Finite Group Databasebeta

← #10844721

Subgroups — #10844721

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82482C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic
3C2×C8×Dic6C211yesmaximal, characteristic, complemented
4#73712C211yesmaximal, characteristic
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70280C212yesmaximal, complemented
7#70257C212yesmaximal, complemented
8#69772C211yesmaximal, characteristic
9#69467C211yesmaximal, characteristic, complemented
10#68900C212yesmaximal, complemented
11#68852C212yesmaximal, complemented
12#25295—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C22×#215C2212yes—
15C2×#875—21no—
16C2×#874C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C2×C3×#103C2211yescharacteristic
19C2×C3×#103C2211yescharacteristic
20C2×C3×C4×C8C2211yescharacteristic
21C3×C4×C4⋊C4C2211yescharacteristic
22#5229C2212yes—
23C2×#821C2212yes—
24C2×#821C2212yes—
25C2×Dic3⋊C8C2212yes—
26C2×C8×Dic3C2212yes—
27#5027—24no—
28#4988C2212yes—
29#4988C2212yes—
30#4982—22no—
31#4972C2212yes—
32C2×#809C2212yes—
33C2×#808C2212yes—
34#4939—21no—
35#4935C2211yescharacteristic
36#4929—21no—
37#4928—21no—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C8×Dic6—24no—
40#4672—22no—
41C3×#340C2211yescharacteristic
42C3×#340C2211yescharacteristic
43#4529C2212yes—
44C2×#460—31no—
45C2×C8×Q8—31no—
46#2582—31no—
47#2548—31no—
48#2065—32no—
49#2055—32no—
50#1968—32no—
51#1957—31no—
52#1948—31no—
53#1934—32no—
54C4×#103D311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×D12—21no—
57C22×D12—21no—
58C22×D12C2312yes—
59C22×Dic6D411yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#215—24no—
72C2×#215—24no—
73C2×#215—42no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4D412yes—
84C2×C4×Dic3C2312yes—
85C22×C3⋊C8C2312yes—
86#875—44no—
87#874—22no—
88#874—22no—
89C4×Dic6—22no—
90C4×Dic6—22no—
91C3×#103—22no—
92C3×#103—22no—
93C3×C4×C8—22no—
94C3×#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#821—24no—
99#821—42no—
100Dic3⋊C8—24no—
101C8×Dic3—24no—
102#809—24no—
103#808—42no—
104C4⋊(C3⋊C8)D414yes—
105C22×SD16—32no—
106C2×#127—31no—
107C2×#124—61no—
108C2×C4×Q8—31no—
109C2×C4×Q8C2×D311yescharacteristic
110#460—64no—
111C8×Q8—64no—
112C2×#106—32no—
113C2×C4⋊C8—31no—
114C2×C4⋊C8—32no—
115C2×#103C2×D311yescharacteristic
116C2×#103C2×D311yescharacteristic
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—32no—
120C2×C4×C8—32no—
121C2×C4×C8C2×D311yescharacteristic
122#398—62no—
123#394—62no—
124#391—32no—
125#391—61no—
126#390—32no—
127#390—32no—
128#390—61no—
129#389—32no—
130#389—61no—
131#384—31no—
132C4×C4⋊C4C2×D311yescharacteristic
133#340—32no—
134#340C2×D311yescharacteristic
135#340C2×D311yescharacteristic
136C23×D3—22no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8#4112yes—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2411yescharacteristic
150C22×Dic3—22no—
151C22×Dic3C2×D412yes—
152C22×Dic3C2×D412yes—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—42no—
160C2×D12—42no—
161C2×D12—42no—
162C2×D12—42no—
163C2×D12—81no—
164C2×D12—81no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6SD1614yescomplemented
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8#4112yes—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—42no—
181C3×C42—42no—
182C3×C42—42no—
183C3×C42C2×D412yes—
184C3×C42#4112yes—
185#215—48no—
186(C2×D3)⋊C4—44no—
187(C2×D3)⋊C4—44no—
188(C2×D3)⋊C4—82no—
189(C2×D3)⋊C4—82no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—42no—
194C4⋊Dic3—81no—
195Dic3⋊C4—24no—
196Dic3⋊C4—82no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8C2×D414yes—
202C22×Q8—31no—
203C22×Q8C22×D311yescharacteristic
204C22×D4—32no—
205C22×D4—61no—
206C22×D4—61no—
207C2×SD16—64no—
208C2×SD16—64no—
209C2×SD16—122no—
210C22×C8—32no—
211C22×C8C22×D311yescharacteristic
212C22×C8C22×D311yescharacteristic
213#127—62no—
214#127—62no—
215#124—124no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—62no—
219C4×Q8—62no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4C22×D311yescharacteristic
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D311yescharacteristic
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—32no—
237C2×C42C22×D311yescharacteristic
238#106—122no—
239C4⋊C8—34no—
240C4⋊C8—64no—
241#103—22no—
242#103—22no—
243#102—64no—
244#102—64no—
245#102—122no—
246C4×C8—22no—
247C4×C8—64no—
248#95—21no—
249#95—62no—
250#95—62no—
251#95—62no—
252C23×C3C22×D411yescharacteristic
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—42no—
258C22×D3—42no—
259C22×D3—42no—
260C3×Q8—22no—
261C3×Q8—22no—
262C3×Q8—41no—
263C3×Q8—41no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—22no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4#12112yes—
279C2×C3×C4#12112yes—
280C2×C3×C4C2×SD1612yes—
281C2×C3×C4#13512yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4H(5,𝔽2)11yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3C2×SD1614yes—
297D12—42no—
298D12—42no—
299D12—42no—
300D12—42no—
301D12—84no—
302D12—84no—
303Dic6—28no—
304C3×C8—22no—
305C3×C8—41no—
306C3⋊C8—24no—
307C24—62no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—34no—
314C2×Q8—62no—
315C2×Q8—62no—
316C2×Q8#23512yes—
317C2×D4—62no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—122no—
326C2×D4—122no—
327C2×D4—241no—
328C2×D4—241no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—62no—
336C22×C4D3×D411yescharacteristic
337C22×C4D3×D411yescharacteristic
338C22×C4C23×D311yescharacteristic
339SD16—128no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—34no—
346C2×C8—62no—
347C2×C8—62no—
348C2×C8#23512yes—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—42no—
356C4⋊C4—64no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—122no—
361C4⋊C4—241no—
362C4⋊C4—242no—
363C22⋊C4—124no—
364C22⋊C4—124no—
365C22⋊C4—242no—
366C22⋊C4—242no—
367C42—42no—
368C42—42no—
369C42—62no—
370C42—62no—
371C42#23512yes—
372C42D3×D412yes—
373C22×C3C2×#12111yescharacteristic
374C22×C3#54811yescharacteristic
375C22×C3#55011yescharacteristic
376C22×C3C22×SD1611yescharacteristic
377C22×C3C2×#13511yescharacteristic
378C22×C3#57611yescharacteristic
379C22×C3#57711yescharacteristic
380C2×D3—82no—
381C2×D3—82no—
382C2×D3—82no—
383C2×D3—82no—
384C2×D3—82no—
385C2×D3—82no—
386C2×D3—82no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—81no—
390C3×C4—81no—
391C3×C4—81no—
392C3×C4#46514yes—
393C3×C4#52911yescharacteristic
394C3×C4#54211yescharacteristic
395C3×C4#54611yescharacteristic
396C3×C4#54811yescharacteristic
397Dic3—24no—
398Dic3—42no—
399Dic3—82no—
400C23—122no—
401C23—122no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23—122no—
406C23—122no—
407C23C2×D3×D411yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—41no—
411Q8—41no—
412Q8—68no—
413D4—122no—
414D4—122no—
415D4—122no—
416D4—122no—
417D4—244no—
418D4—244no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—34no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4#88412yes—
443C2×C4#88412yes—
444C2×C4D3×SD1612yes—
445C2×C4#91912yes—
446C2×C4C2×#23511yescharacteristic
447C2×C4C2×#23511yescharacteristic
448C2×C4C2×#23511yescharacteristic
449C2×C4C2×D3×D411yescharacteristic
450C2×C4#100911yescharacteristic
451C2×C4#101011yescharacteristic
452C8—22no—
453C8—41no—
454C8—64no—
455C2×C3#242611yescharacteristic
456C2×C3C2×#46511yescharacteristic
457C2×C3#306211yescharacteristic
458C2×C3#336211yescharacteristic
459C2×C3#336511yescharacteristic
460C2×C3#339211yescharacteristic
461C2×C3#339611yescharacteristic
462D3—162no—
463C22—242no—
464C22—242no—
465C22—242no—
466C22—242no—
467C22—242no—
468C22—242no—
469C22—242no—
470C22C2×#88411yescharacteristic
471C22#555211yescharacteristic
472C22#555311yescharacteristic
473C22C2×D3×SD1611yescharacteristic
474C22C2×#91911yescharacteristic
475C22#576911yescharacteristic
476C22#577011yescharacteristic
477C4—22no—
478C4—42no—
479C4—64no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—122no—
484C4—242no—
485C4#480714yes—
486C4#547911yescharacteristic
487C4#553411yescharacteristic
488C4#555011yescharacteristic
489C4#555211yescharacteristic
490C3#2529511yescharacteristic, complemented, Sylow
491C2—482no—
492C2#7348511yescharacteristic
493C2C2×#480711yescharacteristic
494C2#7884011yescharacteristic
495C2#7962211yescharacteristic
496C2#7963711yescharacteristic
497C2#7970111yescharacteristic
498C2#7971211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenterC4×#103D3
hypocenterC3#25295
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
1. #82482C2semidirect
3. C2×C8×Dic6C2semidirect
5. C3×C4×#103C2semidirect
6. #70280C2semidirect
7. #70257C2semidirect
9. #69467C2semidirect
10. #68900C2semidirect
11. #68852C2semidirect
54. C4×#103D3semidirect
167. C2×Dic6SD16semidirect
490. C3#25295semidirect

Maximal subgroups

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

subgroupindex
#824822
C2×#50272
C2×C8×Dic62
#737122
C3×C4×#1032
#702802
#702572
#697722
#694672
#689002
#688522
#252953