Finite Group Databasebeta

← #10848708

Subgroups — #10848708

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×#5092C211yesmaximal, characteristic, complemented
3C2×#5093C211yesmaximal, characteristic, complemented
4#82076C211yesmaximal, characteristic
5#71787C211yesmaximal, characteristic, complemented
6#71785C211yesmaximal, characteristic, complemented
7#71700C214yesmaximal
8#71632C214yesmaximal, complemented
9C4×#830C211yesmaximal, characteristic
10#25600—31nomaximal, Sylow
11C3×#533—24no—
12C3×#533—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×#117—21no—
21C22×C3×C42C2211yescharacteristic
22C22×#216C2214yes—
23C2×#924C2212yes—
24C2×#869C2211yescharacteristic
25C2×#835C2214yes—
26#5212—21no—
27#5212—21no—
28C2×#830C2212yes—
29#5093—24no—
30#5092—24no—
31C2×#810C2214yes—
32#4935C2211yescharacteristic
33#4929C2212yes—
34#4928—21no—
35#4928—21no—
36C2×C4×C3⋊C8C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4529C2214yes—
39#4524C2212yes—
40C2×#533D311yescharacteristic
41C2×#494—31no—
42C2×#487—31no—
43#2857—31no—
44#2001—34no—
45#1990—31no—
46#1956—31no—
47#1878—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8—21no—
50C22×C3×Q8C2312yes—
51C23×C3×C4D411yescharacteristic
52C23×C3×C4D411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C22×Dic6C2312yes—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8D414yes—
58C3×C4⋊Q8D414yes—
59C3×#125—28no—
60C3×C22⋊Q8—28no—
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—24no—
63C3×C4×Q8—28no—
64C3×C4×Q8—24no—
65C3×C4×Q8—24no—
66C3×#117—28no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#217—22no—
82C2×#216—28no—
83C2×#216—28no—
84C2×#216—44no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90#924—44no—
91C2×Dic3⋊C4—22no—
92C22×C3⋊C8C2312yes—
93C22×C3⋊C8C2312yes—
94#869—22no—
95#869—22no—
96#835—44no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#830—24no—
100#810—28no—
101C4×C3⋊C8—22no—
102C4×C3⋊C8—22no—
103C22×Q16—34no—
104#533—24no—
105#533—24no—
106C2×C4⋊Q8—31no—
107C2×C4⋊Q8C2×D311yescharacteristic
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×#125—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—32no—
112C2×C22⋊Q8C2×D312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2×D312yes—
115C2×#117—21no—
116C22×C42C2×D311yescharacteristic
117#494—64no—
118#487—64no—
119C2×#103—34no—
120C2×#103—34no—
121C2×C22⋊C8—32no—
122C2×C4×C8—31no—
123C2×C4×C8—31no—
124#391—32no—
125#389—61no—
126#389—61no—
127#384—31no—
128#380—61no—
129#380—61no—
130#340—34no—
131#336—32no—
132C24×C3C2×D411yescharacteristic
133C2×C3×Q8—24no—
134C2×C3×Q8—22no—
135C2×C3×Q8—24no—
136C2×C3×Q8—24no—
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C2×C3×Q8C2×D414yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4Q1614yescomplemented
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—22no—
159C22×Dic3—22no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—24no—
163C2×Dic6—24no—
164C3×C4⋊C4—28no—
165C3×C4⋊C4—28no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4C2×D414yes—
171C3×C22⋊C4—28no—
172C3×C42—28no—
173C3×C42—22no—
174C3×C42—22no—
175C3×C42—21no—
176C3×C42—21no—
177C3×C42C2×D412yes—
178C3×C42C2×D412yes—
179#217—44no—
180#216—416no—
181C4⋊Dic3—24no—
182C4⋊Dic3—42no—
183C4⋊Dic3—42no—
184C4⋊Dic3—42no—
185C4⋊Dic3—42no—
186Dic3⋊C4—44no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8#4114yes—
193C22×Q8—21no—
194C22×Q8—32no—
195C22×Q8C22×D312yes—
196C23×C4#6711yescharacteristic
197C23×C4#6711yescharacteristic
198C23×C4C22×D311yescharacteristic
199C2×Q16—68no—
200C2×Q16—68no—
201C2×Q16—124no—
202C22×C8—32no—
203C22×C8—32no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—62no—
207C4⋊Q8—62no—
208C4⋊Q8#6714yes—
209C4⋊Q8#6714yes—
210#125—28no—
211C22⋊Q8—28no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—124no—
215C4×Q8—28no—
216C4×Q8—24no—
217C4×Q8—24no—
218#117—28no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—32no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4C22×D312yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—62no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42#6711yescharacteristic
237C2×C42#6711yescharacteristic
238C2×C42C22×D311yescharacteristic
239C2×C42C22×D311yescharacteristic
240#103—68no—
241#103—124no—
242C22⋊C8—64no—
243C4×C8—62no—
244C4×C8—62no—
245#95—62no—
246#95—62no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3C2×Q1612yes—
254C23×C3C22×D411yescharacteristic
255C3×Q8—28no—
256C3×Q8—44no—
257C3×Q8—48no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—28no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—44no—
276C2×C3×C4#12012yes—
277C2×C3×C4#12012yes—
278C2×C3×C4C2×Q1612yes—
279C2×C3×C4C2×Q1612yes—
280C2×C3×C4#13512yes—
281C2×C3×C4#13712yes—
282C2×C3×C4#13712yes—
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C2×#4112yes—
286C2×C3×C4H(5,𝔽2)12yes—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293Dic6—44no—
294Dic6—44no—
295C3⋊C8—24no—
296C3⋊C8—42no—
297C24C2×#6711yescharacteristic
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—64no—
305C2×Q8—64no—
306C2×Q8—62no—
307C2×Q8—62no—
308C2×Q8C2×#6714yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4#21614yescomplemented
323C22×C4C2×#6711yescharacteristic
324C22×C4C2×#6711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C22×C4C23×D311yescharacteristic
329Q16—1216no—
330C2×C8—34no—
331C2×C8—62no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—64no—
343C4⋊C4—124no—
344C4⋊C4—122no—
345C4⋊C4—122no—
346C4⋊C4—122no—
347C4⋊C4—122no—
348C4⋊C4C2×#6714yes—
349C22⋊C4—28no—
350C22⋊C4—124no—
351C42—28no—
352C42—22no—
353C42—22no—
354C42—21no—
355C42—21no—
356C42C2×#6712yes—
357C42C2×#6712yes—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3C2×#12011yescharacteristic
366C22×C3#53211yescharacteristic
367C22×C3#53511yescharacteristic
368C22×C3C22×Q1611yescharacteristic
369C22×C3C2×#13511yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C3×C4—22no—
373C3×C4—41no—
374C3×C4—44no—
375C3×C4—41no—
376C3×C4—44no—
377C3×C4#45114yes—
378C3×C4#45214yes—
379C3×C4#54612yes—
380C3×C4#55012yes—
381Dic3—82no—
382Dic3—82no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23C2×#21612yes—
390C23C22×#6711yescharacteristic
391Q8—28no—
392Q8—48no—
393Q8—44no—
394Q8—124no—
395Q8—124no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—28no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4#94212yes—
421C2×C4C2×#21612yes—
422C2×C4C2×#21612yes—
423C2×C4#95012yes—
424C2×C4#95112yes—
425C2×C4#95312yes—
426C2×C4#95312yes—
427C2×C4D3×#4112yes—
428C2×C4#100912yes—
429C2×C4C22×#6711yescharacteristic
430C2×C4C22×#6711yescharacteristic
431C8—64no—
432C8—122no—
433C2×C3—42no—
434C2×C3#243111yescharacteristic
435C2×C3C2×#45111yescharacteristic
436C2×C3C2×#45211yescharacteristic
437C2×C3#317011yescharacteristic
438C2×C3#318511yescharacteristic
439C2×C3#320911yescharacteristic
440C2×C3#321011yescharacteristic
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22C2×#94211yescharacteristic
449C22C22×#21611yescharacteristic
450C22C2×#95011yescharacteristic
451C22C2×#95111yescharacteristic
452C22#583011yescharacteristic
453C22#583211yescharacteristic
454C22C2×#95311yescharacteristic
455C4—22no—
456C4—41no—
457C4—44no—
458C4—41no—
459C4—44no—
460C4—242no—
461C4—242no—
462C4#522814yes—
463C4#523914yes—
464C4#582912yes—
465C4#583412yes—
466C3#2560011yescharacteristic, complemented, Sylow
467C2—42no—
468C2#8334411yescharacteristic
469C2#8334811yescharacteristic
470C2#8335611yescharacteristic
471C2#8337011yescharacteristic
472C2C2×#522811yescharacteristic
473C2C2×#523911yescharacteristic
474C2#8409211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#533D3
hypocenterC3#25600
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. C2×#5092C2semidirect
3. C2×#5093C2semidirect
5. #71787C2semidirect
6. #71785C2semidirect
8. #71632C2semidirect
151. C22×C3×C4Q16semidirect
322. C22×C4#216semidirect
466. C3#25600semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50922
C2×#50932
#820762
#717872
#717852
#717002
#716322
C4×#8302
#256003