Finite Group Databasebeta

← #10831772

Subgroups — #10831772

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5570C211yesmaximal, characteristic
2C2×C3×#441C211yesmaximal, characteristic
3#82481C211yesmaximal, characteristic
4C2×#5021C211yesmaximal, characteristic
5#72065C212yesmaximal
6#71627C211yesmaximal, characteristic
7#70566C212yesmaximal
8#70235C212yesmaximal
9#70232C212yesmaximal
10#69838C211yesmaximal, characteristic
11#69828C211yesmaximal, characteristic
12#24344—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×Q8×Dic3C2212yes—
17#5570—24no—
18#5570—24no—
19C2×#890—22no—
20C2×#890C2212yes—
21C2×#869—21no—
22C2×#869C2211yescharacteristic
23C2×C4×Dic6—21no—
24C2×C4×Dic6C2211yescharacteristic
25C3×#441—24no—
26C2×C3×#106C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5229C2212yes—
31C2×#835C2212yes—
32C2×#835C2212yes—
33#5021—24no—
34#4986—21no—
35#4979—21no—
36#4972C2212yes—
37C2×#807C2212yes—
38#4936C2211yescharacteristic
39#4933—22no—
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41#4551C2212yes—
42#4551C2212yes—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#557—31no—
47C2×#477—31no—
48C2×#441D311yescharacteristic
49#2581—31no—
50#2059—32no—
51#2026—32no—
52#1993—31no—
53#1985—31no—
54#1878—32no—
55#1863—32no—
56#1805—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×Dic6—21no—
60C22×Dic6D411yescharacteristic
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C2×C3×Q16Q814yescomplemented
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8D414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77Q8×Dic3—24no—
78Q8×Dic3—24no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2312yes—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4D412yes—
88C2×C4×Dic3—22no—
89C2×C4×Dic3C2312yes—
90C22×C3⋊C8C2312yes—
91#890—216no—
92#890—24no—
93#890—24no—
94#869—28no—
95#869—22no—
96#869D414yes—
97C4×Dic6—28no—
98C4×Dic6—22no—
99C4×Dic6—22no—
100C3×#106—22no—
101C3×#103—22no—
102C3×#103—22no—
103C3×#97—22no—
104#835—24no—
105#835—24no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111#807—24no—
112C4⋊(C3⋊C8)—22no—
113C22×Q16C2×D311yescharacteristic
114#557—64no—
115#557—64no—
116C2×C4⋊Q8—31no—
117C2×C4⋊Q8—61no—
118C2×#125—32no—
119C2×#125—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×Q8—61no—
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×Q8C2×D311yescharacteristic
125#477—64no—
126#441—24no—
127C2×#107—32no—
128C2×#106C2×D311yescharacteristic
129C2×C4⋊C8—31no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103C2×D311yescharacteristic
133C2×#103C2×D311yescharacteristic
134C2×#97C2×D311yescharacteristic
135#400—61no—
136#395—61no—
137#391—32no—
138#389—32no—
139#387—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8C2×D412yes—
155C2×C3×Q8C2×Q812yes—
156C2×C3×Q8C2×Q812yes—
157C2×C3×Q8#4112yes—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169C2×Dic6—22no—
170C2×Dic6—24no—
171C2×Dic6—42no—
172C2×Dic6—44no—
173C3×Q16—28no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8C2×Q812yes—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4C2×D412yes—
186C3×C42—24no—
187C3×C42—42no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190C4⋊Dic3—28no—
191C4⋊Dic3—28no—
192C4⋊Dic3—22no—
193C4⋊Dic3—24no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3C2×D414yes—
198Dic3⋊C4—28no—
199Dic3⋊C4—24no—
200Dic3⋊C4—44no—
201C4×Dic3—28no—
202C4×Dic3—22no—
203C4×Dic3—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C22×Q8—31no—
208C22×Q8—61no—
209C22×Q8C22×D311yescharacteristic
210C22×Q8C22×D311yescharacteristic
211C2×Q16—22no—
212C2×Q16—22no—
213C2×Q16Dic614yescomplemented
214C22×C8—32no—
215C22×C8C22×D311yescharacteristic
216C22×C8C22×D311yescharacteristic
217C4⋊Q8—34no—
218C4⋊Q8—68no—
219C4⋊Q8—62no—
220#125—64no—
221#125—616no—
222#125—64no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—68no—
227C4×Q8—64no—
228C4×Q8—64no—
229C4×Q8—62no—
230C4×Q8—62no—
231C4×Q8#6714yes—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—32no—
237C2×C4⋊C4—32no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4C22×D311yescharacteristic
244C2×C4⋊C4C22×D311yescharacteristic
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—32no—
248C2×C42—62no—
249C2×C42C22×D311yescharacteristic
250#107—64no—
251#106—22no—
252C4⋊C8—62no—
253#103—22no—
254#103—22no—
255#103—64no—
256#103—64no—
257#97—22no—
258#95—62no—
259#95—62no—
260#95—62no—
261#95—61no—
262#95—61no—
263C23×C3C22×D411yescharacteristic
264C3×Q8—22no—
265C3×Q8—22no—
266C3×Q8—22no—
267C3×Q8—22no—
268C3×Q8—22no—
269C3×Q8—41no—
270C3×Q8—41no—
271C3×Q8C22⋊Q814yes—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4#12012yes—
285C2×C3×C4#12012yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×Q811yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291C2×C3×C42−1+411yescharacteristic
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—24no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—24no—
298C2×Dic3—44no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302Dic6—44no—
303Dic6—48no—
304C3×C8—22no—
305C3×C8—41no—
306C3⋊C8—42no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—64no—
316C2×Q8—62no—
317C2×Q8—122no—
318C2×Q8—124no—
319C2×Q8C2×Dic612yes—
320C2×Q8C2×Dic612yes—
321C2×Q8#23512yes—
322C2×Q8C2×#6712yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—32no—
328C22×C4—32no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4D3×D411yescharacteristic
332C22×C4D3×D411yescharacteristic
333C22×C4C23×D311yescharacteristic
334Q16—28no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—62no—
341C2×C8—62no—
342C2×C8—62no—
343C2×C8C2×Dic612yes—
344C4⋊C4—24no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—34no—
349C4⋊C4—42no—
350C4⋊C4—68no—
351C4⋊C4—68no—
352C4⋊C4—68no—
353C4⋊C4—62no—
354C4⋊C4—64no—
355C4⋊C4—64no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—124no—
360C4⋊C4C2×#6712yes—
361C42—24no—
362C42—42no—
363C42—68no—
364C42—62no—
365C42—62no—
366C42D3×D412yes—
367C42C2×#6712yes—
368C22×C3C2×#12011yescharacteristic
369C22×C3#54811yescharacteristic
370C22×C3D4×Q811yescharacteristic
371C22×C3#57512yes—
372C22×C3#57812yes—
373C3×C4—24no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—81no—
379C3×C4C2×C22⋊Q811yescharacteristic
380C3×C4#53611yescharacteristic
381C3×C4#54811yescharacteristic
382C3×C4D4×Q811yescharacteristic
383Dic3—42no—
384Dic3—44no—
385Dic3—42no—
386Dic3—82no—
387C23C2×D3×D411yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—22no—
391Q8—22no—
392Q8—22no—
393Q8—41no—
394Q8—41no—
395Q8—124no—
396Q8—128no—
397Q8#92414yes—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—64no—
413C2×C4—62no—
414C2×C4—64no—
415C2×C4—62no—
416C2×C4—124no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4#93712yes—
421C2×C4#93712yes—
422C2×C4C22×Dic611yescharacteristic
423C2×C4C2×#23511yescharacteristic
424C2×C4C2×D3×D411yescharacteristic
425C2×C4#101011yescharacteristic
426C2×C4#101011yescharacteristic
427C2×C4C22×#6711yescharacteristic
428C8—22no—
429C8—41no—
430C8—122no—
431C2×C3#240911yescharacteristic
432C2×C3#300511yescharacteristic
433C2×C3#301111yescharacteristic
434C2×C3#337712yes—
435C2×C3#339012yes—
436C22D4×Dic611yescharacteristic
437C22#555211yescharacteristic
438C22#577712yes—
439C22#577912yes—
440C22C2×#93711yescharacteristic
441C4—22no—
442C4—22no—
443C4—24no—
444C4—42no—
445C4—42no—
446C4—81no—
447C4—122no—
448C4—124no—
449C4—122no—
450C4—242no—
451C4D4×Dic611yescharacteristic
452C4#555211yescharacteristic
453C4C2×#92411yescharacteristic
454C4#582511yescharacteristic
455C3#2434411yescharacteristic, complemented, Sylow
456C2#7981812yes—
457C2#7982012yes—
458C2#8234911yescharacteristic
459C2#8398011yescharacteristic
460C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#441C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#441D3
hypocenterC3#24344
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
63. C2×C3×Q16Q8semidirect
213. C2×Q16Dic6semidirect
455. C3#24344semidirect

Maximal subgroups

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

subgroupindex
C2×#55702
C2×C3×#4412
#824812
C2×#50212
#720652
#716272
#705662
#702352
#702322
#698382
#698282
#243443