Finite Group Databasebeta

← #10695826

Subgroups — #10695826

Order 768.

764 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3006C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic, complemented
3#82697C211yesmaximal, characteristic, complemented
4#72139C211yesmaximal, characteristic, complemented
5#71674C211yesmaximal, characteristic, complemented
6#69981C211yesmaximal, characteristic, complemented
7#67686C211yesmaximal, characteristic, complemented
8#9981—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#951—21no—
13C2×#924C2211yescharacteristic
14C3×#449—22no—
15C3×#448—21no—
16C3×#447—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19#5239—22no—
20#5232—21no—
21#5228—21no—
22#5222—21no—
23#5222—22no—
24C2×#834—22no—
25#5218—21no—
26#5211C2211yescharacteristic
27#5211C2211yescharacteristic
28#5207C2211yescharacteristic
29#4983—21no—
30#4949—21no—
31#4947—21no—
32#4537—22no—
33#4467—22no—
34#3007—31no—
35#3006D311yescharacteristic
36#2816—31no—
37#1782—31no—
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×D4—21no—
49C22×C3×D4D411yescharacteristic
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C22×Dic6D411yescharacteristic
54C3×AGL(1,ℤ/8)—44no—
55C2×C3×SD16—22no—
56C2×C3×D8—22no—
57C2×C3×M16—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×#217—21no—
65C2×#217—21no—
66C2×#217—22no—
67C2×#217—22no—
68C2×#217D412yes—
69#951—44no—
70C2×#216—22no—
71C2×#214—22no—
72C2×C4⋊Dic3D411yescharacteristic
73#924—21no—
74#924—21no—
75#924—21no—
76#924—21no—
77C2×Dic3⋊C4—21no—
78C2×Dic3⋊C4—22no—
79C2×#208—21no—
80C3×#103—22no—
81C3×#102—22no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8D412yes—
84#835—22no—
85#834—44no—
86#832—22no—
87C3⋊#95—22no—
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92#830—21no—
93#830D412yes—
94C22×#41C2×D311yescharacteristic
95C2×#137—61no—
96C2×AGL(1,ℤ/8)—21no—
97C2×C22⋊Q8—31no—
98C2×#121C2×D311yescharacteristic
99#452—62no—
100#451—61no—
101#450—61no—
102#449—22no—
103#448—21no—
104#447—21no—
105#417—21no—
106#417—61no—
107C2×#99—62no—
108#407—31no—
109#407C2×D311yescharacteristic
110#399—61no—
111#398—61no—
112#394—62no—
113#394—61no—
114#393—61no—
115#381—31no—
116#381—31no—
117#328—62no—
118#327—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3C2×D411yescharacteristic
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—21no—
137C2×C3×D4—21no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—82no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—41no—
163C2×Dic6—21no—
164C2×Dic6—21no—
165C2×Dic6—42no—
166C3×SD16—44no—
167C3×D8—44no—
168C3×M16—42no—
169C2×C3×C8—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—41no—
174#217—41no—
175#217—41no—
176#217—42no—
177#217—42no—
178#217—82no—
179#217—82no—
180#216—44no—
181#214—44no—
182C4⋊Dic3—21no—
183C4⋊Dic3—21no—
184Dic3⋊C4—41no—
185Dic3⋊C4—41no—
186Dic3⋊C4—82no—
187#208—42no—
188C2×C3⋊C8—22no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—44no—
194C22×Q8—31no—
195C22×Q8#6711yescharacteristic
196C22×D4—21no—
197C22×D4—21no—
198C22×D4#6711yescharacteristic
199C22×D4#6711yescharacteristic
200C23×C4—21no—
201C23×C4C22×D311yescharacteristic
202#137—124no—
203AGL(1,ℤ/8)—44no—
204C2×Q16—62no—
205C2×SD16—22no—
206C2×SD16—62no—
207C2×D8—22no—
208C2×M16—21no—
209C2×M16—61no—
210C22⋊Q8—61no—
211C22⋊Q8—61no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214#121—21no—
215#121—21no—
216#121—21no—
217#121—21no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4#6711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—32no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228#103—22no—
229#103—62no—
230#102—22no—
231#102—62no—
232#99—124no—
233C22⋊C8—21no—
234C22⋊C8—32no—
235C22⋊C8—61no—
236C22⋊C8#6712yes—
237#95—62no—
238#95—61no—
239#95—61no—
240#95—61no—
241#95—61no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—81no—
253C23×C3—81no—
254C23×C3#12011yescharacteristic
255C23×C3#12111yescharacteristic
256C23×C3#12711yescharacteristic
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—42no—
266C3×D4—44no—
267C3×D4—44no—
268C3×D4—81no—
269C3×D4—82no—
270C3×D4—82no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4#12011yescharacteristic
285C2×C3×C4#12011yescharacteristic
286C2×C3×C4#12111yescharacteristic
287C2×C3×C4#12111yescharacteristic
288C2×Dic3—41no—
289C2×Dic3—42no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—81no—
297C2×Dic3—81no—
298C2×Dic3—81no—
299Dic6—42no—
300Dic6—81no—
301C3×C8—42no—
302C3⋊C8—42no—
303C24—21no—
304C24—21no—
305C24D3×D411yescharacteristic
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—21no—
324C2×D4—21no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—82no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—62no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—121no—
344C22×C4D3×D411yescharacteristic
345C22×C4D3×D411yescharacteristic
346C22×C4#23711yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350Q16—124no—
351SD16—44no—
352SD16—124no—
353D8—44no—
354M16—42no—
355M16—122no—
356C2×C8—22no—
357C2×C8—62no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—242no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—121no—
368C22⋊C4—121no—
369C22⋊C4—122no—
370C22⋊C4—122no—
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—84no—
384C22×C3—82no—
385C22×C3—82no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3#39211yescharacteristic
392C22×C3B(4,𝔽2)12yes—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—42no—
398C3×C4—81no—
399Dic3—82no—
400Dic3—81no—
401Dic3—81no—
402Dic3—161no—
403C23—22no—
404C23—22no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—81no—
414C23—81no—
415C23#93711yescharacteristic
416C23#93911yescharacteristic
417C23#94011yescharacteristic
418Q8—22no—
419Q8—22no—
420Q8—41no—
421Q8—41no—
422Q8—122no—
423Q8—241no—
424D4—22no—
425D4—22no—
426D4—41no—
427D4—41no—
428D4—42no—
429D4—44no—
430D4—44no—
431D4—81no—
432D4—82no—
433D4—82no—
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—41no—
447C2×C4—121no—
448C2×C4—122no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—241no—
456C2×C4—241no—
457C2×C4—241no—
458C2×C4#93711yescharacteristic
459C2×C4#93811yescharacteristic
460C2×C4#93911yescharacteristic
461C2×C4#95311yescharacteristic
462C8—42no—
463C8—122no—
464C2×C3—22no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—82no—
468C2×C3—161no—
469C2×C3#200811yescharacteristic
470C2×C3#200911yescharacteristic
471C2×C3#202111yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—42no—
481C22—82no—
482C22—84no—
483C22—82no—
484C22—82no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#503212yes—
491C22#522311yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—81no—
498C4—242no—
499C4—241no—
500C4—241no—
501C4—481no—
502C3#998111yescharacteristic, complemented, Sylow
503C2—22no—
504C2—41no—
505C2—41no—
506C2—82no—
507C2—161no—
508C2#7167111yescharacteristic
509C2#7209911yescharacteristic
510C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3006C2
socleC22×C3#392
hypercenter#3006D3
hypocenterC3#9981
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
2. #84084C2semidirect
3. #82697C2semidirect
4. #72139C2semidirect
5. #71674C2semidirect
6. #69981C2semidirect
7. #67686C2semidirect
502. C3#9981semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840842
#826972
#721392
#716742
#699812
#676862
#99813