Finite Group Databasebeta

← #10845426

Subgroups — #10845426

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2538C211yesmaximal, characteristic
2C2×#5020C211yesmaximal, characteristic, complemented
3#82382C211yesmaximal, characteristic
4C2×#5012C211yesmaximal, characteristic
5#71697C212yesmaximal, complemented
6#71695C212yesmaximal, complemented
7C4×#835C211yesmaximal, characteristic, complemented
8#69582C211yesmaximal, characteristic, complemented
9#69567C211yesmaximal, characteristic, complemented
10#69520C212yesmaximal, complemented
11#69513C212yesmaximal, complemented
12#25348—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2212yes—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#936—21no—
17C22×#208C2212yes—
18C2×#869C2211yescharacteristic
19C3×#396—22no—
20C3×#390—21no—
21C3×#389—21no—
22C3×#384C2211yescharacteristic
23C3×#380C2212yes—
24C2×#835C2211yescharacteristic
25C2×#835C2212yes—
26C2×#835C2211yescharacteristic
27#5212C2212yes—
28#5212C2212yes—
29C2×#830C2212yes—
30#5020—24no—
31#5012—24no—
32#4973—21no—
33C2×#808C2212yes—
34C2×#807C2212yes—
35#4953—22no—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#475—31no—
45C2×#436—31no—
46#2547—31no—
47#2538D311yescharacteristic
48#2077—32no—
49#2036—32no—
50#1990—31no—
51#1947—31no—
52#1894—32no—
53#1809—32no—
54C4×#103—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×Dic6—21no—
59C3×C22⋊Q8—24no—
60C3×C22⋊Q8—42no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4D412yes—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79#936—44no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×C4×Dic3—21no—
85C2×#208—24no—
86C2×#208—24no—
87C2×#208—42no—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91#869—22no—
92#869—22no—
93C3×#95—22no—
94#835—22no—
95#835—24no—
96#835—22no—
97C3⋊#95—21no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#830—24no—
101#808—24no—
102#807—42no—
103C4⋊(C3⋊C8)D414yes—
104C4×C3⋊C8—22no—
105C22×M16—32no—
106C2×C4⋊Q8—31no—
107C2×#124—61no—
108C2×C22⋊Q8C2×D312yes—
109C2×C4×Q8C2×D311yescharacteristic
110C2×C4×D4C2×D311yescharacteristic
111#475—64no—
112#436—64no—
113C2×#107—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#103—32no—
117C2×#103—31no—
118C2×#103—31no—
119C2×C22⋊C8—32no—
120C2×C4×C8—31no—
121#397—62no—
122#396—22no—
123#391—61no—
124#390—21no—
125#389—21no—
126#384C2×D311yescharacteristic
127#380—32no—
128#380—32no—
129#380C2×D312yes—
130C4×C4⋊C4—31no—
131#340—31no—
132#340—32no—
133#340—31no—
134#340—32no—
135#340—32no—
136C24×C3C2×D412yes—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8#4112yes—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C2×Dic6—42no—
163C2×Dic6—42no—
164C2×Dic6—42no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—44no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—81no—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—44no—
176C3×C22⋊C4—82no—
177C3×C42—42no—
178C3×C42C2×D412yes—
179C3×C42#4112yes—
180#217—44no—
181#217—44no—
182C4⋊Dic3—22no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185C4⋊Dic3—22no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—42no—
189C4×Dic3—42no—
190#208—48no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8C2×D414yes—
199C2×C3⋊C8#4112yes—
200C22×Q8—61no—
201C22×Q8C22×D311yescharacteristic
202C22×D4#6711yescharacteristic
203C23×C4C22×D312yes—
204C2×M16—64no—
205C2×M16—64no—
206C2×M16—122no—
207C22×C8—31no—
208C22×C8—32no—
209C22×C8—31no—
210C4⋊Q8—62no—
211C4⋊Q8—62no—
212#124—124no—
213C22⋊Q8—24no—
214C22⋊Q8—42no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4#6711yescharacteristic
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D312yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4#6712yes—
235C2×C42—21no—
236C2×C42—61no—
237C2×C42C22×D311yescharacteristic
238#107—122no—
239#106—64no—
240C4⋊C8—34no—
241#103—62no—
242#103—64no—
243#103—62no—
244C22⋊C8—64no—
245C4×C8—62no—
246#95—22no—
247#95—61no—
248#95—62no—
249#95—62no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3C22×D411yescharacteristic
258C3×Q8—22no—
259C3×Q8—22no—
260C3×Q8—41no—
261C3×Q8—41no—
262C3×D4—44no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4#12112yes—
282C2×C3×C4#12112yes—
283C2×C3×C4AGL(1,ℤ/8)12yes—
284C2×C3×C4#13712yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4H(5,𝔽2)11yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300Dic6—84no—
301C3⋊C8—24no—
302C3⋊C8—22no—
303C3⋊C8—41no—
304C24C2×#6712yes—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—122no—
311C2×Q8—122no—
312C2×Q8—122no—
313C2×Q8#23712yes—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—62no—
329C22×C4C2×#6712yes—
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332C22×C4C23×D311yescharacteristic
333M16—128no—
334C2×C8—34no—
335C2×C8—32no—
336C2×C8—61no—
337C2×C8—62no—
338C2×C8—61no—
339C2×C8—62no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8—61no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—62no—
352C4⋊C4—61no—
353C4⋊C4—62no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—61no—
357C4⋊C4—81no—
358C4⋊C4—122no—
359C22⋊C4—24no—
360C22⋊C4—44no—
361C22⋊C4—82no—
362C22⋊C4—124no—
363C22⋊C4—124no—
364C42—42no—
365C42—122no—
366C42D3×D412yes—
367C42#23712yes—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3C2×#12111yescharacteristic
376C22×C3#54611yescharacteristic
377C22×C3#54711yescharacteristic
378C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
379C22×C3C2×#13711yescharacteristic
380C22×C3#57711yescharacteristic
381C22×C3#57811yescharacteristic
382C3×C4—22no—
383C3×C4—42no—
384C3×C4—42no—
385C3×C4—82no—
386C3×C4—81no—
387C3×C4#46814yes—
388C3×C4#52911yescharacteristic
389C3×C4#54211yescharacteristic
390C3×C4#54611yescharacteristic
391C3×C4#54811yescharacteristic
392Dic3—81no—
393Dic3—81no—
394Dic3—82no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23C22×#6711yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407Q8—244no—
408D4—44no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—122no—
428C2×C4—121no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—122no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4#93912yes—
437C2×C4#93912yes—
438C2×C4#94212yes—
439C2×C4#94912yes—
440C2×C4C2×D3×D411yescharacteristic
441C2×C4C2×#23711yescharacteristic
442C2×C4C2×#23711yescharacteristic
443C2×C4#100411yescharacteristic
444C2×C4D3×#4111yescharacteristic
445C2×C4#101011yescharacteristic
446C8—62no—
447C8—64no—
448C8—121no—
449C2×C3—82no—
450C2×C3#242711yescharacteristic
451C2×C3C2×#46811yescharacteristic
452C2×C3#306311yescharacteristic
453C2×C3#332811yescharacteristic
454C2×C3#333211yescharacteristic
455C2×C3#335311yescharacteristic
456C2×C3#335611yescharacteristic
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22C2×#93911yescharacteristic
465C22#580611yescharacteristic
466C22C2×#94211yescharacteristic
467C22C2×#94911yescharacteristic
468C22#582411yescharacteristic
469C22#583611yescharacteristic
470C22#583811yescharacteristic
471C4—22no—
472C4—42no—
473C4—42no—
474C4—82no—
475C4—81no—
476C4—242no—
477C4—241no—
478C4—241no—
479C4#504914yes—
480C4#559811yescharacteristic
481C4#560011yescharacteristic
482C4#560311yescharacteristic
483C4#561211yescharacteristic
484C3#2534811yescharacteristic, complemented, Sylow
485C2—82no—
486C2#8264711yescharacteristic
487C2C2×#504911yescharacteristic
488C2#8342311yescharacteristic
489C2#8355411yescharacteristic
490C2#8357411yescharacteristic
491C2#8362611yescharacteristic
492C2#8366211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2538D3
hypocenterC3#25348
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#5020C2semidirect
5. #71697C2semidirect
6. #71695C2semidirect
7. C4×#835C2semidirect
8. #69582C2semidirect
9. #69567C2semidirect
10. #69520C2semidirect
11. #69513C2semidirect
484. C3#25348semidirect

Maximal subgroups

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

subgroupindex
C3×#25382
C2×#50202
#823822
C2×#50122
#716972
#716952
C4×#8352
#695822
#695672
#695202
#695132
#253483