Finite Group Databasebeta

← #10693215

Subgroups — #10693215

Order 768.

772 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 107 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×#3003C211yesmaximal, characteristic
2#82919C211yesmaximal, characteristic, complemented
3#72129C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic, complemented
5#71709C211yesmaximal, characteristic, complemented
6#67687C211yesmaximal, characteristic, complemented
7#67685C211yesmaximal, characteristic, complemented
8#9520—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)—21no—
11C3×#525C2211yescharacteristic
12C3×#450—21no—
13C3×#450C412yescomplemented
14C3×#447—21no—
15C3×#447C412yescomplemented
16C3×#418—21no—
17C3×#407C2211yescharacteristic
18#5234—21no—
19C2×#833—21no—
20C2×#833—21no—
21C2×#832—21no—
22C2×#832—21no—
23#5212C2211yescharacteristic
24#5207C2211yescharacteristic
25#5207C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#5012—21no—
28#5011—21no—
29#5011—21no—
30D4×C3⋊C8—21no—
31#5000—21no—
32#4538—21no—
33#4537—21no—
34#3003D311yescharacteristic
35#2976—31no—
36#1786—31no—
37#1783—31no—
38#1513—31no—
39#1305—31no—
40#1293—31no—
41C25×C3—21no—
42C2×C3×#41D411yescharacteristic
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—42no—
49C22×C3×D4—42no—
50C22×C3×D4C2×C412yes—
51C23×C3×C4C2311yescharacteristic
52C3×AGL(1,ℤ/8)—42no—
53C3×AGL(1,ℤ/8)—42no—
54C2×C3×SD16—22no—
55C2×C3×D8—22no—
56C2×C3×M16—21no—
57C3×#123—21no—
58C3×C22⋊Q8C2×C411yescharacteristic
59C3×C22⋊Q8D411yescharacteristic
60C3×#121C2×C411yescharacteristic
61C3×#121D411yescharacteristic
62C3×#120—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C3×#117D411yescharacteristic
66C2×#217—21no—
67C2×C4⋊Dic3—21no—
68C2×#208—21no—
69C2×#208—21no—
70C2×#208D411yescharacteristic
71C2×#208D411yescharacteristic
72C22×C3⋊C8—21no—
73C22×C3⋊C8D411yescharacteristic
74C22×C3⋊C8D411yescharacteristic
75C3×#102—22no—
76C3×#102—22no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8C2×C412yes—
79#833—42no—
80#833—42no—
81#832—42no—
82#832—42no—
83C3⋊#95—21no—
84#830—21no—
85#830—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830—21no—
91#830—21no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)—21no—
94#803—21no—
95C4×C3⋊C8—21no—
96C23×D4C2×D311yescharacteristic
97C2×AGL(1,ℤ/8)—21no—
98#525C2×D311yescharacteristic
99#450—21no—
100#450Dic312yescomplemented
101#447—21no—
102#447Dic312yescomplemented
103#436—61no—
104#435—61no—
105#435—61no—
106C8×D4—61no—
107#433—61no—
108#418—21no—
109C2×#102—61no—
110C2×#102—61no—
111C2×#100—61no—
112C2×#100—61no—
113#408—61no—
114#407—31no—
115#407—31no—
116#407C2×D311yescharacteristic
117C2×C22⋊C8—31no—
118#380—31no—
119#328—61no—
120#327—61no—
121C24×C3—22no—
122C24×C3—44no—
123C24×C3—41no—
124C24×C3—41no—
125C24×C3—81no—
126C24×C3—81no—
127C24×C3—82no—
128C24×C3#4111yescharacteristic
129C3×#41—41no—
130C3×#41—41no—
131C2×C3×Q8C22⋊C411yescharacteristic
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—41no—
147C2×C3×D4—41no—
148C2×C3×D4—41no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4—41no—
153C2×C3×D4—41no—
154C2×C3×D4C22⋊C411yescharacteristic
155C22×C3×C4—21no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4C22×C411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4#4111yescharacteristic
164C22×Dic3—41no—
165C3×SD16—44no—
166C3×D8—44no—
167C3×M16—41no—
168C3×M16—41no—
169C2×C3×C8—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4C22⋊C411yescharacteristic
173C3×C4⋊C4C22⋊C411yescharacteristic
174C3×C22⋊C4—21no—
175C3×C22⋊C4—21no—
176C3×C22⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C42—21no—
180#217—81no—
181C4⋊Dic3—41no—
182C4⋊Dic3—41no—
183#208—41no—
184#208—41no—
185#208—41no—
186#208—41no—
187#208—81no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—41no—
193C2×C3⋊C8—41no—
194C2×C3⋊C8—41no—
195C2×C3⋊C8—81no—
196C25—21no—
197C2×#41#6711yescharacteristic
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4C2×Dic312yes—
206C23×C4C22×D311yescharacteristic
207AGL(1,ℤ/8)—42no—
208AGL(1,ℤ/8)—42no—
209C2×SD16—22no—
210C2×D8—22no—
211C2×M16—21no—
212C2×M16—31no—
213C2×M16—31no—
214C2×M16—61no—
215C2×M16—61no—
216C22×C8—31no—
217C22×C8—31no—
218C22×C8—61no—
219#123—21no—
220C22⋊Q8C2×Dic311yescharacteristic
221C22⋊Q8#6711yescharacteristic
222#121C2×Dic311yescharacteristic
223#121#6711yescharacteristic
224#120—21no—
225C4×D4—21no—
226C4×D4—21no—
227#117#6711yescharacteristic
228C2×C4⋊C4—61no—
229C2×C22⋊C4—61no—
230C4⋊C8—61no—
231C4⋊C8—61no—
232#102—22no—
233#102—22no—
234#102—122no—
235#102—122no—
236#100—122no—
237#100—122no—
238C22⋊C8—21no—
239C22⋊C8—61no—
240C22⋊C8—61no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245C22⋊C8—61no—
246C22⋊C8—61no—
247C22⋊C8C2×Dic312yes—
248#97—61no—
249C4×C8—61no—
250#95—61no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—24no—
254C23×C3—21no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—82no—
264C23×C3—82no—
265C23×C3—82no—
266C23×C3—81no—
267C23×C3—81no—
268C23×C3—82no—
269C23×C3—88no—
270C23×C3—88no—
271C23×C3—81no—
272C23×C3—81no—
273C23×C3C4×D411yescharacteristic
274C23×C3#12111yescharacteristic
275C23×C3#12311yescharacteristic
276C3×Q8—41no—
277C3×D4—24no—
278C3×D4—24no—
279C3×D4—41no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—42no—
284C3×D4—41no—
285C3×D4—41no—
286C3×D4—42no—
287C3×D4—42no—
288C3×D4—81no—
289C3×D4—81no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4C2×C22⋊C411yescharacteristic
305C2×C3×C4C4×D411yescharacteristic
306C2×C3×C4#12011yescharacteristic
307C2×C3×C4#12111yescharacteristic
308C2×Dic3—81no—
309C2×Dic3—81no—
310C2×Dic3—81no—
311C3×C8—42no—
312C3⋊C8—41no—
313C3⋊C8—41no—
314C3⋊C8—41no—
315C3⋊C8—41no—
316C3⋊C8—81no—
317C24—22no—
318C24—44no—
319C24—41no—
320C24—41no—
321C24—81no—
322C24—81no—
323C24—82no—
324C24#23711yescharacteristic
325#41—41no—
326#41—41no—
327C2×Q8#21711yescharacteristic
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4#21711yescharacteristic
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—121no—
355C22×C4D3×D411yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4#23711yescharacteristic
358C22×C4C22×Dic311yescharacteristic
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361SD16—44no—
362D8—44no—
363M16—41no—
364M16—41no—
365M16—121no—
366M16—121no—
367M16—121no—
368M16—121no—
369M16—241no—
370C2×C8—22no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—121no—
376C2×C8—121no—
377C2×C8—121no—
378C2×C8—241no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—121no—
382C4⋊C4—121no—
383C4⋊C4#21711yescharacteristic
384C4⋊C4#21711yescharacteristic
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—21no—
390C22⋊C4—241no—
391C42—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—41no—
403C22×C3—44no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—84no—
411C22×C3—84no—
412C22×C3—84no—
413C22×C3—84no—
414C22×C3—82no—
415C22×C3—82no—
416C22×C3#38611yescharacteristic
417C22×C3#47111yescharacteristic
418C22×C3#47211yescharacteristic
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4B(4,𝔽2)11yescharacteristic
426C3×C4B(4,𝔽2)11yescharacteristic
427Dic3—161no—
428C23—21no—
429C23—21no—
430C23—24no—
431C23—21no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—82no—
441C23—82no—
442C23—81no—
443C23—81no—
444C23—82no—
445C23—82no—
446C23—88no—
447C23—88no—
448C23—81no—
449C23—81no—
450C23D4×Dic311yescharacteristic
451C23#93511yescharacteristic
452C23#93911yescharacteristic
453Q8—41no—
454D4—24no—
455D4—24no—
456D4—41no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—41no—
462D4—41no—
463D4—42no—
464D4—42no—
465D4—81no—
466D4—81no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C4—241no—
484C2×C4D4×Dic311yescharacteristic
485C2×C4#93711yescharacteristic
486C2×C4#93911yescharacteristic
487C2×C4C2×#21711yescharacteristic
488C8—42no—
489C8—121no—
490C8—121no—
491C8—121no—
492C8—121no—
493C8—241no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3—81no—
499C2×C3—84no—
500C2×C3#188911yescharacteristic
501C2×C3#190311yescharacteristic
502C2×C3#190711yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—44no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—82no—
526C22—82no—
527C22#519911yescharacteristic
528C22#520011yescharacteristic
529C22#522011yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—481no—
537C4#503211yescharacteristic
538C4#503211yescharacteristic
539C3#952011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—41no—
543C2—41no—
544C2—81no—
545C2—84no—
546C2#7166911yescharacteristic
547C2#7209011yescharacteristic
548C2#7209111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3003C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#3003D3
hypocenterC3#9520
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3003

Complements

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

NH ≅ G/Ntype
2. #82919C2semidirect
3. #72129C2semidirect
4. #72128C2semidirect
5. #71709C2semidirect
6. #67687C2semidirect
7. #67685C2semidirect
13. C3×#450C4semidirect
15. C3×#447C4semidirect
100. #450Dic3semidirect
102. #447Dic3semidirect
539. C3#9520semidirect

Maximal subgroups

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

subgroupindex
C3×#30032
#829192
#721292
#721282
#717092
#676872
#676852
#95203