Finite Group Databasebeta

← #10695523

Subgroups — #10695523

Order 768.

808 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2615C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic, complemented
3#82631C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#69529C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9934—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#121—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#951—21no—
13C2×#924C2211yescharacteristic
14C3×#393—21no—
15C3×#392—21no—
16C3×#392—22no—
17C3×#380—21no—
18C3×#380C2211yescharacteristic
19#5239—22no—
20C2×#837—22no—
21#5232—21no—
22#5228—21no—
23#5218—21no—
24#5218—21no—
25C4×#217C2211yescharacteristic
26#5207C2211yescharacteristic
27#5207C2211yescharacteristic
28#5042—22no—
29#5040—21no—
30#5033—21no—
31#4970—21no—
32#4537—22no—
33#4527—22no—
34#3193—31no—
35#3007—31no—
36#2615D311yescharacteristic
37#1996—31no—
38#1789—31no—
39#1507—31no—
40#1305—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—22no—
49C22×C3×D4—21no—
50C22×C3×D4D411yescharacteristic
51C22×C3×D4D411yescharacteristic
52C23×C3×C4—21no—
53C23×C3×C4C2311yescharacteristic
54C22×Dic6D411yescharacteristic, complemented
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—44no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4D412yes—
66C2×#217—21no—
67C2×#217—21no—
68#951—44no—
69C2×#216—22no—
70C2×#214—22no—
71C2×#214—22no—
72C2×C4⋊Dic3D411yescharacteristic, complemented
73#924—21no—
74#924—21no—
75#924—21no—
76#924—21no—
77C2×Dic3⋊C4—21no—
78C2×C4×Dic3—21no—
79C2×#208—21no—
80C2×#208D412yescomplemented
81C3×#95—22no—
82C3×#95—21no—
83#837—44no—
84#835—22no—
85#832—22no—
86#832—22no—
87C3⋊#95—21no—
88#830—21no—
89#830—21no—
90#830—21no—
91#830D412yescomplemented
92#810—22no—
93#807—22no—
94C22×#41C2×D311yescharacteristic
95C2×#137—61no—
96C2×C22⋊Q8—31no—
97C2×#121—21no—
98C2×#121C2×D311yescharacteristic
99#484—61no—
100#480—61no—
101#468—62no—
102#452—62no—
103#451—61no—
104#450—61no—
105#450—61no—
106#448—61no—
107C2×#104—62no—
108#417—61no—
109#407—31no—
110#407—31no—
111#393—21no—
112#392—21no—
113#392—22no—
114#380—21no—
115#380C2×D311yescharacteristic
116C4×C22⋊C4—31no—
117#342—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—21no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—44no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—82no—
147C2×C3×D4—82no—
148C2×C3×D4—82no—
149C22×C3×C4—24no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4#4111yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—41no—
166C2×Dic6—21no—
167C2×Dic6—21no—
168C2×Dic6—42no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—42no—
172C3×C22⋊C4—44no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—42no—
176C3×C22⋊C4—82no—
177#217—22no—
178#217—41no—
179#217—41no—
180#217—41no—
181#216—44no—
182#214—44no—
183#214—82no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186Dic3⋊C4—41no—
187Dic3⋊C4—41no—
188C4×Dic3—42no—
189#208—42no—
190#208—42no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—44no—
198C22×Q8—31no—
199C22×Q8#6711yescharacteristic
200C22×D4—21no—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4#6711yescharacteristic
204C22×D4#6711yescharacteristic
205C23×C4—21no—
206C23×C4C22×D311yescharacteristic
207#137—124no—
208C2×Q16—62no—
209C2×SD16—62no—
210C2×SD16—62no—
211C2×M16—32no—
212C2×M16—61no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—61no—
216C22⋊Q8—61no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221#121—44no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4#6711yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4#6712yes—
232C2×C42—61no—
233#107—62no—
234#104—124no—
235#103—62no—
236#103—62no—
237#102—62no—
238#102—62no—
239C22⋊C8—32no—
240C22⋊C8—61no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243#95—22no—
244#95—21no—
245#95—61no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—22no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3#12011yescharacteristic
263C23×C3#12111yescharacteristic
264C23×C3#12711yescharacteristic
265C3×Q8—22no—
266C3×Q8—22no—
267C3×Q8—41no—
268C3×Q8—41no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—41no—
272C3×D4—41no—
273C3×D4—44no—
274C3×D4—44no—
275C3×D4—42no—
276C3×D4—81no—
277C3×D4—82no—
278C3×D4—82no—
279C3×D4—88no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—44no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4#12011yescharacteristic
297C2×C3×C4#12011yescharacteristic
298C2×C3×C4#12111yescharacteristic
299C2×C3×C4#12111yescharacteristic
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—81no—
304C2×Dic3—81no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307Dic6—42no—
308Dic6—42no—
309C3⋊C8—42no—
310C3⋊C8—42no—
311C24—21no—
312C24—21no—
313C24C2×#6711yescharacteristic
314#41—24no—
315#41—24no—
316#41—42no—
317#41—42no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—42no—
325C2×Q8—61no—
326C2×Q8—61no—
327C2×Q8—122no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—82no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—121no—
355C22×C4D3×D411yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4#23711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361Q16—124no—
362SD16—124no—
363SD16—242no—
364M16—122no—
365M16—122no—
366C2×C8—62no—
367C2×C8—62no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—42no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C22⋊C4—44no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—42no—
379C22⋊C4—62no—
380C22⋊C4—82no—
381C22⋊C4—121no—
382C22⋊C4—121no—
383C22⋊C4—121no—
384C42—122no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3#39211yescharacteristic
408C22×C3#45511yescharacteristic
409C22×C3#45611yescharacteristic
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—42no—
414C3×C4—82no—
415C3×C4—81no—
416C3×C4B(4,𝔽2)12yes—
417Dic3—82no—
418Dic3—81no—
419Dic3—81no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23#93811yescharacteristic
437C23#94011yescharacteristic
438C23#95311yescharacteristic
439Q8—22no—
440Q8—22no—
441Q8—41no—
442Q8—41no—
443Q8—122no—
444Q8—122no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—41no—
449D4—44no—
450D4—44no—
451D4—42no—
452D4—81no—
453D4—82no—
454D4—82no—
455D4—88no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—122no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—241no—
476C2×C4—241no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4#93711yescharacteristic
480C2×C4#93711yescharacteristic
481C2×C4#93911yescharacteristic
482C2×C4#93911yescharacteristic
483C8—122no—
484C8—122no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—82no—
490C2×C3—161no—
491C2×C3#200711yescharacteristic
492C2×C3#200811yescharacteristic
493C2×C3#202111yescharacteristic
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—82no—
510C22—82no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#522311yescharacteristic
517C22#524211yescharacteristic
518C22#524611yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—82no—
524C4—81no—
525C4—242no—
526C4—241no—
527C4—241no—
528C4#503212yes—
529C3#993411yescharacteristic, complemented, Sylow
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—82no—
535C2—161no—
536C2#7167111yescharacteristic
537C2#7185811yescharacteristic
538C2#7188111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2615C2
socleC22×C3#392
hypercenter#2615D3
hypocenterC3#9934
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2615

Complements

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

NH ≅ G/Ntype
2. #84084C2semidirect
3. #82631C2semidirect
4. #72138C2semidirect
5. #71673C2semidirect
6. #69529C2semidirect
7. #67626C2semidirect
54. C22×Dic6D4semidirect
72. C2×C4⋊Dic3D4semidirect
80. C2×#208D4semidirect
91. #830D4semidirect
529. C3#9934semidirect

Maximal subgroups

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

subgroupindex
C3×#26152
#840842
#826312
#721382
#716732
#695292
#676262
#99343