yescharacteristic
403C2×C3#189911yescharacteristic
404D3—161no—
405C22—22no—
406C22—22no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22C2×#80511yescharacteristic
421C22#495811yescharacteristic
422C22#496611yescharacteristic
423C22#500111yescharacteristic
424C22#500511yescharacteristic
425C22#500611yescharacteristic
426C22#500611yescharacteristic
427C22#503912yes—
428C22#504912yes—
429C4—22no—
430C4—21no—
431C4—22no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—84no—
437C4—241no—
438C4—241no—
439C4—241no—
440C3#911811yescharacteristic, complemented, Sylow
441C2—22no—
442C2—22no—
443C2—481no—
444C2#6943211yescharacteristic
445C2#6945311yescharacteristic
446C2#6947011yescharacteristic
447C2#6951711yescharacteristic
448C2#6953511yescharacteristic
449C2#6954411yescharacteristic
450C2#6956411yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1434C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1434D3
hypocenterC3#9118
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#4926C2semidirect
3. C3×#1434C2semidirect
5. #67612C2semidirect
18. #4939C4semidirect
19. #4939C4semidirect
20. #4928C4semidirect
21. #4928C4semidirect
29. #1434D3semidirect
440. C3#9118semidirect

Maximal subgroups

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

subgroupindex
#820802
C2×#49262
C3×#14342
C2×#44512
#676122
#91183