Finite Group Databasebeta

← #10691641

Subgroups — #10691641

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2344C211yesmaximal, characteristic, complemented
2C2×#4967C211yesmaximal, characteristic
3#71553C211yesmaximal, characteristic, complemented
4#69509C211yesmaximal, characteristic
5#69456C211yesmaximal, characteristic
6#67625C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9260—31nomaximal, Sylow
9C2×C3×#117—21no—
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#389—21no—
13C3×#387—22no—
14C3×#381—21no—
15C3×#380C2211yescharacteristic
16C3×C4×C4⋊C4—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5212C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#4967—22no—
22#4967—21no—
23#4967—21no—
24C2×#809C412yescomplemented
25C2×#807C412yescomplemented
26C2×#806—21no—
27C2×#806—21no—
28#4529—22no—
29#4451—22no—
30C2×#480—31no—
31#2344D311yescharacteristic, complemented
32#1964—31no—
33#1823—31no—
34#1519—31no—
35#1289—31no—
36#1282—31no—
37C23×C3×C4—21no—
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C22×D12C2×C411yescharacteristic
41C3×#117—42no—
42C3×#117—42no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C412yes—
49C2×C3×C4⋊C4D412yes—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4D411yescharacteristic
54C2×C3×C22⋊C4D411yescharacteristic
55C2×C3×C42—22no—
56C2×C3×C42—21no—
57C2×#217—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3C2×C411yescharacteristic
65C22×C3⋊C8—21no—
66C22×C3⋊C8C2×C412yes—
67C3×#95—22no—
68C3×#95—22no—
69C3×#95—21no—
70C3×#95D412yes—
71C3⋊#95—21no—
72#830—21no—
73#830—21no—
74#830—21no—
75#830—21no—
76#809—22no—
77#809—22no—
78#807—22no—
79#807—22no—
80#806—42no—
81#806—42no—
82C2×#121—31no—
83C2×#117—21no—
84C22×C4⋊C4C2×D311yescharacteristic
85#480—62no—
86#480—61no—
87#480—61no—
88C2×#107—32no—
89C2×#102—32no—
90C2×#99—61no—
91C2×#99—61no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8—31no—
94#389—21no—
95#387—22no—
96#381—21no—
97#380—31no—
98#380C2×D311yescharacteristic
99C4×C4⋊C4—21no—
100C4×C22⋊C4C2×D311yescharacteristic
101#340—62no—
102#327—62no—
103C24×C3C2×D411yescharacteristic
104C23×D3—21no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—42no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C2×#67—41no—
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C22×Dic3—21no—
130C22×Dic3—41no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—42no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—44no—
140C3×C4⋊C4C22⋊C412yes—
141C3×C4⋊C4C22⋊C412yes—
142C3×C22⋊C4—21no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4D812yes—
148C3×C42—42no—
149C3×C42—44no—
150#217—41no—
151#217—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—42no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—42no—
159C2×C3⋊C8—42no—
160C2×C3⋊C8—41no—
161C22×D4—31no—
162C22×D4—61no—
163C23×C4—21no—
164C23×C4C22×D311yescharacteristic
165C22×C8—32no—
166C22×C8—61no—
167#121—61no—
168#121—61no—
169#121—61no—
170#121—61no—
171#117—42no—
172#117—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4C4×D312yes—
181C2×C4⋊C4D1212yes—
182C2×C4⋊C4#6711yescharacteristic
183C2×C4⋊C4#6711yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4#6711yescharacteristic
188C2×C22⋊C4#6711yescharacteristic
189C2×C42—22no—
190C2×C42—21no—
191#107—62no—
192#107—62no—
193#102—62no—
194#102—62no—
195#99—122no—
196#99—122no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201#95—22no—
202#95—22no—
203#95—21no—
204#95—61no—
205#95#6712yes—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3C4×D411yescharacteristic
209C23×C3#12011yescharacteristic
210C23×C3#12111yescharacteristic
211C23×C3C2×D811yescharacteristic
212C23×C3#13511yescharacteristic
213C23×C3AGL(1,ℤ/8)11yescharacteristic
214C23×C3AGL(1,ℤ/8)11yescharacteristic
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—81no—
219C22×D3—81no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4C2×C22⋊C411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12311yescharacteristic
249#67—81no—
250#67—81no—
251#67—81no—
252#67—81no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258C2×Dic3—81no—
259D12—42no—
260D12—81no—
261C3⋊C8—42no—
262C3⋊C8—81no—
263C24—61no—
264C24C2×#6711yescharacteristic
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—122no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—121no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4C2×D1211yescharacteristic
290C22×C4#23511yescharacteristic
291C22×C4#23511yescharacteristic
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C2×C8—62no—
295C2×C8—122no—
296C2×C8—122no—
297C2×C8—121no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—122no—
307C4⋊C4(C2×D3)⋊C412yes—
308C4⋊C4(C2×D3)⋊C412yes—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4#21312yes—
319C42—44no—
320C42—42no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3#38611yescharacteristic
332C22×C3C4×D811yescharacteristic
333C22×C3#44211yescharacteristic
334C22×C3#44711yescharacteristic
335C22×C3#44911yescharacteristic
336C22×C3#45911yescharacteristic
337C22×C3#46311yescharacteristic
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—161no—
342C2×D3—161no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—42no—
347C3×C4—42no—
348C3×C4—42no—
349C3×C4—82no—
350C3×C4—81no—
351C3×C4#41012yes—
352Dic3—81no—
353Dic3—161no—
354C23—41no—
355C23—41no—
356C23—121no—
357C23—121no—
358C23—121no—
359C23—241no—
360C23—241no—
361C23C4×#6711yescharacteristic
362C23#93011yescharacteristic
363C23C2×#21311yescharacteristic
364C23#93211yescharacteristic
365C23#94911yescharacteristic
366C23#95011yescharacteristic
367C23#95311yescharacteristic
368D4—122no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373D4—241no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4—241no—
405C2×C4C2×(C2×D3)⋊C411yescharacteristic
406C2×C4C4×#6711yescharacteristic
407C2×C4#92911yescharacteristic
408C2×C4#93011yescharacteristic
409C8—122no—
410C8—241no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3#187911yescharacteristic
418C2×C3#189911yescharacteristic
419C2×C3#190511yescharacteristic
420D3—161no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—241no—
432C22—241no—
433C22—241no—
434C22—481no—
435C22—481no—
436C22C4×#21311yescharacteristic
437C22#501411yescharacteristic
438C22#501511yescharacteristic
439C22#501611yescharacteristic
440C22#521311yescharacteristic
441C22#521711yescharacteristic
442C22#523811yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—82no—
450C4—81no—
451C4—241no—
452C4—481no—
453C4#500612yes—
454C3#926011yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—481no—
462C2#7159511yescharacteristic
463C2#7161311yescharacteristic
464C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2344C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#2344D3
hypocenterC3#9260
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
1. C3×#2344C2semidirect
3. #71553C2semidirect
6. #67625C2semidirect
7. #67617C2semidirect
24. C2×#809C4semidirect
25. C2×#807C4semidirect
31. #2344D3semidirect
454. C3#9260semidirect

Maximal subgroups

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

subgroupindex
C3×#23442
C2×#49672
#715532
#695092
#694562
#676252
#676172
#92603