Finite Group Databasebeta

← #10698082

Subgroups — #10698082

Order 768.

806 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C3×#2436C211yesmaximal, characteristic, complemented
2#84066C211yesmaximal, characteristic, complemented
3#82627C211yesmaximal, characteristic
4#72133C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#69444C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#10327—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×C3×#124—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#947—21no—
13C2×#930C2211yescharacteristic
14C3×#393—22no—
15C3×#391—21no—
16C3×#386—22no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5231—22no—
19#5229—21no—
20C2×#836—22no—
21#5217—21no—
22#5213—22no—
23C4×#217C2211yescharacteristic
24#5207C2211yescharacteristic
25#5207C2211yescharacteristic
26#5039—22no—
27#5036—21no—
28#5034—21no—
29#4968—21no—
30#4537—22no—
31#4451—22no—
32#3192—31no—
33#2436—31no—
34#2436D311yescharacteristic, complemented
35#1784—31no—
36#1507—31no—
37#1507—31no—
38#1297—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—44no—
46C22×C3×D4—21no—
47C22×C3×D4D411yescharacteristic
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C22×#67—21no—
51C22×D12D411yescharacteristic
52C3×#124—44no—
53C3×#121—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42—21no—
62C2×#217—21no—
63#947—44no—
64C2×#213—22no—
65#930—21no—
66#930—21no—
67#930—21no—
68#930—21no—
69C2×(C2×D3)⋊C4—24no—
70C2×(C2×D3)⋊C4—21no—
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4×Dic3—21no—
73C2×#208—21no—
74C2×#208D412yes—
75C3×#95—24no—
76C3×#95D412yes—
77#836—44no—
78#832—22no—
79C3⋊#95—24no—
80C3⋊#95D412yes—
81#830—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#809—22no—
86#808—22no—
87C23×Q8C2×D311yescharacteristic
88C2×#124—21no—
89C2×#124—61no—
90C2×#121—31no—
91C2×#121C2×D311yescharacteristic
92#481—61no—
93#481—61no—
94#469—62no—
95#447—61no—
96#447—61no—
97C2×#101—62no—
98#407—31no—
99#407—31no—
100#393—22no—
101#393—62no—
102#391—21no—
103#391—61no—
104#386—22no—
105#386—62no—
106C4×C22⋊C4—31no—
107C4×C22⋊C4C2×D311yescharacteristic
108#327—62no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
113C2×C3×Q8—24no—
114C2×C3×Q8—24no—
115C2×C3×Q8—24no—
116C2×C3×Q8—42no—
117C2×C3×Q8—42no—
118C2×C3×Q8—42no—
119C2×C3×Q8—44no—
120C2×C3×Q8—44no—
121C2×C3×Q8—44no—
122C2×C3×Q8—41no—
123C2×C3×Q8—41no—
124C2×C3×Q8—41no—
125C2×C3×Q8—41no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—44no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—44no—
138C22×C3×C4—42no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C2×#67—44no—
146C2×#67—41no—
147C2×#67—41no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—42no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C22⋊C4—48no—
156C3×C22⋊C4—42no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C42—42no—
160#217—42no—
161#213—82no—
162(C2×D3)⋊C4—48no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4×Dic3—42no—
168#208—42no—
169#208—42no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—44no—
179C22×D4—21no—
180C22×D4—31no—
181C22×D4—61no—
182C22×D4#6711yescharacteristic
183C23×C4#6712yes—
184C23×C4C22×D311yescharacteristic
185C2×D8—62no—
186C2×M16—32no—
187C2×M16—61no—
188#124—44no—
189#124—124no—
190#121—21no—
191#121—21no—
192#121—21no—
193#121—21no—
194#121—61no—
195#121—61no—
196#121—61no—
197#121—61no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4#6711yescharacteristic
200C2×C22⋊C4—24no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—64no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C42—21no—
207C2×C42—61no—
208#106—62no—
209#102—62no—
210#102—62no—
211#101—124no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216#95—24no—
217#95—32no—
218#95—64no—
219#95#6712yes—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3—42no—
227C23×C3#12011yescharacteristic
228C23×C3#12011yescharacteristic
229C23×C3#12111yescharacteristic
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—41no—
233C22×D3—42no—
234C3×Q8—44no—
235C3×Q8—44no—
236C3×Q8—44no—
237C3×Q8—44no—
238C3×D4—42no—
239C3×D4—81no—
240C3×D4—82no—
241C3×D4—82no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12711yescharacteristic
264#67—82no—
265#67—82no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271D12—42no—
272D12—81no—
273C3⋊C8—42no—
274C3⋊C8—81no—
275C24—21no—
276C24—61no—
277C24C2×#6711yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—122no—
300C2×D4—124no—
301C2×D4—121no—
302C2×D4—121no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—62no—
311C22×C4—61no—
312C22×C4D3×D411yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4#23711yescharacteristic
316C22×C4C2×#6711yescharacteristic
317C22×C4C2×#6711yescharacteristic
318D8—242no—
319M16—122no—
320M16—122no—
321C2×C8—62no—
322C2×C8—121no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C22⋊C4—48no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—128no—
332C22⋊C4—122no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C42—42no—
336C42—122no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—44no—
345C22×C3—84no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3#39211yescharacteristic
350C22×C3#45412yes—
351C22×C3#45412yes—
352C2×D3—84no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—81no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—44no—
360C3×C4—82no—
361C3×C4—81no—
362C3×C4B(4,𝔽2)12yes—
363Dic3—82no—
364Dic3—81no—
365C23—22no—
366C23—22no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—42no—
372C23—122no—
373C23—122no—
374C23—121no—
375C23—122no—
376C23#93711yescharacteristic
377C23#93911yescharacteristic
378C23#95311yescharacteristic
379Q8—44no—
380Q8—44no—
381Q8—44no—
382Q8—44no—
383D4—42no—
384D4—81no—
385D4—82no—
386D4—82no—
387D4—122no—
388D4—241no—
389D4—242no—
390D4—242no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4#93711yescharacteristic
415C2×C4#93811yescharacteristic
416C2×C4#93911yescharacteristic
417C2×C4#94011yescharacteristic
418C8—122no—
419C8—241no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—41no—
423C2×C3—161no—
424C2×C3#200311yescharacteristic
425C2×C3#200311yescharacteristic
426C2×C3#202111yescharacteristic
427D3—161no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—44no—
436C22—84no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—244no—
441C22—241no—
442C22—241no—
443C22—241no—
444C22#506212yes—
445C22#522311yescharacteristic
446C22#524512yes—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—44no—
451C4—82no—
452C4—81no—
453C4—242no—
454C4—241no—
455C4#503212yes—
456C3#1032711yescharacteristic, complemented, Sylow
457C2—22no—
458C2—22no—
459C2—41no—
460C2—161no—
461C2—481no—
462C2#7167111yescharacteristic
463C2#7173011yescharacteristic
464C2#7186711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2436C2
socleC22×C3#392
prime radical#82627C2
hypercenter#2436D3
hypocenterC3#10327
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
1. C3×#2436C2semidirect
2. #84066C2semidirect
4. #72133C2semidirect
5. #71673C2semidirect
6. #69444C2semidirect
7. #67618C2semidirect
34. #2436D3semidirect
456. C3#10327semidirect

Maximal subgroups

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

subgroupindex
C3×#24362
#840662
#826272
#721332
#716732
#694442
#676182
#103273