Finite Group Databasebeta

← #10693216

Subgroups — #10693216

Order 768.

772 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 93 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#84038C211yesmaximal, characteristic
2#78635C211yesmaximal, characteristic, complemented
3C3×#1783C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic, complemented
5#69979C211yesmaximal, characteristic, complemented
6#67687C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9520—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×#932—21no—
11#5791C2211yescharacteristic
12C2×C3×#100—21no—
13C2×C3×#100—21no—
14C3×#407C2211yescharacteristic
15C3×#407C2211yescharacteristic
16#5218—21no—
17#5218C412yescomplemented
18#5217—21no—
19#5217C412yescomplemented
20#5215—21no—
21C2×#832—21no—
22C2×#832—21no—
23#5212C2211yescharacteristic
24#5207C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5128—21no—
27#5127—21no—
28#4730—21no—
29#4727—21no—
30#67⋊C8—21no—
31#4720—21no—
32#4470—21no—
33#4467—21no—
34#3003—31no—
35#2976—31no—
36#1786—31no—
37#1783D311yescharacteristic, complemented
38#1513—31no—
39#1305—31no—
40#1293—31no—
41C25×C3—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—42no—
48C22×C3×D4—42no—
49C22×C3×D4C2×C412yes—
50C23×C3×C4C2311yescharacteristic
51C2×#235D411yescharacteristic
52C2×C3×M16—21no—
53C2×C3×M16D411yescharacteristic
54C2×C3×M16D411yescharacteristic
55#953—21no—
56C2×#217—21no—
57C2×#214—22no—
58#932—42no—
59#932—42no—
60C2×#213—22no—
61#930C2×C411yescharacteristic
62#930D411yescharacteristic
63#929—21no—
64C4×#67—21no—
65C4×#67—21no—
66#926D411yescharacteristic
67C2×C4⋊Dic3—21no—
68#924C2×C411yescharacteristic
69#924D411yescharacteristic
70C2×#208—21no—
71C22×C3⋊C8D411yescharacteristic
72C22×C3⋊C8D411yescharacteristic
73C2×ΓL(1,𝔽25)—21no—
74C2×D3×C8—21no—
75C3×#100—42no—
76C3×#100—42no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81#832—22no—
82#832—22no—
83#832—42no—
84#832—42no—
85C3⋊#95—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830C2×C412yes—
90(C2×D3)⋊C8—21no—
91(C2×D3)⋊C8—21no—
92#815—21no—
93Dic3⋊C8—21no—
94Dic3⋊C8—21no—
95C8×Dic3—21no—
96C23×D4C2×D311yescharacteristic
97C2×AGL(1,ℤ/8)—61no—
98#525—31no—
99#450—32no—
100#450—61no—
101#447—32no—
102#447—61no—
103#436—61no—
104#435—61no—
105#435—61no—
106C8×D4—61no—
107#433—61no—
108#418—61no—
109C2×#102—61no—
110C2×#102—61no—
111C2×#100—21no—
112C2×#100—21no—
113#408—61no—
114#407—31no—
115#407C2×D311yescharacteristic
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
129C2×C3×D4—22no—
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—41no—
147C2×C3×D4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4C22×C411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C2×#67—21no—
157C2×#67—21no—
158C2×#67—21no—
159C22×Dic3—41no—
160#235—41no—
161#235—41no—
162C2×D12C22⋊C411yescharacteristic
163C2×C4×D3—21no—
164C2×Dic6C22⋊C411yescharacteristic
165C3×M16—41no—
166C3×M16—41no—
167C3×M16—41no—
168C3×M16—41no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—41no—
172#217—21no—
173#217—21no—
174#217—21no—
175#217—81no—
176#214—44no—
177#213—44no—
178(C2×D3)⋊C4—21no—
179(C2×D3)⋊C4—21no—
180C4⋊Dic3—41no—
181C4⋊Dic3—41no—
182C4⋊Dic3C22⋊C411yescharacteristic
183C4⋊Dic3C22⋊C411yescharacteristic
184Dic3⋊C4—21no—
185Dic3⋊C4—21no—
186C4×Dic3—21no—
187#208—41no—
188#208—41no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—41no—
193C2×C3⋊C8—41no—
194ΓL(1,𝔽25)—81no—
195D3×C8—81no—
196C25—21no—
197C2×#41—31no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4C4×D312yes—
206C23×C4C22×D311yescharacteristic
207AGL(1,ℤ/8)—122no—
208AGL(1,ℤ/8)—122no—
209C2×SD16—62no—
210C2×D8—62no—
211C2×M16—21no—
212C2×M16—61no—
213C2×M16—61no—
214C2×M16#6711yescharacteristic
215C2×M16#6711yescharacteristic
216C22×C8—31no—
217C22×C8—31no—
218C22×C8—61no—
219#123—61no—
220C22⋊Q8—31no—
221C22⋊Q8—31no—
222#121—31no—
223#121—31no—
224#120—61no—
225C4×D4—61no—
226C4×D4—61no—
227#117—31no—
228C2×C4⋊C4—61no—
229C2×C22⋊C4—61no—
230C4⋊C8—61no—
231C4⋊C8—61no—
232#102—62no—
233#102—62no—
234#102—122no—
235#102—122no—
236#100—42no—
237#100—42no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—32no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245C22⋊C8—61no—
246C22⋊C8—61no—
247C22⋊C8—61no—
248#97—61no—
249C4×C8—61no—
250#95—61no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—24no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—82no—
263C23×C3—82no—
264C23×C3—82no—
265C23×C3—81no—
266C23×C3—81no—
267C23×C3—82no—
268C23×C3—88no—
269C23×C3—88no—
270C23×C3—81no—
271C23×C3—81no—
272C23×C3C4×D411yescharacteristic
273C23×C3#12111yescharacteristic
274C23×C3#12311yescharacteristic
275C22×D3—21no—
276C3×D4—24no—
277C3×D4—24no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—42no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4C2×C22⋊C411yescharacteristic
294C2×C3×C4C4×D411yescharacteristic
295C2×C3×C4#12011yescharacteristic
296C2×C3×C4#12111yescharacteristic
297#67—41no—
298#67—41no—
299#67—81no—
300#67—81no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—81no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307D12—41no—
308C4×D3—41no—
309C4×D3—41no—
310Dic6—41no—
311C3×C8—41no—
312C3×C8—41no—
313C3×C8—81no—
314C3⋊C8—42no—
315C3⋊C8—41no—
316C3⋊C8—41no—
317C24—22no—
318C24—44no—
319C24—41no—
320C24—41no—
321C24—81no—
322C24—81no—
323C24—82no—
324C24#23511yescharacteristic
325#41—121no—
326#41—121no—
327C2×Q8—31no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—31no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—61no—
354C22×C4—121no—
355C22×C4C2×C4×D311yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4#23711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361SD16—124no—
362D8—124no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—121no—
368M16—121no—
369M16—241no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—41no—
373C2×C8—62no—
374C2×C8—61no—
375C2×C8—61no—
376C2×C8—121no—
377C2×C8—121no—
378C2×C8—241no—
379C4⋊C4—31no—
380C4⋊C4—31no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—121no—
384C4⋊C4—121no—
385C22⋊C4—61no—
386C22⋊C4—61no—
387C22⋊C4—61no—
388C22⋊C4—61no—
389C22⋊C4—61no—
390C22⋊C4—241no—
391C42—61no—
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—44no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3—84no—
408C22×C3—84no—
409C22×C3—84no—
410C22×C3—84no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3#38611yescharacteristic
414C22×C3#47111yescharacteristic
415C22×C3#47211yescharacteristic
416C2×D3—41no—
417C2×D3—41no—
418C2×D3—41no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4B(4,𝔽2)11yescharacteristic
423C3×C4B(4,𝔽2)11yescharacteristic
424Dic3—41no—
425Dic3—41no—
426Dic3—41no—
427Dic3—161no—
428C23—21no—
429C23—21no—
430C23—24no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—61no—
440C23—82no—
441C23—82no—
442C23—81no—
443C23—81no—
444C23—82no—
445C23—82no—
446C23—88no—
447C23—88no—
448C23—81no—
449C23—81no—
450C23#88311yescharacteristic
451C23#88411yescharacteristic
452C23C4×#6711yescharacteristic
453Q8—121no—
454D4—24no—
455D4—24no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—121no—
463D4—121no—
464D4—121no—
465D4—241no—
466D4—241no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C4—241no—
484C2×C4D3×C22⋊C411yescharacteristic
485C2×C4#67⋊C411yescharacteristic
486C2×C4#93711yescharacteristic
487C2×C4#93811yescharacteristic
488C8—41no—
489C8—41no—
490C8—81no—
491C8—122no—
492C8—121no—
493C8—121no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3—84no—
499C2×C3#188911yescharacteristic
500C2×C3#190311yescharacteristic
501C2×C3#190711yescharacteristic
502D3—81no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—44no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—84no—
519C22—84no—
520C22—84no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22—121no—
525C22—121no—
526C22—121no—
527C22#474811yescharacteristic
528C22#474911yescharacteristic
529C22#495611yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—121no—
534C4—121no—
535C4—121no—
536C4—481no—
537C4#503211yescharacteristic
538C4#503211yescharacteristic
539C3#952011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—41no—
543C2—41no—
544C2—84no—
545C2—241no—
546C2#6949911yescharacteristic
547C2#7008711yescharacteristic
548C2#7008811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1783C2
socleC22×C3#386
prime radical#84038C2
cosocleC23×C3×C4C23
hypercenter#1783D3
hypocenterC3#9520
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
2. #78635C2semidirect
3. C3×#1783C2semidirect
4. #72128C2semidirect
5. #69979C2semidirect
6. #67687C2semidirect
7. #67684C2semidirect
17. #5218C4semidirect
19. #5217C4semidirect
37. #1783D3semidirect
539. C3#9520semidirect

Maximal subgroups

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

subgroupindex
#840382
#786352
C3×#17832
#721282
#699792
#676872
#676842
#95203