Finite Group Databasebeta

← #10844693

Subgroups — #10844693

Order 768.

850 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C8×Q8C211yesmaximal, characteristic, complemented
2#82500C211yesmaximal, characteristic
3#82479C211yesmaximal, characteristic, complemented
4C2×#4841C211yesmaximal, characteristic
5#70488C212yesmaximal, complemented
6#70482C212yesmaximal
7#70454C211yesmaximal, characteristic, complemented
8#69774C212yesmaximal
9#69771C211yesmaximal, characteristic, complemented
10#69767C212yesmaximal, complemented
11C4×#821C211yesmaximal, characteristic
12#25293—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15C22×#204C2212yes—
16C2×C4×D12C2211yescharacteristic
17C2×#869C2211yescharacteristic
18C3×C8×Q8—24no—
19C2×C3×C4⋊C8C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C2×C3×C4×C8C2212yes—
23#5229—21no—
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#817C2212yes—
27C2×#816C2212yes—
28C2×#816C2212yes—
29C2×#816C2212yes—
30#4994—22no—
31#4987—21no—
32#4986—22no—
33#4977—21no—
34#4939C2212yes—
35#4935C2211yescharacteristic
36C4×C4⋊Dic3C2211yescharacteristic
37#4929C2212yes—
38#4929C2212yes—
39#4928C2212yes—
40#4841—24no—
41#4551C2212yes—
42#4551C2211yescharacteristic
43#4551C2211yescharacteristic
44C2×#461—31no—
45C2×C8×Q8D311yescharacteristic, complemented
46#2568—31no—
47#2538—31no—
48#2066—32no—
49#2055—32no—
50#1968—32no—
51#1956—31no—
52#1948—31no—
53#1933—32no—
54C4×#102—31no—
55C22×C3×Q8D411yescharacteristic
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×Dic6C2312yes—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2311yescharacteristic
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C4⋊C4D412yes—
66C2×C3×C42C2312yes—
67C2×C3×C42C2311yescharacteristic
68#946—44no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—22no—
81C2×C4×Dic3—21no—
82C2×#204—24no—
83C2×#204—24no—
84C2×#204—42no—
85C4×D12—22no—
86C4×D12—22no—
87#869—22no—
88#869—22no—
89C3×C4⋊C8—24no—
90C3×C4⋊C8D414yes—
91C3×C4×C8—24no—
92C3×C4×C8—22no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—21no—
99#821—22no—
100#821—22no—
101#817—42no—
102#816—24no—
103#816—24no—
104#816—42no—
105C22×SD16—32no—
106C2×C4⋊Q8—31no—
107C2×C22⋊Q8—61no—
108C2×C4×Q8C2×D311yescharacteristic
109C2×C4×D4—31no—
110#461—64no—
111C8×Q8—24no—
112C2×#106—32no—
113C2×C4⋊C8C2×D312yes—
114C2×C4⋊C8C2×D311yescharacteristic
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118C2×#102—31no—
119C2×#102—31no—
120C2×C4×C8C2×D312yes—
121C2×C4×C8C2×D311yescharacteristic
122#400—62no—
123#396—62no—
124#391—32no—
125#391—32no—
126#391—61no—
127#390—32no—
128#389—32no—
129#384—31no—
130#382—61no—
131#380—61no—
132C4×C4⋊C4—31no—
133#340—32no—
134#340—31no—
135#340—31no—
136C23×D3—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8SD1614yescomplemented
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—21no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12#4112yes—
156C2×C4×D3—42no—
157C2×C4×D3—42no—
158C2×C4×D3—42no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8C2×D414yes—
173C2×C3×C8#4112yes—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—22no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42C2×D412yes—
179C3×C42#4112yes—
180(C2×D3)⋊C4—42no—
181(C2×D3)⋊C4—44no—
182(C2×D3)⋊C4—82no—
183C4⋊Dic3—21no—
184C4⋊Dic3—22no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—22no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—42no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4⋊Dic3—81no—
195Dic3⋊C4—44no—
196C4×Dic3—42no—
197#204—48no—
198C22×Q8—32no—
199C22×Q8D1211yescharacteristic
200C22×D4—31no—
201C23×C4—61no—
202C2×SD16—64no—
203C2×SD16—64no—
204C2×SD16—122no—
205C22×C8C22×D311yescharacteristic
206C22×C8C22×D311yescharacteristic
207C22×C8C22×D312yes—
208C4⋊Q8—62no—
209C4⋊Q8—62no—
210C22⋊Q8—124no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×D4—62no—
214C4×D4—62no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4D1211yescharacteristic
225C2×C4⋊C4D1212yes—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C42—61no—
230C2×C42C22×D312yes—
231C2×C42C22×D311yescharacteristic
232#106—122no—
233C4⋊C8—24no—
234C4⋊C8#6714yes—
235#103—64no—
236#103—64no—
237#103—122no—
238#102—62no—
239#102—62no—
240C4×C8—24no—
241C4×C8—22no—
242#95—61no—
243#95—62no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—62no—
248C23×C3C22×D411yescharacteristic
249C22×D3—22no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C3×Q8—28no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4#12112yes—
265C2×C3×C4#12112yes—
266C2×C3×C4C2×SD1614yes—
267C2×C3×C4C2×SD1612yes—
268C2×C3×C4#13512yes—
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4H(5,𝔽2)11yescharacteristic
274C2×C3×C42−1+411yescharacteristic
275C2×Dic3—41no—
276C2×Dic3—42no—
277C2×Dic3—41no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—41no—
281C2×Dic3—42no—
282C2×Dic3—41no—
283C2×Dic3—42no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—42no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290D12—22no—
291D12—22no—
292D12—41no—
293D12—41no—
294C4×D3—84no—
295Dic6—42no—
296Dic6—42no—
297Dic6—42no—
298Dic6—42no—
299C3×C8—24no—
300C3×C8—22no—
301C3×C8—41no—
302C24—61no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—62no—
306C2×Q8—62no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8#20414yescomplemented
312C2×D4—32no—
313C2×D4—61no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—62no—
321C22×C4—61no—
322C22×C4—62no—
323C22×C4—122no—
324C22×C4—122no—
325C22×C4—122no—
326C22×C4C2×D1212yes—
327C22×C4C2×D1212yes—
328C22×C4C2×D1211yescharacteristic
329C22×C4C2×D1211yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—128no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8#23912yes—
340C2×C8C2×#6714yes—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—61no—
344C4⋊C4—61no—
345C4⋊C4—62no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—62no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—124no—
352C4⋊C4—122no—
353C4⋊C4—122no—
354C4⋊C4—122no—
355C4⋊C4—241no—
356C22⋊C4—124no—
357C22⋊C4—122no—
358C22⋊C4—242no—
359C42—22no—
360C42—22no—
361C42—122no—
362C42#23512yes—
363C42C2×#6712yes—
364C22×C3C2×#12111yescharacteristic
365C22×C3#54811yescharacteristic
366C22×C3#55011yescharacteristic
367C22×C3C22×SD1611yescharacteristic
368C22×C3C2×#13511yescharacteristic
369C22×C3#57711yescharacteristic
370C22×C3#57811yescharacteristic
371C2×D3—42no—
372C2×D3—42no—
373C2×D3—42no—
374C2×D3—81no—
375C2×D3—81no—
376C2×D3—81no—
377C2×D3—81no—
378C3×C4—24no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4#46514yes—
382C3×C4#52911yescharacteristic
383C3×C4#54211yescharacteristic
384C3×C4#54611yescharacteristic
385C3×C4#54811yescharacteristic
386Dic3—81no—
387Dic3—82no—
388Dic3—81no—
389Dic3—82no—
390Dic3—81no—
391C23—62no—
392C23—121no—
393C23—121no—
394C23—121no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23C22×D1211yescharacteristic
399Q8—28no—
400Q8—122no—
401Q8—122no—
402Q8—122no—
403Q8—122no—
404D4—62no—
405D4—62no—
406D4—121no—
407D4—121no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—122no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—122no—
425C2×C4—121no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—121no—
429C2×C4—122no—
430C2×C4—244no—
431C2×C4C2×#20414yes—
432C2×C4C2×#20412yes—
433C2×C4#90412yes—
434C2×C4#93012yes—
435C2×C4#93012yes—
436C2×C4C2×#23511yescharacteristic
437C2×C4C2×#23911yescharacteristic
438C2×C4#100711yescharacteristic
439C2×C4D3×#4111yescharacteristic
440C2×C4#100911yescharacteristic
441C2×C4C22×#6711yescharacteristic
442C8—22no—
443C8—24no—
444C8—41no—
445C2×C3#242611yescharacteristic
446C2×C3C2×#46511yescharacteristic
447C2×C3#306311yescharacteristic
448C2×C3#335911yescharacteristic
449C2×C3#336011yescharacteristic
450C2×C3#339211yescharacteristic
451C2×C3#339311yescharacteristic
452D3—82no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22#557611yescharacteristic
461C22#557711yescharacteristic
462C22C22×#20411yescharacteristic
463C22C2×#90411yescharacteristic
464C22#575111yescharacteristic
465C22#575311yescharacteristic
466C22C2×#93011yescharacteristic
467C4—24no—
468C4—22no—
469C4—42no—
470C4—241no—
471C4—242no—
472C4—241no—
473C4—241no—
474C4—242no—
475C4#511414yes—
476C4#558011yescharacteristic
477C4#558211yescharacteristic
478C4#581711yescharacteristic
479C4#582911yescharacteristic
480C3#2529311yescharacteristic, complemented, Sylow
481C2—242no—
482C2#7991111yescharacteristic
483C2#7991211yescharacteristic
484C2#7992511yescharacteristic
485C2#7992611yescharacteristic
486C2#8249511yescharacteristic
487C2C2×#511411yescharacteristic
488C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C8×Q8C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×C8×Q8D3
hypocenterC3#25293
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C8×Q8C2semidirect
3. #82479C2semidirect
5. #70488C2semidirect
7. #70454C2semidirect
9. #69771C2semidirect
10. #69767C2semidirect
45. C2×C8×Q8D3semidirect
139. C2×C3×Q8SD16semidirect
311. C2×Q8#204semidirect
480. C3#25293semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×Q82
#825002
#824792
C2×#48412
#704882
#704822
#704542
#697742
#697712
#697672
C4×#8212
#252933