Finite Group Databasebeta

← #10698097

Subgroups — #10698097

Order 768.

1044 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 87 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#82653C211yesmaximal, characteristic
2#82631C211yesmaximal, characteristic, complemented
3#73588C211yesmaximal, characteristic, complemented
4C3×#1507C211yesmaximal, characteristic, complemented
5#69988C211yesmaximal, characteristic, complemented
6#69980C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#10329—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#940—21no—
12C2×#937—22no—
13C2×#924C2211yescharacteristic
14C2×#875—21no—
15C3×#407C2211yescharacteristic
16C3×C4×C22⋊C4C2211yescharacteristic
17#5218—21no—
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20C2×#823—22no—
21#5042—22no—
22#5040—21no—
23#5033—21no—
24#4988—21no—
25#4970—21no—
26#4960—22no—
27#4955—22no—
28#4939—21no—
29#4672—22no—
30C3×#327—22no—
31#4467—22no—
32#3193—31no—
33#2435—31no—
34#2433—31no—
35#1783—31no—
36#1508—31no—
37#1507D311yescharacteristic, complemented
38#1297—31no—
39C24×D3D412yes—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—24no—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—44no—
51C22×Dic6D411yescharacteristic, complemented
52C2×C3×M16—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—21no—
61C2×#217—21no—
62C2×#217—21no—
63#940—44no—
64#937—48no—
65C2×#214—22no—
66C2×(C2×D3)⋊C4—24no—
67C2×(C2×D3)⋊C4—24no—
68C2×(C2×D3)⋊C4—24no—
69C2×C4⋊Dic3D411yescharacteristic, complemented
70#924—21no—
71#924—21no—
72#924—21no—
73#924—21no—
74C2×Dic3⋊C4—21no—
75C2×C4×Dic3—21no—
76C2×#208D412yescomplemented
77#875—44no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×#95D412yescomplemented
81#832—22no—
82C3⋊#95D412yes—
83#830—21no—
84#830—21no—
85#823—44no—
86#810—22no—
87#807—22no—
88C23×D4—31no—
89C2×#127—61no—
90C2×#124—61no—
91C2×C22⋊Q8—31no—
92C2×#121C2×D311yescharacteristic
93C2×#120—62no—
94#484—61no—
95#480—61no—
96#468—62no—
97#450—61no—
98#448—61no—
99C2×#100—62no—
100#407—31no—
101#407C2×D311yescharacteristic
102#394—62no—
103#392—62no—
104#390—61no—
105#390—61no—
106#379—62no—
107C4×C22⋊C4—31no—
108C4×C22⋊C4C2×D311yescharacteristic
109#328—62no—
110#327—22no—
111C24×C3—21no—
112C24×C3C2×D411yescharacteristic
113C23×D3—22no—
114C23×D3—22no—
115C23×D3—22no—
116C23×D3—44no—
117C23×D3—44no—
118C23×D3—44no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—44no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—42no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C2×#67—48no—
135C2×#67—48no—
136C2×#67—84no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C2×D12—24no—
141C2×D12—24no—
142C2×D12—24no—
143C2×D12—42no—
144C2×D12—42no—
145C2×D12—42no—
146C2×D12—44no—
147C2×D12—44no—
148C2×D12—44no—
149C2×D12—41no—
150C2×D12—41no—
151C2×D12—41no—
152C2×D12—41no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6—42no—
156C3×M16—42no—
157C2×C3×C8—41no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C22⋊C4—42no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C42—42no—
164#217—24no—
165#217—41no—
166#217—41no—
167#214—82no—
168(C2×D3)⋊C4—48no—
169(C2×D3)⋊C4—48no—
170(C2×D3)⋊C4—84no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173Dic3⋊C4—41no—
174Dic3⋊C4—41no—
175C4×Dic3—42no—
176#208—42no—
177C2×C3⋊C8—22no—
178C25—32no—
179C22×Q8—31no—
180C22×D4—21no—
181C22×D4—64no—
182C22×D4—61no—
183C22×D4—61no—
184C22×D4—61no—
185C22×D4—61no—
186C22×D4—61no—
187C22×D4—61no—
188C22×D4—124no—
189C22×D4D1211yescharacteristic
190C23×C4C22×D311yescharacteristic
191C2×SD16—62no—
192C2×M16—21no—
193C2×M16—32no—
194#127—124no—
195#124—124no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204#120—128no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4D1211yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—64no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—64no—
215C2×C42—21no—
216C2×C42—61no—
217#107—62no—
218#103—62no—
219#102—62no—
220#100—124no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225#95—32no—
226#95#6712yescomplemented
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3#12011yescharacteristic
235C23×C3#12011yescharacteristic
236C23×C3#12111yescharacteristic
237C22×D3—24no—
238C22×D3—24no—
239C22×D3—24no—
240C22×D3—44no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C22×D3—44no—
245C22×D3—42no—
246C22×D3—44no—
247C22×D3—44no—
248C22×D3—816no—
249C22×D3—82no—
250C22×D3—82no—
251C3×D4—42no—
252C3×D4—81no—
253C3×D4—82no—
254C3×D4—82no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4#12011yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12711yescharacteristic
270#67—816no—
271C2×Dic3—24no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—42no—
278D12—44no—
279D12—44no—
280D12—44no—
281D12—44no—
282Dic6—42no—
283Dic6—81no—
284C3×C8—81no—
285C3⋊C8—42no—
286C24—21no—
287C24—62no—
288C24—62no—
289C24—62no—
290C24—124no—
291C24—124no—
292C24—124no—
293C24D3×D411yescharacteristic
294C2×Q8—61no—
295C2×Q8—61no—
296C2×Q8—122no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—64no—
304C2×D4—64no—
305C2×D4—64no—
306C2×D4—122no—
307C2×D4—122no—
308C2×D4—122no—
309C2×D4—124no—
310C2×D4—124no—
311C2×D4—124no—
312C2×D4—128no—
313C2×D4—128no—
314C2×D4—121no—
315C2×D4—121no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—244no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—62no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4C2×D1211yescharacteristic
326C22×C4#23511yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C2×#6711yescharacteristic
331SD16—242no—
332M16—42no—
333M16—122no—
334C2×C8—41no—
335C2×C8—62no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—61no—
339C4⋊C4—61no—
340C4⋊C4—121no—
341C4⋊C4—121no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—64no—
346C22⋊C4—128no—
347C22⋊C4—128no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C22⋊C4—244no—
351C42—42no—
352C42—122no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—44no—
361C22×C3—84no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3#39211yescharacteristic
366C22×C3#45312yes—
367C22×C3#45512yes—
368C2×D3—48no—
369C2×D3—48no—
370C2×D3—44no—
371C2×D3—84no—
372C2×D3—84no—
373C2×D3—84no—
374C2×D3—82no—
375C2×D3—82no—
376C2×D3—84no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—82no—
381C3×C4—81no—
382C3×C4B(4,𝔽2)12yes—
383Dic3—44no—
384Dic3—81no—
385Dic3—81no—
386C23—22no—
387C23—22no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—42no—
393C23—64no—
394C23—64no—
395C23—64no—
396C23—124no—
397C23—122no—
398C23—122no—
399C23—122no—
400C23—124no—
401C23—124no—
402C23—122no—
403C23—124no—
404C23—242no—
405C23—242no—
406C23—2416no—
407C23#88211yescharacteristic
408C23#88411yescharacteristic
409C23#93711yescharacteristic
410Q8—122no—
411Q8—241no—
412D4—42no—
413D4—81no—
414D4—82no—
415D4—82no—
416D4—124no—
417D4—124no—
418D4—124no—
419D4—124no—
420D4—2416no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—64no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4#88211yescharacteristic
440C2×C4#88411yescharacteristic
441C2×C4#93011yescharacteristic
442C2×C4#94011yescharacteristic
443C8—81no—
444C8—122no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—41no—
448C2×C3—161no—
449C2×C3#200211yescharacteristic
450C2×C3#200411yescharacteristic
451C2×C3#202111yescharacteristic
452D3—84no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—44no—
461C22—84no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—128no—
466C22—128no—
467C22—124no—
468C22—244no—
469C22—244no—
470C22—244no—
471C22—244no—
472C22—242no—
473C22—242no—
474C22#482512yes—
475C22#496011yescharacteristic
476C22#519812yes—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—82no—
481C4—81no—
482C4—124no—
483C4—241no—
484C4—241no—
485C4#474112yes—
486C3#1032911yescharacteristic, complemented, Sylow
487C2—22no—
488C2—22no—
489C2—41no—
490C2—161no—
491C2—244no—
492C2#7003711yescharacteristic
493C2#7032411yescharacteristic
494C2#7038611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#392
prime radical#82653C2
hypercenter#1507D3
hypocenterC3#10329
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #82631C2semidirect
3. #73588C2semidirect
4. C3×#1507C2semidirect
5. #69988C2semidirect
6. #69980C2semidirect
7. #67618C2semidirect
37. #1507D3semidirect
51. C22×Dic6D4semidirect
69. C2×C4⋊Dic3D4semidirect
76. C2×#208D4semidirect
80. C3×#95D4semidirect
226. #95#67semidirect
486. C3#10329semidirect

Maximal subgroups

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

subgroupindex
#826532
#826312
#735882
C3×#15072
#699882
#699802
#676182
#103293