Finite Group Databasebeta

← #10830145

Subgroups — #10830145

Order 768.

1036 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic
3#82481C211yesmaximal, characteristic
4C2×#5030C211yesmaximal, characteristic
5#72073C212yesmaximal
6#72065C212yesmaximal
7#71786C211yesmaximal, characteristic
8#70486C211yesmaximal, characteristic
9#70473C211yesmaximal, characteristic
10#70284C212yesmaximal
11#70237C212yesmaximal
12#24203—31nomaximal, Sylow
13C22×C3×Q16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×Q8×Dic3C2212yes—
17C2×#944C2212yes—
18C22×#216C2211yescharacteristic
19#5721—24no—
20#5721—24no—
21C2×#890—21no—
22C2×#889—22no—
23C2×#887—21no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#462—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29#5229C2212yes—
30C2×#835C2212yes—
31C2×#835C2211yescharacteristic
32C2×#835C2212yes—
33C2×#816C2212yes—
34#5030—24no—
35#4986—21no—
36#4979—21no—
37#4972C2212yes—
38C2×#810C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4936C2211yescharacteristic
41#4933—22no—
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#557—31no—
47C2×#462D311yescharacteristic
48C2×#441—31no—
49#2581—31no—
50#2043—32no—
51#2016—32no—
52#1985—31no—
53#1960—31no—
54#1878—32no—
55#1865—32no—
56#1805—31no—
57C22×C3×Q8C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C22×Dic6D411yescharacteristic
60C22×Dic6C2311yescharacteristic
61C2×C3×Q16—24no—
62C2×C3×Q16—24no—
63C2×C3×Q16—42no—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75Q8×Dic3—24no—
76Q8×Dic3—24no—
77#944—24no—
78#944—24no—
79C2×#216—22no—
80C2×#216—22no—
81C2×#216—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3D411yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4D412yes—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3C2312yes—
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96#890—28no—
97#889—216no—
98#887—28no—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C4×Dic6D414yes—
103C3×C4⋊C8—22no—
104C3×#103—24no—
105#835—22no—
106#835—24no—
107#835—42no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#816—42no—
114#810—22no—
115#808—22no—
116#803—22no—
117C22×Q16—31no—
118C22×Q16C2×D312yes—
119#557—64no—
120#557—64no—
121C2×C4⋊Q8—32no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#125—62no—
124C2×#125—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—31no—
128C2×C4×Q8—61no—
129C2×C4×Q8C2×D311yescharacteristic
130#462—24no—
131#441—64no—
132C2×#106—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—31no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#103C2×D312yes—
140C2×#97—31no—
141#400—61no—
142#395—61no—
143#391—32no—
144#389—32no—
145#387—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C2×C3×Q8—22no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8C2×D414yes—
161C2×C3×Q8#4112yes—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3C2×D412yes—
172C22×Dic3C2×D412yes—
173C2×Dic6—22no—
174C2×Dic6—24no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6C2×D412yes—
181C3×Q16—48no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—42no—
190C3×C42—42no—
191C3×C42C2×D412yes—
192C3×C42C2×D412yes—
193#216—28no—
194C4⋊Dic3—28no—
195C4⋊Dic3—22no—
196C4⋊Dic3—21no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3C2×D412yes—
200Dic3⋊C4—28no—
201Dic3⋊C4—28no—
202Dic3⋊C4—24no—
203Dic3⋊C4—24no—
204Dic3⋊C4—44no—
205C4×Dic3—28no—
206C4×Dic3—24no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8#4112yes—
215C22×Q8—31no—
216C22×Q8—31no—
217C22×Q8C22×D311yescharacteristic
218C22×Q8C22×D312yes—
219C2×Q16—24no—
220C2×Q16—24no—
221C2×Q16—42no—
222C2×Q16—62no—
223C2×Q16—62no—
224C2×Q16—62no—
225C22×C8—31no—
226C22×C8—31no—
227C22×C8C22×D312yes—
228C4⋊Q8—22no—
229C4⋊Q8—64no—
230C4⋊Q8—64no—
231C4⋊Q8#6714yes—
232#125—616no—
233#125—68no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—34no—
237C4×Q8—68no—
238C4×Q8—64no—
239C4×Q8—64no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×Q8—62no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4C22×D311yescharacteristic
256C2×C42—21no—
257C2×C42—32no—
258C2×C42—62no—
259C2×C42—61no—
260C2×C42C22×D311yescharacteristic
261#106—62no—
262C4⋊C8—22no—
263#103—24no—
264#103—62no—
265#103—62no—
266#103—64no—
267#103—122no—
268#103—122no—
269#97—62no—
270#95—62no—
271#95—62no—
272#95—62no—
273#95—61no—
274#95—61no—
275C23×C3C22×D411yescharacteristic
276C3×Q8—22no—
277C3×Q8—22no—
278C3×Q8—24no—
279C3×Q8—24no—
280C3×Q8—42no—
281C3×Q8—42no—
282C3×Q8—41no—
283C3×Q8—41no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4#12012yes—
295C2×C3×C4#12012yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4H(5,𝔽2)11yescharacteristic
301C2×C3×C42−1+411yescharacteristic
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—24no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3#13714yes—
315Dic6—28no—
316Dic6—22no—
317Dic6—22no—
318Dic6—41no—
319Dic6—41no—
320C3×C8—42no—
321C3⋊C8—22no—
322C3⋊C8—41no—
323C2×Q8—22no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—22no—
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—32no—
334C2×Q8—64no—
335C2×Q8—62no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8#23712yes—
342C2×Q8C2×#6714yes—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—32no—
347C22×C4—32no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4D3×D411yescharacteristic
352C22×C4D3×D411yescharacteristic
353C22×C4C23×D311yescharacteristic
354Q16—48no—
355Q16—68no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—32no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8—61no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—32no—
370C4⋊C4—42no—
371C4⋊C4—68no—
372C4⋊C4—68no—
373C4⋊C4—68no—
374C4⋊C4—64no—
375C4⋊C4—62no—
376C4⋊C4—64no—
377C4⋊C4—61no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—124no—
381C42—42no—
382C42—68no—
383C42—64no—
384C42—62no—
385C42—62no—
386C42D3×D412yes—
387C42C2×#6712yes—
388C22×C3C2×#12011yescharacteristic
389C22×C3#54811yescharacteristic
390C22×C3#55011yescharacteristic
391C22×C3C2×#13712yes—
392C22×C3#57812yes—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—44no—
396C3×C4—42no—
397C3×C4—81no—
398C3×C4#52511yescharacteristic
399C3×C4#53711yescharacteristic
400C3×C4#54611yescharacteristic
401C3×C4#54711yescharacteristic
402Dic3—24no—
403Dic3—42no—
404Dic3—42no—
405Dic3—42no—
406Dic3—82no—
407C23C2×D3×D411yescharacteristic
408Q8—22no—
409Q8—24no—
410Q8—24no—
411Q8—22no—
412Q8—41no—
413Q8—42no—
414Q8—42no—
415Q8—41no—
416Q8—68no—
417Q8—62no—
418Q8—62no—
419Q8—121no—
420Q8—121no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—34no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—64no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—122no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4#93712yes—
445C2×C4#93712yes—
446C2×C4C2×D3×D411yescharacteristic
447C2×C4C2×#23711yescharacteristic
448C2×C4#100411yescharacteristic
449C2×C4D3×#4111yescharacteristic
450C2×C4#101011yescharacteristic
451C2×C4C22×#6711yescharacteristic
452C8—42no—
453C8—62no—
454C8—121no—
455C2×C3#240711yescharacteristic
456C2×C3#300411yescharacteristic
457C2×C3#301111yescharacteristic
458C2×C3#337412yes—
459C2×C3#339812yes—
460C22#559811yescharacteristic
461C22#559911yescharacteristic
462C22#576612yes—
463C22D3×#13712yes—
464C22C2×#93711yescharacteristic
465C4—22no—
466C4—22no—
467C4—44no—
468C4—42no—
469C4—64no—
470C4—81no—
471C4—122no—
472C4—122no—
473C4—122no—
474C4—242no—
475C4#560611yescharacteristic
476C4#560811yescharacteristic
477C4#580611yescharacteristic
478C4#582411yescharacteristic
479C3#2420311yescharacteristic, complemented, Sylow
480C2#8046612yes—
481C2#8046912yes—
482C2#8270111yescharacteristic
483C2#8388411yescharacteristic
484C2#8397911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#462C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#462D3
hypocenterC3#24203
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
479. C3#24203semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×C3×#4622
#824812
C2×#50302
#720732
#720652
#717862
#704862
#704732
#702842
#702372
#242033