Finite Group Databasebeta

← #10695824

Subgroups — #10695824

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×#2816C211yesmaximal, characteristic, complemented
2#82735C211yesmaximal, characteristic, complemented
3#82624C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic, complemented
5#71674C211yesmaximal, characteristic, complemented
6#69537C211yesmaximal, characteristic, complemented
7#67629C211yesmaximal, characteristic, complemented
8#9981—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C2×#942—21no—
13C2×#932—21no—
14C3×#399—21no—
15C3×#398—21no—
16C3×#394—21no—
17C3×#394—22no—
18C3×#393—21no—
19C3×#381C2211yescharacteristic
20#5211C2211yescharacteristic
21#5207C2211yescharacteristic
22#5207C2211yescharacteristic
23#5050—21no—
24#5047—22no—
25#5046—21no—
26#5040—21no—
27#5037—22no—
28#5036—21no—
29#4966—21no—
30#4966—21no—
31C2×#806—22no—
32#4538—22no—
33#4537—22no—
34#3007—31no—
35#3006—31no—
36#2816D311yescharacteristic, complemented
37#1782—31no—
38#1511—31no—
39#1510—31no—
40#1306—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×#67—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—44no—
51C22×D12D411yescharacteristic
52C22×C4×D3—21no—
53C22×Dic6D411yescharacteristic
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4D412yes—
71C2×#217—21no—
72C2×#216—22no—
73#942—44no—
74C2×#215—22no—
75C2×#214—22no—
76#932—44no—
77C2×#213—22no—
78C2×#208—21no—
79C2×#208—21no—
80C3×#95—22no—
81C3×#95—21no—
82C3×#95—21no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85#830—21no—
86#830—21no—
87#830D412yes—
88#830D412yes—
89#810—22no—
90#810—22no—
91#809—22no—
92#809—22no—
93#806—44no—
94C22×#41—31no—
95C2×#137—61no—
96C2×AGL(1,ℤ/8)—61no—
97C2×C22⋊Q8C2×D311yescharacteristic
98C2×#121C2×D311yescharacteristic
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—62no—
108#407—31no—
109#407—31no—
110#399—21no—
111#398—21no—
112#394—21no—
113#394—22no—
114#393—21no—
115#381—31no—
116#381C2×D311yescharacteristic
117#328—62no—
118#327—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—42no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—42no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—82no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C2×#67—24no—
143C2×#67—42no—
144C2×#67—41no—
145C2×#67—41no—
146C22×Dic3—21no—
147C22×Dic3—41no—
148#235—24no—
149#235—24no—
150#235—42no—
151#235—42no—
152#235—42no—
153#235—42no—
154#235—42no—
155#235—42no—
156C2×D12—21no—
157C2×D12—21no—
158C2×D12—42no—
159C2×C4×D3—42no—
160C2×C4×D3—42no—
161C2×C4×D3—41no—
162C2×C4×D3—41no—
163C2×Dic6—21no—
164C2×Dic6—21no—
165C2×Dic6—42no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—82no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—42no—
178C3×C22⋊C4—82no—
179C3×C22⋊C4—82no—
180#217—42no—
181#216—44no—
182#215—44no—
183#214—44no—
184#213—44no—
185#208—42no—
186#208—42no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×#41—61no—
190C2×#41—61no—
191C2×#41—61no—
192C2×#41—61no—
193C2×#41—124no—
194C22×Q8—31no—
195C22×Q8#6711yescharacteristic
196C22×D4—21no—
197C22×D4—31no—
198C22×D4—61no—
199C22×D4#6711yescharacteristic
200C23×C4—61no—
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—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214#121—21no—
215#121—21no—
216#121—21no—
217#121—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4#6711yescharacteristic
221C2×C4⋊C4#6711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4#6712yes—
228#103—62no—
229#103—62no—
230#102—62no—
231#102—62no—
232#99—124no—
233C22⋊C8—32no—
234C22⋊C8—32no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—61no—
241#95—61no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3#12011yescharacteristic
250C23×C3#12111yescharacteristic
251C23×C3#12711yescharacteristic
252C22×D3—22no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—81no—
256C22×D3—81no—
257C3×Q8—42no—
258C3×Q8—81no—
259C3×D4—42no—
260C3×D4—81no—
261C3×D4—82no—
262C3×D4—82no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283#67—44no—
284#67—44no—
285C2×Dic3—22no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—81no—
289C2×Dic3—81no—
290C2×Dic3—81no—
291D12—22no—
292D12—22no—
293D12—41no—
294D12—41no—
295C4×D3—44no—
296C4×D3—44no—
297Dic6—22no—
298Dic6—22no—
299Dic6—41no—
300Dic6—41no—
301C3⋊C8—42no—
302C3⋊C8—42no—
303C24—21no—
304C24—61no—
305C24C2×#6711yescharacteristic
306#41—64no—
307#41—64no—
308#41—122no—
309#41—122no—
310#41—122no—
311#41—122no—
312#41—122no—
313#41—122no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—42no—
317C2×Q8—61no—
318C2×Q8—61no—
319C2×Q8—122no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—64no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—82no—
329C2×D4—122no—
330C2×D4—122no—
331C2×D4—121no—
332C2×D4—121no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—42no—
338C22×C4—61no—
339C22×C4—122no—
340C22×C4—122no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4D3×D411yescharacteristic
345C22×C4D3×D411yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4#23711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350Q16—124no—
351SD16—124no—
352SD16—124no—
353D8—124no—
354M16—122no—
355M16—122no—
356C2×C8—62no—
357C2×C8—62no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—82no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—42no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—122no—
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—82no—
382C22×C3—84no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3#39211yescharacteristic
387C22×C3B(4,𝔽2)12yes—
388C2×D3—42no—
389C2×D3—82no—
390C2×D3—81no—
391C2×D3—81no—
392C2×D3—82no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—82no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4—81no—
401Dic3—42no—
402Dic3—161no—
403C23—22no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—62no—
411C23—121no—
412C23—121no—
413C23—241no—
414C23—241no—
415C23#93811yescharacteristic
416C23#94011yescharacteristic
417C23#95311yescharacteristic
418Q8—42no—
419Q8—62no—
420Q8—62no—
421Q8—81no—
422Q8—121no—
423Q8—121no—
424D4—42no—
425D4—62no—
426D4—62no—
427D4—81no—
428D4—82no—
429D4—82no—
430D4—121no—
431D4—121no—
432D4—124no—
433D4—124no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—62no—
451C2×C4—124no—
452C2×C4—124no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—241no—
456C2×C4—241no—
457C2×C4—241no—
458C2×C4#93711yescharacteristic
459C2×C4#93711yescharacteristic
460C2×C4#93911yescharacteristic
461C2×C4#93911yescharacteristic
462C8—122no—
463C8—122no—
464C2×C3—22no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—161no—
468C2×C3#200811yescharacteristic
469C2×C3#200911yescharacteristic
470C2×C3#202111yescharacteristic
471D3—82no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—82no—
481C22—84no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—122no—
486C22—242no—
487C22—242no—
488C22—241no—
489C22—241no—
490C22#522311yescharacteristic
491C22#524412yes—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—82no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—122no—
501C4—481no—
502C3#998111yescharacteristic, complemented, Sylow
503C2—22no—
504C2—41no—
505C2—41no—
506C2—161no—
507C2—242no—
508C2#7185111yescharacteristic
509C2#7185811yescharacteristic
510C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2816C2
socleC22×C3#392
prime radical#82624C2
hypercenter#2816D3
hypocenterC3#9981
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2816

Complements

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

NH ≅ G/Ntype
1. C3×#2816C2semidirect
2. #82735C2semidirect
4. #71689C2semidirect
5. #71674C2semidirect
6. #69537C2semidirect
7. #67629C2semidirect
36. #2816D3semidirect
502. C3#9981semidirect

Maximal subgroups

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

subgroupindex
C3×#28162
#827352
#826242
#716892
#716742
#695372
#676292
#99813