Finite Group Databasebeta

← #10695827

Subgroups — #10695827

Order 768.

764 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 79 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#84084C211yesmaximal, characteristic, complemented
2#84082C211yesmaximal, characteristic
3#74145C211yesmaximal, characteristic, complemented
4C3×#1782C211yesmaximal, characteristic, complemented
5#69533C211yesmaximal, characteristic, complemented
6#69531C211yesmaximal, characteristic, complemented
7#67629C211yesmaximal, characteristic, complemented
8#9981—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×#951—21no—
11C2×#949—21no—
12C2×#930C2211yescharacteristic
13C2×#924C2211yescharacteristic
14C2×C3×#99—22no—
15C3×#381C2211yescharacteristic
16C3×#381C2211yescharacteristic
17#5239—22no—
18#5238—22no—
19#5232—21no—
20#5232—21no—
21#5228—21no—
22#5227—21no—
23#5218—21no—
24#5217—21no—
25#5207C2211yescharacteristic
26#5207C2211yescharacteristic
27#4961—22no—
28#4677—21no—
29#4673—21no—
30#4672—21no—
31#4661—21no—
32#4452—22no—
33#4451—22no—
34#3007—31no—
35#3006—31no—
36#2816—31no—
37#1782D311yescharacteristic, complemented
38#1511—31no—
39#1510—31no—
40#1306—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4D411yescharacteristic
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C22×#67—21no—
52C22×D12D411yescharacteristic
53C22×Dic6D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4D412yes—
56C2×#217—21no—
57#951—44no—
58#949—44no—
59C2×#216—22no—
60C2×#215—22no—
61C2×#214—22no—
62C2×#213—22no—
63#930—21no—
64#930—21no—
65#930—21no—
66#930—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4⋊Dic3D411yescharacteristic
73#924—21no—
74#924—21no—
75#924—21no—
76#924—21no—
77C2×Dic3⋊C4—21no—
78C2×#208—21no—
79C2×#208—21no—
80C3×#99—44no—
81C3×#95—21no—
82C3×#95—21no—
83C3×#95—21no—
84C3×#95—21no—
85#835—22no—
86#835—22no—
87#832—22no—
88#832—22no—
89C3⋊#95—22no—
90#830—21no—
91#830—21no—
92#830D412yes—
93#830D412yes—
94C22×#41C2×D311yescharacteristic
95C2×#137—61no—
96C2×AGL(1,ℤ/8)—61no—
97C2×C22⋊Q8—31no—
98C2×#121—31no—
99#452—62no—
100#451—61no—
101#450—61no—
102#449—62no—
103#448—61no—
104#447—61no—
105#417—61no—
106#417—61no—
107C2×#99—22no—
108#407—31no—
109#407—31no—
110#399—61no—
111#398—61no—
112#394—61no—
113#394—62no—
114#393—61no—
115#381C2×D311yescharacteristic
116#381C2×D311yescharacteristic
117#328—62no—
118#327—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C3×#41—24no—
123C3×#41—24no—
124C3×#41—42no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—42no—
133C2×C3×D4—24no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4#4111yescharacteristic
154C2×#67—41no—
155C2×#67—41no—
156C2×#67—82no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12—42no—
163C2×Dic6—21no—
164C2×Dic6—21no—
165C2×Dic6—42no—
166C3×C22⋊C4—42no—
167C3×C22⋊C4—42no—
168#217—41no—
169#217—41no—
170#216—44no—
171#215—44no—
172#214—44no—
173#213—44no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—82no—
177(C2×D3)⋊C4—82no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
182C4⋊Dic3—82no—
183Dic3⋊C4—41no—
184Dic3⋊C4—41no—
185#208—42no—
186#208—42no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—44no—
194C22×Q8—31no—
195C22×Q8D1211yescharacteristic
196C22×D4—21no—
197C22×D4—31no—
198C22×D4—61no—
199C22×D4D1211yescharacteristic
200C23×C4—21no—
201C23×C4C22×D311yescharacteristic
202#137—124no—
203AGL(1,ℤ/8)—124no—
204C2×Q16—62no—
205C2×SD16—62no—
206C2×SD16—62no—
207C2×D8—62no—
208C2×M16—61no—
209C2×M16—61no—
210C22⋊Q8—61no—
211C22⋊Q8—61no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214#121—61no—
215#121—61no—
216#121—61no—
217#121—61no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—62no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—61no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4#6712yes—
228#103—62no—
229#103—62no—
230#102—62no—
231#102—62no—
232#99—44no—
233C22⋊C8—32no—
234C22⋊C8—32no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—62no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—81no—
249C23×C3—81no—
250C23×C3#12011yescharacteristic
251C23×C3#12111yescharacteristic
252C23×C3#12711yescharacteristic
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—41no—
256C22×D3—42no—
257C3×Q8—22no—
258C3×Q8—22no—
259C3×Q8—41no—
260C3×Q8—41no—
261C3×D4—22no—
262C3×D4—22no—
263C3×D4—41no—
264C3×D4—41no—
265C3×D4—44no—
266C3×D4—44no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—44no—
275C2×C3×C4—44no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—81no—
282C2×C3×C4—81no—
283C2×C3×C4—81no—
284C2×C3×C4#12011yescharacteristic
285C2×C3×C4#12011yescharacteristic
286C2×C3×C4#12111yescharacteristic
287C2×C3×C4#12111yescharacteristic
288#67—82no—
289#67—82no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297D12—42no—
298D12—81no—
299Dic6—42no—
300Dic6—81no—
301C3⋊C8—42no—
302C3⋊C8—42no—
303C24—21no—
304C24—61no—
305C24C2×#6711yescharacteristic
306#41—24no—
307#41—24no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—42no—
312#41—42no—
313#41—42no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—42no—
317C2×Q8—61no—
318C2×Q8—61no—
319C2×Q8—122no—
320C2×D4—24no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—122no—
330C2×D4—121no—
331C2×D4—121no—
332C2×D4—242no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4C2×D1211yescharacteristic
345C22×C4#23511yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350Q16—124no—
351SD16—124no—
352SD16—124no—
353D8—124no—
354M16—122no—
355M16—122no—
356C2×C8—62no—
357C2×C8—62no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—242no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—121no—
368C22⋊C4—121no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C22⋊C4—242no—
372C22⋊C4—242no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—42no—
382C22×C3—82no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3#39211yescharacteristic
388C22×C3B(4,𝔽2)12yes—
389C2×D3—84no—
390C2×D3—81no—
391C2×D3—81no—
392C2×D3—81no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—42no—
398C3×C4—82no—
399C3×C4—161no—
400Dic3—81no—
401Dic3—81no—
402Dic3—81no—
403C23—22no—
404C23—22no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—81no—
410C23—81no—
411C23—122no—
412C23—122no—
413C23—121no—
414C23—122no—
415C23#93011yescharacteristic
416C23#93711yescharacteristic
417C23#94011yescharacteristic
418Q8—22no—
419Q8—22no—
420Q8—41no—
421Q8—41no—
422Q8—122no—
423Q8—241no—
424D4—22no—
425D4—22no—
426D4—41no—
427D4—41no—
428D4—44no—
429D4—44no—
430D4—122no—
431D4—241no—
432D4—242no—
433D4—242no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—122no—
458C2×C4#88211yescharacteristic
459C2×C4#88211yescharacteristic
460C2×C4#88411yescharacteristic
461C2×C4#88411yescharacteristic
462C8—122no—
463C8—122no—
464C2×C3—22no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—82no—
468C2×C3#200811yescharacteristic
469C2×C3#200911yescharacteristic
470C2×C3#202111yescharacteristic
471D3—161no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—42no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—81no—
485C22—81no—
486C22—244no—
487C22—241no—
488C22—241no—
489C22—241no—
490C22#496011yescharacteristic
491C22#503212yes—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—82no—
498C4—161no—
499C4—241no—
500C4—241no—
501C4—241no—
502C3#998111yescharacteristic, complemented, Sylow
503C2—22no—
504C2—41no—
505C2—41no—
506C2—82no—
507C2—481no—
508C2#6950011yescharacteristic
509C2#7005011yescharacteristic
510C2#7005311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1782C2
socleC22×C3#392
prime radical#84082C2
hypercenter#1782D3
hypocenterC3#9981
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #84084C2semidirect
3. #74145C2semidirect
4. C3×#1782C2semidirect
5. #69533C2semidirect
6. #69531C2semidirect
7. #67629C2semidirect
37. #1782D3semidirect
502. C3#9981semidirect

Maximal subgroups

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

subgroupindex
#840842
#840822
#741452
C3×#17822
#695332
#695312
#676292
#99813