Finite Group Databasebeta

← #10690966

Subgroups — #10690966

Order 768.

692 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73999C211yesmaximal, characteristic
2C2×#4650C211yesmaximal, characteristic, complemented
3C2×#4470C211yesmaximal, characteristic
4#69910C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9156—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×Dic3⋊C4—21no—
11C22×C4×Dic3C2211yescharacteristic
12C2×#886—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4988C412yescomplemented
20#4984C412yescomplemented
21#4958—21no—
22#4670—22no—
23#4650—22no—
24#4650—22no—
25#4550—22no—
26#4470—22no—
27#4451—22no—
28#2854—31no—
29C2×#382—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—42no—
56C2×Dic3⋊C4—42no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3C2×C412yes—
62C2×C4×Dic3D412yes—
63C22×C3⋊C8—21no—
64#886—42no—
65#886—42no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95D412yes—
71C3⋊#95—24no—
72C3⋊#95—21no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77C2×#123—61no—
78C2×#121—31no—
79C22×C4⋊C4—61no—
80C22×C42—31no—
81C2×C22⋊C8—31no—
82C2×C22⋊C8C2×D311yescharacteristic
83#390—32no—
84#389—32no—
85#388—62no—
86#386—61no—
87#382—62no—
88#382—62no—
89#380C2×D311yescharacteristic
90C2×#95—31no—
91C2×#95—61no—
92C2×#95C2×D311yescharacteristic
93#336—62no—
94#327—62no—
95#327—62no—
96C24×C3C2×D411yescharacteristic
97C23×D3—21no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4—42no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×#67—41no—
112C2×#67—41no—
113C2×#67—41no—
114C2×#67—41no—
115C2×#67—41no—
116C2×#67—41no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3—42no—
126C22×Dic3—42no—
127C22×Dic3—42no—
128C22×Dic3—41no—
129C22×Dic3—41no—
130C22×Dic3C22⋊C412yes—
131C22×Dic3C22⋊C412yes—
132C22×Dic3D812yes—
133C22×Dic3SD1612yes—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
137C2×C3×C8—42no—
138C2×C3×C8—41no—
139C3×C4⋊C4—42no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142(C2×D3)⋊C4—44no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—44no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—44no—
149Dic3⋊C4—48no—
150C4×Dic3—22no—
151C4×Dic3—22no—
152C4×Dic3—22no—
153C4×Dic3—22no—
154C2×C3⋊C8—42no—
155C2×C3⋊C8—41no—
156C22×D4—31no—
157C22×D4—61no—
158C23×C4—21no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—61no—
162C23×C4C22×D311yescharacteristic
163C22×C8—21no—
164C22×C8—61no—
165#123—122no—
166#123—122no—
167#121—61no—
168#121—61no—
169#121—61no—
170#121—61no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—62no—
173C2×C4⋊C4—64no—
174C2×C4⋊C4—122no—
175C2×C4⋊C4—122no—
176C2×C4⋊C4C4×D312yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C42—32no—
182C2×C42—32no—
183C2×C42—61no—
184C2×C42—61no—
185C2×C42—61no—
186C2×C42—61no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190C22⋊C8—61no—
191#95—21no—
192#95—21no—
193#95—32no—
194#95—62no—
195#95—64no—
196#95—61no—
197#95D1212yes—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3C2×C22⋊C411yescharacteristic
201C23×C3#12011yescharacteristic
202C23×C3#12311yescharacteristic
203C23×C3C2×D811yescharacteristic
204C23×C3C2×SD1611yescharacteristic
205C23×C3AGL(1,ℤ/8)11yescharacteristic
206C23×C3#13711yescharacteristic
207C22×D3—41no—
208C22×D3—42no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—42no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—82no—
225C2×C3×C4—82no—
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230#67—81no—
231#67—81no—
232#67—81no—
233#67—81no—
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—82no—
252C2×Dic3—82no—
253C2×Dic3#10214yes—
254D12—81no—
255D12—81no—
256C3×C8—81no—
257C3⋊C8—81no—
258C24—61no—
259C24D3×D411yescharacteristic
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—122no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—32no—
274C22×C4—32no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—62no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4—122no—
291C22×C4—121no—
292C22×C4—121no—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4#23511yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4#23911yescharacteristic
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—122no—
302C2×C8—121no—
303C4⋊C4—42no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—124no—
307C4⋊C4—128no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—124no—
311C22⋊C4—124no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—62no—
315C42—62no—
316C42—62no—
317C42—62no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3#38611yescharacteristic
327C22×C3#41012yes—
328C22×C3C2×#10211yescharacteristic
329C22×C3#41711yescharacteristic
330C22×C3#42112yes—
331C22×C3#44711yescharacteristic
332C22×C3#44811yescharacteristic
333C22×C3#48011yescharacteristic
334C22×C3#48111yescharacteristic
335C2×D3—81no—
336C2×D3—81no—
337C2×D3—81no—
338C2×D3—82no—
339C2×D3—82no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—82no—
345C3×C4—82no—
346Dic3—24no—
347Dic3—42no—
348Dic3—82no—
349Dic3—81no—
350C23—22no—
351C23—22no—
352C23—121no—
353C23—122no—
354C23—121no—
355C23—121no—
356C23—122no—
357C23D3×C22⋊C411yescharacteristic
358C23#88211yescharacteristic
359C23#89411yescharacteristic
360C23D3×D811yescharacteristic
361C23D3×SD1611yescharacteristic
362C23#91411yescharacteristic
363C23#91811yescharacteristic
364D4—241no—
365D4—241no—
366D4—241no—
367D4—241no—
368D4—241no—
369D4—241no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—34no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—122no—
401C2×C4—121no—
402C2×C4—242no—
403C2×C4—242no—
404C2×C4C4×D1211yescharacteristic
405C2×C4#88411yescharacteristic
406C2×C4#89311yescharacteristic
407C2×C4#89511yescharacteristic
408C8—81no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3#151212yes—
415C2×C3#151712yes—
416C2×C3#187411yescharacteristic
417C2×C3#189711yescharacteristic
418C2×C3#189911yescharacteristic
419D3—161no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22—242no—
432C22—242no—
433C22#466511yescharacteristic
434C22#474012yes—
435C22D3×#10211yescharacteristic
436C22#477311yescharacteristic
437C22#480311yescharacteristic
438C22#480611yescharacteristic
439C22#482112yes—
440C22#486311yescharacteristic
441C22#488211yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—64no—
447C4—82no—
448C4—82no—
449C4—122no—
450C4—242no—
451C4—241no—
452C3#915611yescharacteristic, complemented, Sylow
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—481no—
458C2#6828212yes—
459C2#6831212yes—
460C2#6875711yescharacteristic
461C2#6888011yescharacteristic
462C2#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#9156
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×#4650C2semidirect
4. #69910C2semidirect
6. C3×#1294C2semidirect
7. #67624C2semidirect
19. #4988C4semidirect
20. #4984C4semidirect
34. #1294D3semidirect
452. C3#9156semidirect

Maximal subgroups

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

subgroupindex
#739992
C2×#46502
C2×#44702
#699102
C2×#44512
C3×#12942
#676242
#91563