Finite Group Databasebeta

← #10822746

Subgroups — #10822746

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5825C211yesmaximal, characteristic
2C3×#2311C211yesmaximal, characteristic
3C2×#5234C211yesmaximal, characteristic
4C2×#5004C211yesmaximal, characteristic, complemented
5C2×#4998C211yesmaximal, characteristic, complemented
6C22×#810C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#69527C214yesmaximal, complemented
9#69526C214yesmaximal, complemented
10#23487—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15#5825—24no—
16#5825—24no—
17C2×#944—21no—
18C2×#936—21no—
19C2×#935—22no—
20C2×#926C2211yescharacteristic
21C2×#924C2214yes—
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C3×#391—21no—
25C3×#390—21no—
26C3×#386—22no—
27C3×#380C2214yes—
28#5234—24no—
29C2×#830C2212yes—
30C2×#830C2212yes—
31#5004—24no—
32#4998—24no—
33C2×#810—24no—
34C2×#810C2214yes—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#808C2212yes—
38C2×#807C2212yes—
39#4529C2214yes—
40C2×#536—31no—
41C2×#427—31no—
42C2×#419—31no—
43C22×#103—31no—
44C2×#408—31no—
45#2311D311yescharacteristic
46#2036—34no—
47#2035—34no—
48#1736—31no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C22×C3×Q8D411yescharacteristic
56C22×C3×D4D412yes—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—21no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—21no—
67C22×Dic6—22no—
68C22×Dic6D414yes—
69C22×Dic6C2312yes—
70C3×#117—22no—
71C3×#117—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C42—21no—
80C2×#217—24no—
81C2×#217—21no—
82#944—28no—
83#936—28no—
84#935—216no—
85#926—22no—
86#926D414yes—
87C2×C4⋊Dic3C2312yes—
88#924—28no—
89#924—28no—
90C2×Dic3⋊C4—24no—
91C2×C4×Dic3—21no—
92C2×#208—22no—
93C2×#208—22no—
94C2×#208—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8D414yes—
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C3×#95—24no—
102#830—24no—
103#830—24no—
104#810—28no—
105#810—24no—
106#810—24no—
107#808—24no—
108#807—24no—
109C22×#41C2×D311yescharacteristic
110C23×Q8—31no—
111C22×M16—31no—
112C23×C8—31no—
113#536—64no—
114#536—64no—
115C2×C4⋊Q8—61no—
116C2×#124—61no—
117C2×#123—62no—
118C2×C22⋊Q8—34no—
119C2×#117—31no—
120C2×#117C2×D311yescharacteristic
121C22×C4⋊C4C2×D311yescharacteristic
122#427—64no—
123C2×#107—32no—
124C2×#106—32no—
125#419—64no—
126C2×#103—32no—
127C2×#103—32no—
128C2×#103—34no—
129C2×#103—64no—
130#408—64no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133#391—21no—
134#390—21no—
135#386—22no—
136#380C2×D314yes—
137#340—34no—
138C24×C3C2×D412yes—
139C24×C3C2×D411yescharacteristic
140C3×#41—44no—
141C3×#41—44no—
142C3×#41—44no—
143C3×#41—44no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—42no—
162C22×C3×C4—82no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4#4112yes—
169C22×C3×C4#4112yes—
170C22×C3×C4#4112yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—24no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—42no—
176C22×Dic3—42no—
177C2×Dic6—28no—
178C2×Dic6—28no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—24no—
184C2×Dic6—22no—
185C2×Dic6—44no—
186C2×Dic6—42no—
187C2×Dic6—42no—
188C2×Dic6—44no—
189C2×Dic6C2×D414yes—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—42no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—84no—
197C3×C42—42no—
198#217—216no—
199#217—24no—
200C4⋊Dic3—22no—
201C4⋊Dic3C2×D414yes—
202Dic3⋊C4—216no—
203C4×Dic3—24no—
204#208—44no—
205C2×C3⋊C8—28no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8C2×D412yes—
214C2×C3⋊C8C2×D412yes—
215C2×#41—22no—
216C2×#41—22no—
217C2×#41—22no—
218C2×#41—22no—
219C2×#41—22no—
220C2×#41—22no—
221C22×Q8—34no—
222C22×Q8—32no—
223C22×Q8—62no—
224C22×Q8—61no—
225C22×Q8—62no—
226C22×Q8—62no—
227C22×Q8—62no—
228C22×Q8—62no—
229C22×Q8#6711yescharacteristic
230C22×D4#6712yes—
231C22×D4#6711yescharacteristic
232C23×C4—21no—
233C23×C4—61no—
234C23×C4C22×D312yes—
235C23×C4C22×D311yescharacteristic
236C2×M16—62no—
237C2×M16—62no—
238C2×M16—62no—
239C22×C8—34no—
240C22×C8—31no—
241C22×C8—31no—
242C22×C8—32no—
243C22×C8—62no—
244C22×C8—62no—
245C4⋊Q8—68no—
246#124—68no—
247#123—616no—
248C22⋊Q8—68no—
249C22⋊Q8—68no—
250#117—22no—
251#117—22no—
252#117—34no—
253#117—62no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4C22×D312yes—
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—64no—
265C2×C42—21no—
266C2×C42—61no—
267#107—64no—
268#106—64no—
269#103—64no—
270#103—64no—
271#103—68no—
272C22⋊C8—64no—
273C22⋊C8—64no—
274#95—24no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3#12012yes—
283C23×C3#12112yes—
284C23×C3#12312yes—
285C23×C3C22×D411yescharacteristic
286C3×Q8—44no—
287C3×D4—48no—
288C3×D4—44no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—44no—
300C2×C3×C4—42no—
301C2×C3×C4—84no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12114yes—
304C2×C3×C4#12114yes—
305C2×C3×C4#12112yes—
306C2×C3×C4#12314yes—
307C2×C3×C4#12312yes—
308C2×C3×C4#12414yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—48no—
322Dic6—24no—
323Dic6—24no—
324Dic6—42no—
325Dic6—42no—
326Dic6—48no—
327C3⋊C8—22no—
328C3⋊C8—22no—
329C3⋊C8—42no—
330C24C2×#6712yes—
331C24C2×#6711yescharacteristic
332#41—44no—
333#41—44no—
334#41—44no—
335#41—44no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—34no—
340C2×Q8—64no—
341C2×Q8—62no—
342C2×Q8—68no—
343C2×Q8—68no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×Q8—124no—
349C2×Q8—124no—
350C2×Q8—122no—
351C2×Q8—122no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—64no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—82no—
371C22×C4—122no—
372C22×C4—122no—
373C22×C4#23512yes—
374C22×C4#23512yes—
375C22×C4D3×D412yes—
376C22×C4D3×D412yes—
377C22×C4D3×D412yes—
378C22×C4#23712yes—
379C22×C4C2×#6712yes—
380C22×C4C2×#6711yescharacteristic
381C22×C4C23×D311yescharacteristic
382M16—124no—
383C2×C8—32no—
384C2×C8—32no—
385C2×C8—68no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—34no—
398C4⋊C4—42no—
399C4⋊C4—616no—
400C4⋊C4—62no—
401C22⋊C4—42no—
402C22⋊C4—616no—
403C22⋊C4—64no—
404C22⋊C4—84no—
405C42—42no—
406C42—64no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—44no—
412C22×C3—42no—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3#39414yes—
417C22×C3C2×#12011yescharacteristic
418C22×C3C2×#12111yescharacteristic
419C22×C3C2×#12311yescharacteristic
420C22×C3#57712yes—
421C22×C3#57812yes—
422C3×C4—42no—
423C3×C4—82no—
424C3×C4—82no—
425C3×C4#39414yes—
426C3×C4C2×#12111yescharacteristic
427C3×C4C2×#12111yescharacteristic
428C3×C4C2×#12311yescharacteristic
429C3×C4C2×#12411yescharacteristic
430Dic3—44no—
431Dic3—44no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23#92912yes—
440C23#93712yes—
441C23#93912yes—
442C23C22×#6711yescharacteristic
443Q8—44no—
444Q8—64no—
445Q8—64no—
446Q8—128no—
447Q8—122no—
448Q8—122no—
449D4—48no—
450D4—44no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—64no—
464C2×C4—64no—
465C2×C4—84no—
466C2×C4—128no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4#88314yes—
472C2×C4#88414yes—
473C2×C4#88414yes—
474C2×C4#88514yes—
475C2×C4#92912yes—
476C2×C4#93712yes—
477C2×C4#93912yes—
478C2×C4C2×#23511yescharacteristic
479C2×C4C2×#23511yescharacteristic
480C2×C4C2×D3×D411yescharacteristic
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×D3×D411yescharacteristic
483C2×C4C2×#23711yescharacteristic
484C8—62no—
485C8—62no—
486C8—122no—
487C2×C3—24no—
488C2×C3—82no—
489C2×C3C2×#39411yescharacteristic
490C2×C3#301312yes—
491C2×C3#305711yescharacteristic
492C2×C3#305911yescharacteristic
493C2×C3#309712yes—
494C22—24no—
495C22—22no—
496C22—22no—
497C22—24no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22#495914yes—
504C22C2×#92911yescharacteristic
505C22C2×#93711yescharacteristic
506C22C2×#93911yescharacteristic
507C22#583612yes—
508C22#583812yes—
509C4—42no—
510C4—82no—
511C4—82no—
512C4—124no—
513C4—124no—
514C4#495914yes—
515C4C2×#88311yescharacteristic
516C4C2×#88411yescharacteristic
517C4C2×#88411yescharacteristic
518C4C2×#88511yescharacteristic
519C3#2348711yescharacteristic, complemented, Sylow
520C2—24no—
521C2—82no—
522C2C2×#495911yescharacteristic
523C2#8257312yes—
524C2#8337711yescharacteristic
525C2#8339911yescharacteristic
526C2#8342712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2311C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#2311D3
hypocenterC3#23487
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
4. C2×#5004C2semidirect
5. C2×#4998C2semidirect
6. C22×#810C2semidirect
7. #71558C2semidirect
8. #69527C2semidirect
9. #69526C2semidirect
519. C3#23487semidirect

Maximal subgroups

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

subgroupindex
C2×#58252
C3×#23112
C2×#52342
C2×#50042
C2×#49982
C22×#8102
#715582
#695272
#695262
#234873