Finite Group Databasebeta

← #10690926

Subgroups — #10690926

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73999C211yesmaximal, characteristic
2C2×#4649C211yesmaximal, characteristic, complemented
3C2×#4470C211yesmaximal, characteristic
4#69910C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9148—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×Dic3⋊C4—21no—
11C22×C4×Dic3—21no—
12C22×C4×Dic3C2211yescharacteristic
13C2×#886—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4988C412yescomplemented
20#4984C412yescomplemented
21#4958—21no—
22#4670—22no—
23#4649—22no—
24#4649—22no—
25#4550—22no—
26#4470—22no—
27#4451—22no—
28#2854—31no—
29C2×C4×C4⋊C4—31no—
30#1817—31no—
31C2×#327—31no—
32C2×#327—31no—
33#1294—31no—
34#1294D311yescharacteristic, complemented
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C22×#67—21no—
38C23×Dic3—21no—
39C23×Dic3D411yescharacteristic
40C23×Dic3D411yescharacteristic
41C22×D12C2×C411yescharacteristic
42C22×C3×C8—21no—
43C2×C3×C4⋊C4C2×C412yes—
44C2×C3×C22⋊C4—21no—
45#930—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×Dic3⋊C4—24no—
55C2×Dic3⋊C4—28no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—24no—
61C2×C4×Dic3—42no—
62C2×C4×Dic3—42no—
63C2×C4×Dic3C2×C412yes—
64C2×C4×Dic3D412yes—
65C22×C3⋊C8—21no—
66#886—42no—
67#886—42no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95D412yes—
73C3⋊#95—21no—
74C3⋊#95—22no—
75C3⋊#95D412yes—
76#830—21no—
77#830—21no—
78C2×#123—61no—
79C2×#121—31no—
80C22×C4⋊C4—61no—
81C22×C42—31no—
82C22×C42—61no—
83C2×C22⋊C8—31no—
84C2×C22⋊C8C2×D311yescharacteristic
85#390—32no—
86#389—32no—
87#388—62no—
88#386—61no—
89#380C2×D311yescharacteristic
90C4×C4⋊C4—62no—
91C4×C4⋊C4—62no—
92C2×#95—31no—
93C2×#95C2×D311yescharacteristic
94#336—62no—
95#327—62no—
96#327—62no—
97C24×C3C2×D411yescharacteristic
98C23×D3—21no—
99C22×C3×C4—24no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—42no—
103C22×C3×C4—42no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—41no—
113C2×#67—41no—
114C2×#67—41no—
115C2×#67—41no—
116C2×#67—41no—
117C2×#67—41no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—24no—
125C22×Dic3—24no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—42no—
129C22×Dic3—41no—
130C22×Dic3—41no—
131C22×Dic3C22⋊C412yes—
132C22×Dic3C22⋊C412yes—
133C22×Dic3D812yes—
134C22×Dic3SD1612yes—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—42no—
138C2×C3×C8—42no—
139C2×C3×C8—41no—
140C3×C4⋊C4—42no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—44no—
146(C2×D3)⋊C4—44no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—44no—
150Dic3⋊C4—216no—
151C4×Dic3—22no—
152C4×Dic3—22no—
153C4×Dic3—22no—
154C4×Dic3—22no—
155C4×Dic3—48no—
156C2×C3⋊C8—42no—
157C2×C3⋊C8—41no—
158C22×D4—31no—
159C22×D4—61no—
160C23×C4—21no—
161C23×C4—31no—
162C23×C4—31no—
163C23×C4—61no—
164C23×C4C22×D311yescharacteristic
165C22×C8—21no—
166C22×C8—61no—
167#123—122no—
168#123—122no—
169#121—61no—
170#121—61no—
171#121—61no—
172#121—61no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—64no—
175C2×C4⋊C4—68no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4C4×D312yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—62no—
181C2×C22⋊C4—61no—
182C2×C42—32no—
183C2×C42—32no—
184C2×C42—61no—
185C2×C42—61no—
186C2×C42—61no—
187C2×C42—61no—
188C2×C42—64no—
189C2×C42—122no—
190C2×C42—122no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195#95—21no—
196#95—21no—
197#95—32no—
198#95—62no—
199#95—61no—
200#95D1212yes—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3C2×C22⋊C411yescharacteristic
204C23×C3#12011yescharacteristic
205C23×C3#12311yescharacteristic
206C23×C3C2×D811yescharacteristic
207C23×C3C2×SD1611yescharacteristic
208C23×C3AGL(1,ℤ/8)11yescharacteristic
209C23×C3#13711yescharacteristic
210C22×D3—41no—
211C22×D3—42no—
212C22×D3—41no—
213C22×D3—41no—
214C22×D3—42no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—44no—
224C2×C3×C4—44no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233#67—81no—
234#67—81no—
235#67—81no—
236#67—81no—
237C2×Dic3—24no—
238C2×Dic3—24no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—24no—
248C2×Dic3—42no—
249C2×Dic3—44no—
250C2×Dic3—44no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3#10214yes—
256C2×Dic3#10414yes—
257D12—81no—
258D12—81no—
259C3×C8—81no—
260C3⋊C8—81no—
261C24—61no—
262C24D3×D411yescharacteristic
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—122no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—32no—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—32no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—64no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—64no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—122no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4C2×C4×D311yescharacteristic
297C22×C4C2×D1211yescharacteristic
298C22×C4#23511yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4#23911yescharacteristic
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—122no—
305C2×C8—121no—
306C4⋊C4—42no—
307C4⋊C4—616no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—124no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—124no—
314C22⋊C4—124no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C42—62no—
318C42—62no—
319C42—62no—
320C42—62no—
321C42—128no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3#38611yescharacteristic
331C22×C3#41012yes—
332C22×C3C2×#10211yescharacteristic
333C22×C3#41711yescharacteristic
334C22×C3C2×#10412yes—
335C22×C3#44711yescharacteristic
336C22×C3#44811yescharacteristic
337C22×C3#48011yescharacteristic
338C22×C3#48111yescharacteristic
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—82no—
343C2×D3—82no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—44no—
349C3×C4—82no—
350Dic3—24no—
351Dic3—24no—
352Dic3—44no—
353Dic3—81no—
354C23—22no—
355C23—22no—
356C23—121no—
357C23—122no—
358C23—121no—
359C23—121no—
360C23—122no—
361C23D3×C22⋊C411yescharacteristic
362C23#88211yescharacteristic
363C23#89411yescharacteristic
364C23D3×D811yescharacteristic
365C23D3×SD1611yescharacteristic
366C23#91411yescharacteristic
367C23#91811yescharacteristic
368D4—241no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373D4—241no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—34no—
378C2×C4—34no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—64no—
391C2×C4—64no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—64no—
401C2×C4—122no—
402C2×C4—124no—
403C2×C4—124no—
404C2×C4—122no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4C4×D1211yescharacteristic
409C2×C4#88411yescharacteristic
410C2×C4#89311yescharacteristic
411C2×C4#89511yescharacteristic
412C8—81no—
413C8—241no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3#150212yes—
419C2×C3#150912yes—
420C2×C3#187411yescharacteristic
421C2×C3#189611yescharacteristic
422C2×C3#189911yescharacteristic
423D3—161no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—241no—
433C22—241no—
434C22—241no—
435C22—242no—
436C22—242no—
437C22#466511yescharacteristic
438C22#474012yes—
439C22D3×#10211yescharacteristic
440C22#477311yescharacteristic
441C22#480311yescharacteristic
442C22#480611yescharacteristic
443C22D3×#10412yes—
444C22#486311yescharacteristic
445C22#488211yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—44no—
451C4—64no—
452C4—64no—
453C4—82no—
454C4—124no—
455C4—241no—
456C3#914811yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—481no—
462C2#6827712yes—
463C2#6831112yes—
464C2#6875611yescharacteristic
465C2#6888011yescharacteristic
466C2#6895411yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9148
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
2. C2×#4649C2semidirect
4. #69910C2semidirect
6. C3×#1294C2semidirect
7. #67624C2semidirect
19. #4988C4semidirect
20. #4984C4semidirect
34. #1294D3semidirect
456. C3#9148semidirect

Maximal subgroups

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

subgroupindex
#739992
C2×#46492
C2×#44702
#699102
C2×#44512
C3×#12942
#676242
#91483