Finite Group Databasebeta

← #10848845

Subgroups — #10848845

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5518C211yesmaximal, characteristic, complemented
2C3×#2620C211yesmaximal, characteristic
3C2×#5060C211yesmaximal, characteristic, complemented
4C2×#5056C211yesmaximal, characteristic, complemented
5#71700C212yesmaximal, complemented
6#71685C212yesmaximal
7#71582C211yesmaximal, characteristic, complemented
8#69690C212yesmaximal, complemented
9#69660C212yesmaximal, complemented
10#69583C211yesmaximal, characteristic, complemented
11#69579C211yesmaximal, characteristic, complemented
12#25612—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×#216C2212yes—
18C22×#214C2212yes—
19C2×#924C2212yes—
20#5518—24no—
21#5518—24no—
22C2×#889—22no—
23C2×#879—22no—
24C2×#870C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2212yes—
27C3×#396—22no—
28C3×#394—22no—
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#384C2211yescharacteristic
32C2×#835C2212yes—
33C2×#832C2212yes—
34C2×#830C2212yes—
35#5060—24no—
36#5056—24no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#803C2211yescharacteristic
40C2×C4×C3⋊C8C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C2×#541—31no—
45C2×#497—31no—
46C2×#488—31no—
47#2620D311yescharacteristic
48#2001—32no—
49#2000—32no—
50#1992—31no—
51#1956—31no—
52#1880—32no—
53#1878—32no—
54#1750—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×Dic6C2312yes—
60C3×#124—22no—
61C3×#124—22no—
62C3×C22⋊Q8—42no—
63C3×#121—42no—
64C3×#117—22no—
65C3×#117—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#217—22no—
79C2×#216—24no—
80C2×#216—24no—
81C2×#216—42no—
82C2×#214—24no—
83C2×#214—24no—
84C2×#214—42no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3C2312yes—
87#924—24no—
88#924—24no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—22no—
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2312yes—
95#889—216no—
96#879—216no—
97#870—22no—
98#870D414yescomplemented
99#869—22no—
100#869D414yescomplemented
101C4×Dic6—24no—
102C4×Dic6—24no—
103C3×#95—22no—
104#835—24no—
105#832—24no—
106#830—24no—
107#810—42no—
108#810—42no—
109#803—22no—
110C4×C3⋊C8—22no—
111C22×Q16—32no—
112C22×SD16—32no—
113#541—64no—
114#541—64no—
115C2×C4⋊Q8—31no—
116C2×#126—62no—
117C2×#125—31no—
118C2×#125—62no—
119C2×#124C2×D311yescharacteristic
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8C2×D311yescharacteristic
122C2×#121C2×D311yescharacteristic
123C2×C4×Q8—32no—
124C2×#117C2×D311yescharacteristic
125#497—64no—
126#488—64no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#102—32no—
131C2×C22⋊C8—32no—
132C2×#97—31no—
133C2×C4×C8—31no—
134#396—22no—
135#394—22no—
136#391C2×D311yescharacteristic
137#390C2×D311yescharacteristic
138#384C2×D311yescharacteristic
139#340—32no—
140#340—32no—
141#336—32no—
142C24×C3—21no—
143C24×C3C2×D411yescharacteristic
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—42no—
153C2×C3×D4—82no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6C2×D414yes—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—82no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—82no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42C2×D412yes—
186C3×C42C2×D412yes—
187#217—28no—
188#216—48no—
189#214—48no—
190C4⋊Dic3—28no—
191C4⋊Dic3—22no—
192C4⋊Dic3C2×D414yes—
193Dic3⋊C4—28no—
194Dic3⋊C4—28no—
195Dic3⋊C4—28no—
196C4×Dic3—28no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8#4114yes—
203C22×Q8—32no—
204C22×Q8C22×D311yescharacteristic
205C22×D4—21no—
206C22×D4C22×D311yescharacteristic
207C23×C4C22×D311yescharacteristic
208C2×Q16—64no—
209C2×Q16—64no—
210C2×Q16—122no—
211C2×SD16—64no—
212C2×SD16—64no—
213C2×SD16—122no—
214C22×C8—32no—
215C22×C8—32no—
216C4⋊Q8—34no—
217C4⋊Q8—62no—
218#126—616no—
219#125—34no—
220#125—616no—
221#125—62no—
222#124—22no—
223#124—22no—
224C22⋊Q8—42no—
225C22⋊Q8—64no—
226C22⋊Q8—64no—
227#121—42no—
228C4×Q8—64no—
229C4×Q8—64no—
230#117—22no—
231#117—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4#6712yes—
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4#6712yes—
248C2×C42—62no—
249C2×C42C22×D311yescharacteristic
250C2×C42C22×D311yescharacteristic
251#103—64no—
252#103—122no—
253#103—122no—
254#102—64no—
255C22⋊C8—64no—
256#97—62no—
257C4×C8—62no—
258#95—22no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3C22×D411yescharacteristic
270C3×Q8—42no—
271C3×Q8—42no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—84no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4#12014yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C2×#4112yes—
296C2×C3×C4H(5,𝔽2)12yes—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303Dic6—24no—
304Dic6—24no—
305Dic6—42no—
306Dic6—42no—
307C3⋊C8—24no—
308C3⋊C8—42no—
309C24—21no—
310C24C2×#6711yescharacteristic
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—34no—
316C2×Q8—62no—
317C2×Q8—62no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—62no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—42no—
326C2×D4—82no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4C2×#6712yes—
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6712yes—
339C22×C4C23×D311yescharacteristic
340Q16—128no—
341SD16—128no—
342C2×C8—34no—
343C2×C8—62no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—34no—
352C4⋊C4—42no—
353C4⋊C4—68no—
354C4⋊C4—68no—
355C4⋊C4—68no—
356C4⋊C4—68no—
357C4⋊C4—62no—
358C4⋊C4—82no—
359C22⋊C4—24no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C22⋊C4—68no—
364C22⋊C4—82no—
365C42—21no—
366C42—21no—
367C42—68no—
368C42D3×D412yes—
369C42D3×D412yes—
370C22×C3—42no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—42no—
375C22×C3—82no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—81no—
379C22×C3—82no—
380C22×C3C2×#12011yescharacteristic
381C22×C3#53211yescharacteristic
382C22×C3#53511yescharacteristic
383C22×C3#57712yes—
384C22×C3#57812yes—
385C3×C4—24no—
386C3×C4—41no—
387C3×C4—42no—
388C3×C4—81no—
389C3×C4—81no—
390C3×C4—81no—
391C3×C4#54612yes—
392C3×C4#55012yes—
393Dic3—44no—
394Dic3—44no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—22no—
399C23—22no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23C22×#6711yescharacteristic
406Q8—42no—
407Q8—42no—
408Q8—64no—
409Q8—64no—
410Q8—122no—
411Q8—122no—
412D4—42no—
413D4—42no—
414D4—84no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4#93714yes—
439C2×C4C2×D3×D411yescharacteristic
440C2×C4C2×D3×D411yescharacteristic
441C2×C4#100412yes—
442C2×C4D3×#4112yes—
443C8—64no—
444C8—122no—
445C2×C3—81no—
446C2×C3—161no—
447C2×C3#243111yescharacteristic
448C2×C3#301312yes—
449C2×C3#317812yes—
450C2×C3#322211yescharacteristic
451C2×C3#322311yescharacteristic
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—42no—
457C22—82no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—82no—
462C22C2×#93711yescharacteristic
463C22#580511yescharacteristic
464C22#583211yescharacteristic
465C22#583612yes—
466C22#583812yes—
467C4—24no—
468C4—41no—
469C4—42no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4—124no—
474C4—124no—
475C4#559912yes—
476C4#561212yes—
477C3#2561211yescharacteristic, complemented, Sylow
478C2—81no—
479C2—161no—
480C2#8265211yescharacteristic
481C2#8339311yescharacteristic
482C2#8341411yescharacteristic
483C2#8342712yes—
484C2#8345812yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2620C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2620D3
hypocenterC3#25612
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2620

Complements

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

NH ≅ G/Ntype
1. C2×#5518C2semidirect
3. C2×#5060C2semidirect
4. C2×#5056C2semidirect
5. #71700C2semidirect
7. #71582C2semidirect
8. #69690C2semidirect
9. #69660C2semidirect
10. #69583C2semidirect
11. #69579C2semidirect
98. #870D4semidirect
100. #869D4semidirect
477. C3#25612semidirect

Maximal subgroups

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

subgroupindex
C2×#55182
C3×#26202
C2×#50602
C2×#50562
#717002
#716852
#715822
#696902
#696602
#695832
#695792
#256123