Finite Group Databasebeta

← #10698064

Subgroups — #10698064

Order 768.

664 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82080C211yesmaximal, characteristic
2#82075C211yesmaximal, characteristic, complemented
3#78612C211yesmaximal, characteristic
4C3×#1817C211yesmaximal, characteristic, complemented
5#69946C211yesmaximal, characteristic, complemented
6#69944C211yesmaximal, characteristic
7#67676C211yesmaximal, characteristic, complemented
8#10323—31nomaximal, Sylow
9C22×C3×C42C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#929—21no—
12C2×#924C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15#5213—21no—
16#5212C2211yescharacteristic
17#5212C2211yescharacteristic
18#5211—21no—
19#5210—21no—
20#5114—22no—
21#4940—22no—
22#4939—21no—
23#4933—22no—
24#4929—21no—
25#4928—21no—
26#4928—21no—
27#4738—21no—
28#4735—21no—
29#4731—21no—
30#4721—21no—
31C3×#336—22no—
32#4470—22no—
33#4467—22no—
34#3187—31no—
35#2855—31no—
36#2854—31no—
37#1817D311yescharacteristic, complemented
38#1520—31no—
39#1519—31no—
40#1289—31no—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×D12D411yescharacteristic
46C22×Dic6D411yescharacteristic
47C22×C3×C8—21no—
48C22×C3×C8D412yes—
49C2×C3×C42—21no—
50C2×C3×C42—21no—
51C2×C3×C42—21no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62#929—42no—
63#929—42no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×C4⋊Dic3D411yescharacteristic
71#924—21no—
72#924—21no—
73#924—21no—
74#924—21no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—21no—
78C2×#204—22no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95D412yes—
88C3⋊#95D412yes—
89#821—22no—
90#817—22no—
91#816—22no—
92C2×#123—61no—
93C2×C22⋊Q8—31no—
94C2×#121—31no—
95C22×C42C2×D311yescharacteristic
96#483—61no—
97#481—61no—
98#465—62no—
99#450—61no—
100#448—61no—
101C2×C22⋊C8C2×D311yescharacteristic
102C2×C22⋊C8C2×D311yescharacteristic
103#391—61no—
104#390—61no—
105#389—61no—
106#389—61no—
107#388—62no—
108#387—62no—
109#386—61no—
110#385—61no—
111#381—61no—
112#380—31no—
113#380—31no—
114#336—22no—
115#328—62no—
116#327—62no—
117C24×C3C2×D411yescharacteristic
118C23×D3—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4SD1612yescomplemented
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C2×#67—41no—
138C2×#67—41no—
139C2×#67—41no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—42no—
151C2×Dic6—21no—
152C2×Dic6—21no—
153C2×Dic6—42no—
154C2×C3×C8—22no—
155C2×C3×C8—42no—
156C2×C3×C8—42no—
157C2×C3×C8—41no—
158C3×C42—41no—
159C3×C42—41no—
160C3×C42—41no—
161C3×C42—41no—
162#217—41no—
163#217—41no—
164#217—41no—
165#217—41no—
166#217—41no—
167#217—41no—
168(C2×D3)⋊C4—44no—
169(C2×D3)⋊C4—44no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—41no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4⋊Dic3—42no—
177C4⋊Dic3—42no—
178Dic3⋊C4—44no—
179Dic3⋊C4—44no—
180Dic3⋊C4—41no—
181Dic3⋊C4—41no—
182#204—82no—
183C22×Q8—31no—
184C22×D4—31no—
185C22×D4—61no—
186C23×C4D1211yescharacteristic
187C23×C4D1211yescharacteristic
188C23×C4C22×D311yescharacteristic
189C2×SD16—62no—
190C22×C8—21no—
191C22×C8#6712yes—
192#123—122no—
193#123—122no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198#121—61no—
199#121—61no—
200#121—61no—
201#121—61no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—61no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221#106—62no—
222#103—62no—
223#102—62no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228#95—32no—
229#95—32no—
230#95—62no—
231#95—62no—
232#95—62no—
233#95—62no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3#12011yescharacteristic
237C23×C3#12011yescharacteristic
238C23×C3#12111yescharacteristic
239C23×C3C2×SD1611yescharacteristic
240C23×C3#13511yescharacteristic
241C23×C3AGL(1,ℤ/8)11yescharacteristic
242C23×C3#13711yescharacteristic
243C22×D3—42no—
244C22×D3—42no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12711yescharacteristic
272#67—81no—
273#67—81no—
274#67—81no—
275#67—81no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—42no—
286C2×Dic3—41no—
287C2×Dic3—42no—
288C2×Dic3—41no—
289D12—42no—
290D12—81no—
291Dic6—42no—
292Dic6—81no—
293C3×C8—42no—
294C3×C8—81no—
295C24—61no—
296C24C2×D1211yescharacteristic
297C2×Q8—61no—
298C2×Q8—61no—
299C2×Q8—122no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—122no—
303C2×D4—121no—
304C2×D4—121no—
305C2×D4—121no—
306C2×D4—121no—
307C2×D4—121no—
308C2×D4—121no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4#20412yescomplemented
326C22×C4#23511yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4D3×D411yescharacteristic
331C22×C4C2×#6711yescharacteristic
332SD16—242no—
333C2×C8—22no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C4⋊C4—61no—
338C4⋊C4—61no—
339C4⋊C4—61no—
340C4⋊C4—61no—
341C4⋊C4—124no—
342C4⋊C4—124no—
343C4⋊C4—122no—
344C4⋊C4—122no—
345C4⋊C4—121no—
346C4⋊C4—121no—
347C22⋊C4—124no—
348C22⋊C4—124no—
349C22⋊C4—121no—
350C22⋊C4—121no—
351C22⋊C4—121no—
352C22⋊C4—121no—
353C22⋊C4—121no—
354C22⋊C4—121no—
355C22⋊C4—121no—
356C22⋊C4—121no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3#39211yescharacteristic
372C22×C3#44811yescharacteristic
373C22×C3#44911yescharacteristic
374C22×C3#45011yescharacteristic
375C22×C3#45211yescharacteristic
376C22×C3#46511yescharacteristic
377C22×C3#46811yescharacteristic
378C2×D3—82no—
379C2×D3—82no—
380C2×D3—81no—
381C2×D3—81no—
382C2×D3—81no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—42no—
387C3×C4—42no—
388C3×C4#45812yes—
389Dic3—82no—
390Dic3—82no—
391Dic3—81no—
392Dic3—81no—
393Dic3—81no—
394C23—41no—
395C23—41no—
396C23—122no—
397C23—122no—
398C23—121no—
399C23—121no—
400C23—121no—
401C23#88211yescharacteristic
402C23#88211yescharacteristic
403C23C2×#20411yescharacteristic
404C23#90411yescharacteristic
405C23#90811yescharacteristic
406C23#90911yescharacteristic
407C23#93011yescharacteristic
408Q8—122no—
409Q8—241no—
410D4—122no—
411D4—241no—
412D4—241no—
413D4—241no—
414D4—241no—
415D4—241no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—122no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—122no—
449C2×C4#88411yescharacteristic
450C2×C4#88411yescharacteristic
451C2×C4#93711yescharacteristic
452C2×C4#94011yescharacteristic
453C8—42no—
454C8—81no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3#200011yescharacteristic
462C2×C3#201011yescharacteristic
463C2×C3#202211yescharacteristic
464D3—161no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—242no—
476C22—242no—
477C22—241no—
478C22—241no—
479C22—241no—
480C22#473111yescharacteristic
481C22#473311yescharacteristic
482C22#473411yescharacteristic
483C22#473811yescharacteristic
484C22#496011yescharacteristic
485C22#511411yescharacteristic
486C22#513411yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—242no—
493C4—242no—
494C4—241no—
495C4—241no—
496C4—241no—
497C4#505312yes—
498C3#1032311yescharacteristic, complemented, Sylow
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—481no—
506C2#6953911yescharacteristic
507C2#6996211yescharacteristic
508C2#6999311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1817C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1817D3
hypocenterC3#10323
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #82075C2semidirect
4. C3×#1817C2semidirect
5. #69946C2semidirect
7. #67676C2semidirect
37. #1817D3semidirect
130. C22×C3×C4SD16semidirect
325. C22×C4#204semidirect
498. C3#10323semidirect

Maximal subgroups

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

subgroupindex
#820802
#820752
#786122
C3×#18172
#699462
#699442
#676762
#103233