Finite Group Databasebeta

← #10698974

Subgroups — #10698974

Order 768.

574 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2#71764C211yesmaximal, characteristic, complemented
3C2×#4544C211yesmaximal, characteristic, complemented
4#71614C211yesmaximal, characteristic, complemented
5#71604C211yesmaximal, characteristic, complemented
6C2×#4479C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10472—31nomaximal, Sylow
9C22×C3×D8—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#459—22no—
13C3×#459—22no—
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×#102—21no—
16C2×#832—21no—
17C2×#832—21no—
18C2×#832—21no—
19C2×#830—21no—
20C2×#830—21no—
21#4935C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4544—22no—
26#4529—21no—
27#4479—22no—
28C2×#459D311yescharacteristic
29#1957—31no—
30#1954—31no—
31#1802—31no—
32C2×#335—31no—
33C2×#331—31no—
34#1296—31no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C2×C3×D8—24no—
41C2×C3×D8—42no—
42C2×C3×D8—42no—
43C22×C3×C8—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127D412yes—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60C22×C3⋊C8D411yescharacteristic
61C22×C3⋊C8D411yescharacteristic
62C3×C4⋊C8—21no—
63C3×C4⋊C8D412yes—
64C3×#102—24no—
65#832—42no—
66#832—42no—
67#832—42no—
68#830—42no—
69#830—42no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C22×D8—21no—
77C2×#127C2×D311yescharacteristic
78C2×C4×D4C2×D311yescharacteristic
79#459—22no—
80#459—22no—
81C2×C4⋊C8—31no—
82C2×C4⋊C8—31no—
83C2×C4⋊C8C2×D311yescharacteristic
84C2×#102—21no—
85C2×#102—61no—
86C2×#102—61no—
87C2×#102—61no—
88C2×C22⋊C8—61no—
89C2×C22⋊C8—61no—
90C2×C4×C8—31no—
91#384—31no—
92#340—61no—
93#335—62no—
94#331—62no—
95C24×C3—21no—
96C24×C3—41no—
97C2×C3×D4—21no—
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—21no—
101C2×C3×D4—22no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—81no—
117C2×C3×D4—81no—
118C2×C3×D4D812yes—
119C22×C3×C4—21no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×Dic3—41no—
130C3×D8—48no—
131C2×C3×C8—22no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—81no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8D812yescomplemented
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4#6711yescharacteristic
163C23×C4—21no—
164C2×D8—24no—
165C2×D8—42no—
166C2×D8—42no—
167C22×C8—21no—
168C22×C8—31no—
169C22×C8—31no—
170C22×C8—61no—
171C22×C8—61no—
172#127—21no—
173#127—21no—
174#127—21no—
175#127#6712yes—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4#6711yescharacteristic
184C2×C22⋊C4—21no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8#6712yes—
192#102—24no—
193#102—122no—
194#102—122no—
195#102—122no—
196C22⋊C8—122no—
197C22⋊C8—122no—
198C4×C8—61no—
199C4×C8—61no—
200C23×C3—22no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3—81no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3#12111yescharacteristic
215C3×D4—22no—
216C3×D4—22no—
217C3×D4—42no—
218C3×D4—41no—
219C3×D4—42no—
220C3×D4—42no—
221C3×D4—42no—
222C3×D4—41no—
223C3×D4—82no—
224C3×D4—82no—
225C3×D4—81no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12311yescharacteristic
242C2×C3×C4#12311yescharacteristic
243C2×C3×C4#12411yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C2×D811yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4#13711yescharacteristic
252C2×Dic3—81no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C3×C8—42no—
256C3⋊C8—22no—
257C3⋊C8—41no—
258C3⋊C8—81no—
259C3⋊C8—81no—
260C24—21no—
261C24—41no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4#21312yes—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—121no—
292C22×C4D3×D411yescharacteristic
293C22×C4#23711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295D8—48no—
296C2×C8—22no—
297C2×C8—32no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—121no—
306C2×C8—121no—
307C2×C8—121no—
308C2×C8—121no—
309C2×C8—121no—
310C2×C8—121no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—121no—
314C4⋊C4—121no—
315C4⋊C4—121no—
316C4⋊C4—121no—
317C4⋊C4—241no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42#23711yescharacteristic
321C42#23711yescharacteristic
322C42C2×#6711yescharacteristic
323C42C2×#6711yescharacteristic
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—82no—
332C22×C3—82no—
333C22×C3—82no—
334C22×C3—161no—
335C22×C3—161no—
336C22×C3—161no—
337C22×C3—161no—
338C22×C3#39411yescharacteristic
339C22×C3#45911yescharacteristic
340C22×C3#46311yescharacteristic
341C22×C3#46611yescharacteristic
342C22×C3#46811yescharacteristic
343C22×C3#47111yescharacteristic
344C22×C3#47211yescharacteristic
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—81no—
348C3×C4#44711yescharacteristic
349C3×C4#45211yescharacteristic
350C3×C4#45911yescharacteristic
351C3×C4#46311yescharacteristic
352C3×C4#46611yescharacteristic
353C3×C4#46911yescharacteristic
354C3×C4#48011yescharacteristic
355C3×C4#48211yescharacteristic
356Dic3—161no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23#93911yescharacteristic
372D4—22no—
373D4—22no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—41no—
378D4—42no—
379D4—41no—
380D4—82no—
381D4—82no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—241no—
396C2×C4—241no—
397C2×C4—241no—
398C2×C4D3×D811yescharacteristic
399C2×C4#91111yescharacteristic
400C2×C4#91111yescharacteristic
401C2×C4#91211yescharacteristic
402C2×C4C2×#21311yescharacteristic
403C2×C4#93211yescharacteristic
404C2×C4#93511yescharacteristic
405C2×C4#93511yescharacteristic
406C2×C4#93611yescharacteristic
407C2×C4#93811yescharacteristic
408C2×C4#93911yescharacteristic
409C2×C4#95011yescharacteristic
410C2×C4#95111yescharacteristic
411C2×C4#95311yescharacteristic
412C8—42no—
413C8—62no—
414C8—121no—
415C8—241no—
416C8—241no—
417C2×C3—82no—
418C2×C3—162no—
419C2×C3#166711yescharacteristic
420C2×C3#168111yescharacteristic
421C2×C3#168511yescharacteristic
422C2×C3#169711yescharacteristic
423C2×C3#203311yescharacteristic
424C2×C3#205111yescharacteristic
425C2×C3#205411yescharacteristic
426C22—42no—
427C22—42no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22#503811yescharacteristic
441C22#504211yescharacteristic
442C22#515011yescharacteristic
443C22#515311yescharacteristic
444C22#519911yescharacteristic
445C22#520011yescharacteristic
446C22#522211yescharacteristic
447C4—21no—
448C4—21no—
449C4—81no—
450C4—481no—
451C4#503311yescharacteristic
452C4#503511yescharacteristic
453C4#515011yescharacteristic
454C4#515311yescharacteristic
455C4#515711yescharacteristic
456C4#515911yescharacteristic
457C4#521711yescharacteristic
458C4#523911yescharacteristic
459C3#1047211yescharacteristic, complemented, Sylow
460C2—82no—
461C2—162no—
462C2#7100511yescharacteristic
463C2#7100711yescharacteristic
464C2#7100811yescharacteristic
465C2#7101011yescharacteristic
466C2#7167911yescharacteristic
467C2#7204011yescharacteristic
468C2#7209311yescharacteristic
subgroupG/H
centerC23#939
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3#121
hypercenterC2×#459D3
hypocenterC3#10472
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. #71764C2semidirect
3. C2×#4544C2semidirect
4. #71614C2semidirect
5. #71604C2semidirect
6. C2×#4479C2semidirect
7. #67702C2semidirect
158. C2×C3⋊C8D8semidirect
459. C3#10472semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
#717642
C2×#45442
#716142
#716042
C2×#44792
#677022
#104723