Finite Group Databasebeta

← #10830001

Subgroups — #10830001

Order 768.

1050 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5567C211yesmaximal, characteristic
2C2×C3×C4×Q16C211yesmaximal, characteristic
3#82482C211yesmaximal, characteristic
4C2×#5031C211yesmaximal, characteristic
5#72073C212yesmaximal
6#71626C211yesmaximal, characteristic
7#70642C212yesmaximal
8#70284C212yesmaximal
9#70281C212yesmaximal
10#69788C211yesmaximal, characteristic
11#69771C211yesmaximal, characteristic
12#24191—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#944C2212yes—
17C22×#216C2212yes—
18#5567—24no—
19#5567—24no—
20C2×#889—22no—
21C2×#887C2212yes—
22C2×#870—21no—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6—21no—
25C2×C4×Dic6C2211yescharacteristic
26C3×C4×Q16—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31#5229C2212yes—
32C2×#835C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35#5031—24no—
36#4982—22no—
37#4972C2212yes—
38C2×#810C2212yes—
39#4935C2211yescharacteristic
40#4929—21no—
41#4928—21no—
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#557—31no—
47C2×#462—31no—
48C2×C4×Q16D311yescharacteristic
49#2582—31no—
50#2043—32no—
51#2016—32no—
52#1984—31no—
53#1956—31no—
54#1878—32no—
55#1863—32no—
56#1803—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×Dic6D411yescharacteristic
60C22×Dic6C2312yes—
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C2×C3×Q16—22no—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8D414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77#944—24no—
78#944—24no—
79C2×#216—24no—
80C2×#216—24no—
81C2×#216—42no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4D412yes—
93C2×C4×Dic3—22no—
94C2×C4×Dic3C2312yes—
95C22×C3⋊C8C2312yes—
96#889—216no—
97#887—24no—
98#887—24no—
99#870—28no—
100#869—22no—
101#869D414yes—
102C4×Dic6—28no—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×#107—22no—
106C3×#103—22no—
107C3×#103—22no—
108C3×C4×C8—22no—
109#835—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#816—24no—
114#816—42no—
115#810—24no—
116C4⋊(C3⋊C8)—22no—
117C22×Q16—32no—
118C22×Q16C2×D311yescharacteristic
119#557—64no—
120#557—64no—
121C2×C4⋊Q8—32no—
122C2×C4⋊Q8—31no—
123C2×#125—61no—
124C2×#125—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—61no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×Q8C2×D311yescharacteristic
130#462—64no—
131C4×Q16—24no—
132C2×#107C2×D311yescharacteristic
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#103C2×D311yescharacteristic
139C2×#103C2×D311yescharacteristic
140C2×C4×C8C2×D311yescharacteristic
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8C2×D412yes—
161C2×C3×Q8#4112yes—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3C2×D412yes—
172C22×Dic3C2×D412yes—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6Q1614yescomplemented
181C2×Dic6C2×D414yes—
182C3×Q16—28no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8#4112yes—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4C2×D412yes—
195C3×C42—24no—
196C3×C42—42no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#216—48no—
200C4⋊Dic3—24no—
201C4⋊Dic3—22no—
202C4⋊Dic3—24no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—42no—
206C4⋊Dic3—81no—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209Dic3⋊C4—28no—
210Dic3⋊C4—24no—
211Dic3⋊C4—82no—
212C4×Dic3—28no—
213C4×Dic3—22no—
214C4×Dic3—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C22×Q8—31no—
219C22×Q8—32no—
220C22×Q8C22×D311yescharacteristic
221C22×Q8C22×D311yescharacteristic
222C2×Q16—22no—
223C2×Q16—22no—
224C2×Q16—22no—
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—122no—
228C22×C8—32no—
229C22×C8C22×D311yescharacteristic
230C22×C8C22×D311yescharacteristic
231C4⋊Q8—34no—
232C4⋊Q8—64no—
233C4⋊Q8—64no—
234C4⋊Q8—62no—
235#125—68no—
236#125—616no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—68no—
241C4×Q8—64no—
242C4×Q8—64no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8#6714yes—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—32no—
264C2×C42—62no—
265C2×C42C22×D311yescharacteristic
266#107—22no—
267C4⋊C8—62no—
268#103—22no—
269#103—22no—
270#103—64no—
271#103—64no—
272#103—122no—
273#103—122no—
274C4×C8—22no—
275#95—62no—
276#95—62no—
277#95—62no—
278C23×C3C22×D411yescharacteristic
279C3×Q8—22no—
280C3×Q8—22no—
281C3×Q8—22no—
282C3×Q8—22no—
283C3×Q8—41no—
284C3×Q8—41no—
285C3×Q8—41no—
286C3×Q8—41no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4#12012yes—
302C2×C3×C4#12012yes—
303C2×C3×C4C2×Q1612yes—
304C2×C3×C4#13712yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—24no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3C2×Q1614yes—
323Dic6—28no—
324Dic6—24no—
325Dic6—24no—
326Dic6—42no—
327Dic6—42no—
328C3×C8—22no—
329C3×C8—41no—
330C3⋊C8—42no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—34no—
342C2×Q8—34no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×Q8—62no—
349C2×Q8—62no—
350C2×Q8#23512yes—
351C2×Q8C2×#6712yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4C23×D311yescharacteristic
363Q16—28no—
364Q16—128no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—62no—
371C2×C8—62no—
372C2×C8—62no—
373C2×C8#23512yes—
374C4⋊C4—24no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—42no—
379C4⋊C4—68no—
380C4⋊C4—68no—
381C4⋊C4—68no—
382C4⋊C4—64no—
383C4⋊C4—62no—
384C4⋊C4—64no—
385C4⋊C4—64no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—122no—
389C4⋊C4—241no—
390C4⋊C4—242no—
391C4⋊C4C2×#6712yes—
392C42—24no—
393C42—42no—
394C42—68no—
395C42—62no—
396C42—62no—
397C42D3×D412yes—
398C42C2×#6712yes—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54811yescharacteristic
401C22×C3#55011yescharacteristic
402C22×C3C22×Q1611yescharacteristic
403C22×C3C2×#13711yescharacteristic
404C22×C3#57711yescharacteristic
405C22×C3#57811yescharacteristic
406C3×C4—22no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—81no—
411C3×C4#45114yes—
412C3×C4#52511yescharacteristic
413C3×C4#53711yescharacteristic
414C3×C4#54611yescharacteristic
415C3×C4#54711yescharacteristic
416Dic3—24no—
417Dic3—44no—
418Dic3—42no—
419Dic3—82no—
420C23C2×D3×D411yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—22no—
424Q8—22no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428Q8—41no—
429Q8—68no—
430Q8—64no—
431Q8—64no—
432Q8—122no—
433Q8—122no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—34no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—64no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4D3×Q1612yes—
461C2×C4#91812yes—
462C2×C4#93712yes—
463C2×C4#93712yes—
464C2×C4C2×#23511yescharacteristic
465C2×C4C2×#23511yescharacteristic
466C2×C4C2×D3×D411yescharacteristic
467C2×C4#100911yescharacteristic
468C2×C4#101011yescharacteristic
469C2×C4C22×#6711yescharacteristic
470C8—22no—
471C8—41no—
472C8—122no—
473C2×C3#240711yescharacteristic
474C2×C3C2×#45111yescharacteristic
475C2×C3#301311yescharacteristic
476C2×C3#336311yescharacteristic
477C2×C3#337411yescharacteristic
478C2×C3#339511yescharacteristic
479C2×C3#340011yescharacteristic
480C22#555211yescharacteristic
481C22#555311yescharacteristic
482C22C2×D3×Q1611yescharacteristic
483C22C2×#91811yescharacteristic
484C22#577011yescharacteristic
485C22#577111yescharacteristic
486C22C2×#93711yescharacteristic
487C4—22no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—64no—
492C4—81no—
493C4—122no—
494C4—124no—
495C4—242no—
496C4#518614yes—
497C4#555011yescharacteristic
498C4#555111yescharacteristic
499C4#579111yescharacteristic
500C4#583311yescharacteristic
501C3#2419111yescharacteristic, complemented, Sylow
502C2#7969911yescharacteristic
503C2#7970211yescharacteristic
504C2#7971111yescharacteristic
505C2#7971311yescharacteristic
506C2#8240211yescharacteristic
507C2C2×#518611yescharacteristic
508C2#8391311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×Q16C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×C4×Q16D3
hypocenterC3#24191
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
180. C2×Dic6Q16semidirect
501. C3#24191semidirect

Maximal subgroups

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

subgroupindex
C2×#55672
C2×C3×C4×Q162
#824822
C2×#50312
#720732
#716262
#706422
#702842
#702812
#697882
#697712
#241913