Finite Group Databasebeta

← #10690527

Subgroups — #10690527

Order 768.

610 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4685C211yesmaximal, characteristic
2C4×#897C211yesmaximal, characteristic
3#70475C211yesmaximal, characteristic
4C2×#4484C211yesmaximal, characteristic, complemented
5C2×#4456C211yesmaximal, characteristic, complemented
6C3×#1279C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9073—31nomaximal, Sylow
9C2×#897C411yescharacteristic
10C2×C42×D3C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×(C2×D3)⋊C8—21no—
16C2×(C2×D3)⋊C8—21no—
17C2×Dic3⋊C8—21no—
18#4987C411yescharacteristic
19#4984C411yescharacteristic
20#4973C411yescharacteristic
21C4×(C2×D3)⋊C4—21no—
22C4×C4⋊Dic3—21no—
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4685—22no—
26#4685—22no—
27#4668—21no—
28#4653—21no—
29#4550—21no—
30#4484—22no—
31#4456—22no—
32C2×#431—31no—
33C4×C22⋊Q8—31no—
34#1797—31no—
35C2×Q8⋊C8—31no—
36C2×Q8⋊C8—31no—
37#1279—31no—
38#1279D311yescharacteristic, complemented
39C22×C4×D3C2×C411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3C2×C411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61#897—22no—
62#897C814yescomplemented
63C42×D3—21no—
64C42×D3—21no—
65C42×D3—21no—
66C42×D3—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4×C8—21no—
74C3×C4×C8D412yes—
75C3×#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80(C2×D3)⋊C8—24no—
81(C2×D3)⋊C8—42no—
82Dic3⋊C8—24no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4×C3⋊C8—21no—
86C4×C3⋊C8D412yes—
87C2×C22⋊Q8—31no—
88C2×C4×Q8—31no—
89C22×C42—31no—
90#431—62no—
91#431—62no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8—61no—
94C2×C4⋊C8C2×D311yescharacteristic
95C2×C22⋊C8—61no—
96C2×C22⋊C8—61no—
97C2×C4×C8—31no—
98C2×C4×C8C2×D311yescharacteristic
99#391—31no—
100#389—31no—
101#387—61no—
102#385—61no—
103#380—31no—
104C4×C4⋊C4—61no—
105C4×C4⋊C4C2×D311yescharacteristic
106C4×C22⋊C4—61no—
107#336—61no—
108Q8⋊C8—62no—
109Q8⋊C8—62no—
110C23×D3C22⋊C411yescharacteristic
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×C4×D3—21no—
133C2×C4×D3—41no—
134C2×C4×D3—41no—
135C2×C4×D3C2×C812yes—
136C2×C4×D3M1612yes—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—21no—
142C2×Dic6C2×C812yes—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4C2×C812yes—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158C3×C42#4111yescharacteristic
159(C2×D3)⋊C4—24no—
160(C2×D3)⋊C4—42no—
161C4⋊Dic3—24no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—42no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C4×Dic3—21no—
172C4×Dic3—21no—
173C4×Dic3—41no—
174C4×Dic3—41no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C22×Q8—31no—
182C23×C4—31no—
183C23×C4—31no—
184C23×C4—31no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—61no—
188C22×C8—61no—
189C22⋊Q8—34no—
190C22⋊Q8—62no—
191C4×Q8—61no—
192C4×Q8—61no—
193C4×Q8—61no—
194C4×Q8—61no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4C4×D311yescharacteristic
200C2×C4⋊C4C4×D311yescharacteristic
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C42—21no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42—61no—
210C2×C42—61no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—64no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218C22⋊C8—64no—
219C22⋊C8—122no—
220C4×C8—21no—
221C4×C8—32no—
222C4×C8—61no—
223C4×C8D1212yes—
224#95—21no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95—61no—
229C23×C3C2×C22⋊C411yescharacteristic
230C22×D3—21no—
231C22×D3—21no—
232C22×D3—21no—
233C22×D3—21no—
234C22×D3—21no—
235C22×D3—21no—
236C22×D3C22⋊C812yes—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×C3×C4C22×C811yescharacteristic
252C2×C3×C4C2×M1611yescharacteristic
253C2×C3×C4#13111yescharacteristic
254C2×C3×C4#13111yescharacteristic
255C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
256C2×C3×C4#13711yescharacteristic
257C2×C3×C4#13711yescharacteristic
258C2×C3×C4#13711yescharacteristic
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—22no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3C22⋊C812yes—
274C4×D3—22no—
275C4×D3—22no—
276C4×D3—22no—
277C4×D3—22no—
278C4×D3—42no—
279C4×D3—42no—
280C4×D3—41no—
281C4×D3—41no—
282C4×D3—41no—
283C4×D3—41no—
284Dic6—22no—
285Dic6—22no—
286Dic6—41no—
287Dic6—41no—
288C3×C8—42no—
289C3×C8—81no—
290C3⋊C8—42no—
291C3⋊C8—81no—
292C24—31no—
293C2×Q8—32no—
294C2×Q8—61no—
295C2×Q8—61no—
296C2×Q8—61no—
297C2×Q8—61no—
298C2×Q8—61no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4C2×C4×D311yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4D3×D411yescharacteristic
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—62no—
332C2×C8—121no—
333C2×C8—121no—
334C2×C8—121no—
335C2×C8—121no—
336C2×C8—121no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—64no—
341C4⋊C4—61no—
342C4⋊C4—61no—
343C4⋊C4—122no—
344C4⋊C4—121no—
345C4⋊C4—121no—
346C4⋊C4D3×C812yes—
347C22⋊C4—64no—
348C22⋊C4—122no—
349C42—41no—
350C42—41no—
351C42—61no—
352C42—61no—
353C42—61no—
354C42—61no—
355C42—61no—
356C42—61no—
357C42—121no—
358C42—121no—
359C42C2×D1211yescharacteristic
360C42#23511yescharacteristic
361C42D3×D411yescharacteristic
362C42#23911yescharacteristic
363C22×C3#38611yescharacteristic
364C22×C3C2×C22⋊C811yescharacteristic
365C22×C3#40811yescharacteristic
366C22×C3#41711yescharacteristic
367C22×C3#41911yescharacteristic
368C22×C3#42111yescharacteristic
369C22×C3#42111yescharacteristic
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—41no—
374C2×D3—41no—
375C2×D3—41no—
376C2×D3—41no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—42no—
380C3×C4—81no—
381C3×C4C8×D411yescharacteristic
382C3×C4C8×D411yescharacteristic
383C3×C4#43511yescharacteristic
384C3×C4#43511yescharacteristic
385C3×C4#46811yescharacteristic
386C3×C4#46811yescharacteristic
387C3×C4#47011yescharacteristic
388C3×C4#47011yescharacteristic
389Dic3—22no—
390Dic3—42no—
391Dic3—41no—
392Dic3—41no—
393Dic3—81no—
394C23—32no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23—61no—
400C23—61no—
401C23D3×C22⋊C411yescharacteristic
402Q8—62no—
403Q8—62no—
404Q8—121no—
405Q8—121no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—32no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—62no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4C4×D1211yescharacteristic
440C2×C4#88211yescharacteristic
441C2×C4#88411yescharacteristic
442C2×C4#89311yescharacteristic
443C2×C4#89411yescharacteristic
444C2×C4#89511yescharacteristic
445C2×C4C2×D3×C811yescharacteristic
446C2×C4#90111yescharacteristic
447C2×C4D3×M1611yescharacteristic
448C2×C4#90711yescharacteristic
449C2×C4#91111yescharacteristic
450C2×C4#91511yescharacteristic
451C2×C4#91511yescharacteristic
452C2×C4#91811yescharacteristic
453C8—42no—
454C8—81no—
455C8—122no—
456C8—241no—
457C2×C3#147711yescharacteristic
458C2×C3#147911yescharacteristic
459C2×C3#149511yescharacteristic
460C2×C3#149711yescharacteristic
461C2×C3#186911yescharacteristic
462C2×C3#189411yescharacteristic
463C2×C3#189711yescharacteristic
464D3—42no—
465C22—62no—
466C22—62no—
467C22—62no—
468C22—121no—
469C22—121no—
470C22—121no—
471C22—121no—
472C22#466511yescharacteristic
473C22D3×C22⋊C811yescharacteristic
474C22#472711yescharacteristic
475C22#477111yescharacteristic
476C22#480211yescharacteristic
477C22#482111yescharacteristic
478C22#482111yescharacteristic
479C4—21no—
480C4—21no—
481C4—42no—
482C4—62no—
483C4—81no—
484C4—121no—
485C4—122no—
486C4—121no—
487C4—241no—
488C4C8×D1211yescharacteristic
489C4#470611yescharacteristic
490C4#478011yescharacteristic
491C4#481611yescharacteristic
492C4#483411yescharacteristic
493C4#483611yescharacteristic
494C4#486711yescharacteristic
495C4#488611yescharacteristic
496C3#907311yescharacteristic, complemented, Sylow
497C2—122no—
498C2#6816611yescharacteristic
499C2#6817811yescharacteristic
500C2#6823111yescharacteristic
501C2#6824311yescharacteristic
502C2#6866811yescharacteristic
503C2#6875711yescharacteristic
504C2#6888411yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9073
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
4. C2×#4484C2semidirect
5. C2×#4456C2semidirect
6. C3×#1279C2semidirect
7. #67619C2semidirect
38. #1279D3semidirect
62. #897C8semidirect
496. C3#9073semidirect

Maximal subgroups

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

subgroupindex
C2×#46852
C4×#8972
#704752
C2×#44842
C2×#44562
C3×#12792
#676192
#90733