Finite Group Databasebeta

← #10695528

Subgroups — #10695528

Order 768.

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

normal subgroups only

This group has 750 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×#3195C211yesmaximal, characteristic, complemented
2#82820C211yesmaximal, characteristic
3#82735C211yesmaximal, characteristic
4#71701C211yesmaximal, characteristic, complemented
5#71688C211yesmaximal, characteristic, complemented
6#69991C211yesmaximal, characteristic
7#67684C211yesmaximal, characteristic, complemented
8#9935—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#946—21no—
13C2×#942—21no—
14C3×#483—21no—
15C3×#483—21no—
16C3×#470—22no—
17C3×#451—21no—
18C3×#451—21no—
19C3×#407C2211yescharacteristic
20#5212C2211yescharacteristic
21C4×#217C2211yescharacteristic
22#5207C2211yescharacteristic
23C2×#825—22no—
24#5050—21no—
25#5047—22no—
26#5046—21no—
27#4991—21no—
28#4987—21no—
29#4979—21no—
30#4966—21no—
31#4959—22no—
32#4538—22no—
33#4526—22no—
34#3195D311yescharacteristic, complemented
35#3007—31no—
36#2616—31no—
37#1996—31no—
38#1789—31no—
39#1508—31no—
40#1305—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—44no—
50C22×D12D411yescharacteristic
51C22×C4×D3—21no—
52C22×Dic6D411yescharacteristic
53C2×C3×Q16—22no—
54C2×C3×M16D412yes—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C3×C22⋊Q8—21no—
62C3×C22⋊Q8—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×#217—21no—
70C2×#217D412yes—
71#946—44no—
72C2×#216—22no—
73#942—44no—
74C2×#215—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3—21no—
78C2×Dic3⋊C4—22no—
79C2×C4×Dic3—21no—
80C2×#208—21no—
81C3×#106—22no—
82C3×#103—22no—
83C3×#103—22no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8—21no—
86C3⋊#95—22no—
87C3⋊#95—21no—
88C3⋊#95—21no—
89#830—21no—
90#830D412yes—
91#825—44no—
92#810—22no—
93#809—22no—
94C22×#41—31no—
95C2×#137—61no—
96C2×C22⋊Q8—61no—
97C2×C22⋊Q8C2×D311yescharacteristic
98C2×C22⋊Q8C2×D311yescharacteristic
99#483—21no—
100#483—21no—
101#470—22no—
102#452—62no—
103#451—21no—
104#451—21no—
105#451—61no—
106#450—61no—
107C2×#104—62no—
108#417—61no—
109#407—31no—
110#407C2×D311yescharacteristic
111#395—61no—
112#394—62no—
113#394—61no—
114#380—31no—
115#380—61no—
116C4×C22⋊C4—31no—
117#342—62no—
118#328—62no—
119C24×C3C2×D411yescharacteristic
120C23×D3—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—42no—
126C2×C3×Q8—42no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—42no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—42no—
139C2×#67—42no—
140C2×#67—41no—
141C2×#67—41no—
142C22×Dic3—21no—
143C22×Dic3—22no—
144C22×Dic3—41no—
145#235—24no—
146#235—24no—
147#235—42no—
148#235—42no—
149#235—42no—
150#235—42no—
151#235—42no—
152#235—42no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—42no—
156C2×C4×D3—24no—
157C2×C4×D3—42no—
158C2×C4×D3—41no—
159C2×C4×D3—41no—
160C2×Dic6—21no—
161C2×Dic6—21no—
162C2×Dic6—42no—
163C3×Q16—82no—
164C3×M16—42no—
165C2×C3×C8—22no—
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—41no—
173C3×C4⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178#217—22no—
179#217—41no—
180#217—42no—
181#216—44no—
182#215—44no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—82no—
185C4⋊Dic3—42no—
186Dic3⋊C4—44no—
187C4×Dic3—42no—
188#208—42no—
189C2×C3⋊C8—22no—
190C2×#41—61no—
191C2×#41—61no—
192C2×#41—61no—
193C2×#41—61no—
194C2×#41—124no—
195C22×Q8—31no—
196C22×Q8#6711yescharacteristic
197C22×Q8#6711yescharacteristic
198C22×D4—31no—
199C22×D4—61no—
200C23×C4—61no—
201C23×C4C22×D311yescharacteristic
202#137—124no—
203C2×Q16—22no—
204C2×Q16—62no—
205C2×SD16—62no—
206C2×M16—61no—
207C2×M16#6712yes—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—124no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4#6711yescharacteristic
222C2×C4⋊C4#6711yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—32no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C42—61no—
230#106—22no—
231#104—124no—
232#103—22no—
233#103—22no—
234#103—62no—
235#102—62no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—32no—
239C22⋊C8—61no—
240#95—62no—
241#95—61no—
242#95—61no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3#12011yescharacteristic
248C23×C3#12111yescharacteristic
249C23×C3#12711yescharacteristic
250C22×D3—22no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C3×Q8—42no—
258C3×Q8—42no—
259C3×Q8—42no—
260C3×Q8—81no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4#12011yescharacteristic
277C2×C3×C4#12011yescharacteristic
278C2×C3×C4#12111yescharacteristic
279C2×C3×C4#12111yescharacteristic
280#67—44no—
281#67—44no—
282C2×Dic3—22no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
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—61no—
304C24D3×D411yescharacteristic
305#41—64no—
306#41—64no—
307#41—122no—
308#41—122no—
309#41—122no—
310#41—122no—
311#41—122no—
312#41—122no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—61no—
320C2×Q8—61no—
321C2×Q8—122no—
322C2×D4—61no—
323C2×D4—61no—
324C2×D4—122no—
325C2×D4—122no—
326C2×D4—122no—
327C2×D4—121no—
328C2×D4—121no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—42no—
334C22×C4—64no—
335C22×C4—61no—
336C22×C4—62no—
337C22×C4—122no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4D3×D411yescharacteristic
342C22×C4D3×D411yescharacteristic
343C22×C4#23711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345C22×C4C2×#6711yescharacteristic
346C22×C4C2×#6711yescharacteristic
347Q16—82no—
348Q16—124no—
349SD16—124no—
350M16—42no—
351M16—122no—
352C2×C8—22no—
353C2×C8—62no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—122no—
363C4⋊C4—124no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—62no—
369C22⋊C4—124no—
370C22⋊C4—121no—
371C22⋊C4—122no—
372C22⋊C4—242no—
373C42—122no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—81no—
383C22×C3—81no—
384C22×C3#39211yescharacteristic
385C22×C3#45511yescharacteristic
386C22×C3#45611yescharacteristic
387C2×D3—42no—
388C2×D3—42no—
389C2×D3—42no—
390C2×D3—81no—
391C2×D3—81no—
392C2×D3—81no—
393C2×D3—81no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4B(4,𝔽2)12yes—
402Dic3—42no—
403Dic3—82no—
404Dic3—82no—
405C23—21no—
406C23—21no—
407C23—41no—
408C23—41no—
409C23—62no—
410C23—121no—
411C23—121no—
412C23—121no—
413C23—121no—
414C23—121no—
415C23—121no—
416C23#93711yescharacteristic
417C23#93911yescharacteristic
418C23#94011yescharacteristic
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—62no—
423Q8—62no—
424Q8—81no—
425Q8—121no—
426Q8—121no—
427D4—62no—
428D4—62no—
429D4—121no—
430D4—121no—
431D4—124no—
432D4—124no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—62no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—124no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—124no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—241no—
458C2×C4—241no—
459C2×C4#93711yescharacteristic
460C2×C4#93811yescharacteristic
461C2×C4#93911yescharacteristic
462C2×C4#95311yescharacteristic
463C8—42no—
464C8—122no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3#200711yescharacteristic
470C2×C3#200911yescharacteristic
471C2×C3#202111yescharacteristic
472D3—82no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—81no—
482C22—81no—
483C22—122no—
484C22—122no—
485C22—122no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22—241no—
490C22#520211yescharacteristic
491C22#520511yescharacteristic
492C22#522311yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—122no—
501C4—242no—
502C4—242no—
503C4#524412yes—
504C3#993511yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—41no—
508C2—41no—
509C2—242no—
510C2#7185111yescharacteristic
511C2#7211011yescharacteristic
512C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3195C2
socleC22×C3#392
prime radicalC22×#235C22
hypercenter#3195D3
hypocenterC3#9935
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
1. C3×#3195C2semidirect
4. #71701C2semidirect
5. #71688C2semidirect
7. #67684C2semidirect
34. #3195D3semidirect
504. C3#9935semidirect

Maximal subgroups

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

subgroupindex
C3×#31952
#828202
#827352
#717012
#716882
#699912
#676842
#99353