Finite Group Databasebeta

← #10844723

Subgroups — #10844723

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
1C3×#2582C211yesmaximal, characteristic, complemented
2#82476C211yesmaximal, characteristic
3C2×#5027C211yesmaximal, characteristic
4C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
5C4×#835C211yesmaximal, characteristic, complemented
6#69680C212yesmaximal, complemented
7#69676C212yesmaximal, complemented
8#69629C212yesmaximal, complemented
9#69615C212yesmaximal, complemented
10#69580C211yesmaximal, characteristic
11#69568C211yesmaximal, characteristic, complemented
12#25295—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947—21no—
16C22×#215C2212yes—
17C2×#874C2211yescharacteristic
18C3×#398—22no—
19C3×#391—21no—
20C3×#391C2212yes—
21C3×#389—21no—
22C3×#389C2212yes—
23C3×#384C2211yescharacteristic
24C2×#835C2211yescharacteristic
25C2×#835C2211yescharacteristic
26#5027—24no—
27Q8×C3⋊C8—24no—
28#4991—22no—
29#4988—21no—
30C2×#809C2212yes—
31C2×#809C2212yes—
32C2×#809C2212yes—
33C2×#808C2212yes—
34#4939C2212yes—
35#4939C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C2×C4×C3⋊C8C2212yes—
40C2×C4×C3⋊C8C2211yescharacteristic
41#4529C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#460—31no—
45C2×C8×Q8—31no—
46#2582D311yescharacteristic, complemented
47#2548—31no—
48#2065—32no—
49#2055—32no—
50#1968—32no—
51#1957—31no—
52#1948—31no—
53#1934—32no—
54C4×#103—31no—
55C22×C3×Q8D411yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C22×D12—21no—
58C22×D12—21no—
59C22×D12C2312yes—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42C2312yes—
74C2×C3×C42C2311yescharacteristic
75#947—44no—
76C2×#215—24no—
77C2×#215—24no—
78C2×#215—42no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4×Dic3—21no—
86C22×C3⋊C8C2311yescharacteristic
87C22×C3⋊C8C2312yes—
88C22×C3⋊C8C2311yescharacteristic
89#874—22no—
90#874—22no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94#835—22no—
95#835—22no—
96C3⋊#95—21no—
97#809—24no—
98#809—24no—
99#809—42no—
100#808—42no—
101C4⋊(C3⋊C8)—24no—
102C4⋊(C3⋊C8)D414yes—
103C4×C3⋊C8—22no—
104C4×C3⋊C8—24no—
105C22×SD16—32no—
106C2×#127—31no—
107C2×#124—61no—
108C2×C4×Q8C2×D311yescharacteristic
109C2×C4×Q8C2×D311yescharacteristic
110#460—64no—
111C8×Q8—64no—
112C2×#106—32no—
113C2×C4⋊C8—31no—
114C2×C4⋊C8—32no—
115C2×#103—31no—
116C2×#103—31no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—32no—
120C2×C4×C8—32no—
121C2×C4×C8—31no—
122#398—22no—
123#394—62no—
124#391—21no—
125#391C2×D312yes—
126#390—32no—
127#390—32no—
128#390—61no—
129#389—21no—
130#389C2×D312yes—
131#384C2×D311yescharacteristic
132C4×C4⋊C4—31no—
133#340—32no—
134#340—31no—
135#340—31no—
136C23×D3—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8SD1614yescomplemented
145C2×C3×Q8#4112yes—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—21no—
155C22×Dic3—21no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—42no—
163C2×D12—42no—
164C2×D12—42no—
165C2×D12—42no—
166C2×D12—81no—
167C2×D12—81no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—81no—
177C3×C4⋊C4—82no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42—42no—
181C3×C42C2×D412yes—
182C3×C42#4112yes—
183#215—48no—
184(C2×D3)⋊C4—44no—
185(C2×D3)⋊C4—44no—
186(C2×D3)⋊C4—82no—
187(C2×D3)⋊C4—82no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4×Dic3—42no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8C2×D414yes—
201C2×C3⋊C8#4112yes—
202C22×Q8#6711yescharacteristic
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×C8—31no—
212C22×C8—31no—
213#127—62no—
214#127—62no—
215#124—124no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4#6711yescharacteristic
227C2×C4⋊C4#6712yes—
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D312yes—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C42—21no—
235C2×C42—61no—
236C2×C42C22×D312yes—
237C2×C42C22×D311yescharacteristic
238#106—122no—
239C4⋊C8—34no—
240C4⋊C8—64no—
241#103—62no—
242#103—62no—
243#102—64no—
244#102—64no—
245#102—122no—
246C4×C8—64no—
247C4×C8—62no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—61no—
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—28no—
261C3×Q8—22no—
262C3×Q8—22no—
263C3×Q8—41no—
264C3×Q8—41no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4#12112yes—
282C2×C3×C4#12112yes—
283C2×C3×C4C2×SD1614yes—
284C2×C3×C4C2×SD1612yes—
285C2×C3×C4#13512yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4H(5,𝔽2)11yescharacteristic
291C2×C3×C42−1+411yescharacteristic
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298D12—42no—
299D12—42no—
300D12—42no—
301D12—42no—
302D12—84no—
303D12—84no—
304C3⋊C8—24no—
305C3⋊C8—22no—
306C3⋊C8—41no—
307C24—62no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8#21514yescomplemented
316C2×Q8#23912yes—
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—22no—
331C22×C4—21no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4C2×#6712yes—
335C22×C4C2×#6711yescharacteristic
336C22×C4C2×#6712yes—
337C22×C4C2×#6711yescharacteristic
338C22×C4C23×D311yescharacteristic
339SD16—128no—
340C2×C8—34no—
341C2×C8—32no—
342C2×C8—61no—
343C2×C8—62no—
344C2×C8—61no—
345C2×C8—62no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8—61no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—61no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—81no—
362C4⋊C4—82no—
363C22⋊C4—124no—
364C22⋊C4—124no—
365C22⋊C4—242no—
366C22⋊C4—242no—
367C42—22no—
368C42—22no—
369C42—42no—
370C42—122no—
371C42D3×D412yes—
372C42#23712yes—
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—24no—
388C3×C4—22no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—82no—
392C3×C4—81no—
393C3×C4#46514yes—
394C3×C4#52911yescharacteristic
395C3×C4#54211yescharacteristic
396C3×C4#54611yescharacteristic
397C3×C4#54811yescharacteristic
398Dic3—81no—
399Dic3—81no—
400C23—122no—
401C23—122no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23—122no—
406C23—122no—
407C23C22×#6711yescharacteristic
408Q8—28no—
409Q8—22no—
410Q8—22no—
411Q8—41no—
412Q8—41no—
413D4—122no—
414D4—122no—
415D4—122no—
416D4—122no—
417D4—244no—
418D4—244no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4#93912yes—
442C2×C4#93912yes—
443C2×C4C2×#21514yes—
444C2×C4C2×#21512yes—
445C2×C4#95012yes—
446C2×C4C2×D3×D411yescharacteristic
447C2×C4C2×#23711yescharacteristic
448C2×C4C2×#23911yescharacteristic
449C2×C4#100711yescharacteristic
450C2×C4D3×#4111yescharacteristic
451C2×C4#100911yescharacteristic
452C8—62no—
453C8—64no—
454C8—121no—
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×#93911yescharacteristic
471C22C22×#21511yescharacteristic
472C22#581711yescharacteristic
473C22C2×#95011yescharacteristic
474C22#583411yescharacteristic
475C22#583511yescharacteristic
476C22#583711yescharacteristic
477C4—24no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—82no—
482C4—81no—
483C4—241no—
484C4—241no—
485C4#504814yes—
486C4#562411yescharacteristic
487C4#563011yescharacteristic
488C4#563311yescharacteristic
489C4#564311yescharacteristic
490C3#2529511yescharacteristic, complemented, Sylow
491C2—482no—
492C2C2×#504811yescharacteristic
493C2#8275511yescharacteristic
494C2#8342211yescharacteristic
495C2#8368511yescharacteristic
496C2#8369711yescharacteristic
497C2#8370311yescharacteristic
498C2#8372011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2582C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2582D3
hypocenterC3#25295
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
1. C3×#2582C2semidirect
4. C2×Q8×C3⋊C8C2semidirect
5. C4×#835C2semidirect
6. #69680C2semidirect
7. #69676C2semidirect
8. #69629C2semidirect
9. #69615C2semidirect
11. #69568C2semidirect
46. #2582D3semidirect
144. C2×C3×Q8SD16semidirect
315. C2×Q8#215semidirect
490. C3#25295semidirect

Maximal subgroups

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

subgroupindex
C3×#25822
#824762
C2×#50272
C2×Q8×C3⋊C82
C4×#8352
#696802
#696762
#696292
#696152
#695802
#695682
#252953