Finite Group Databasebeta

← #10823683

Subgroups — #10823683

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85811C211yesmaximal, characteristic
2C3×#2327C211yesmaximal, characteristic, complemented
3C4×#942C212yesmaximal, complemented
4#71795C212yesmaximal, complemented
5C4×#836C211yesmaximal, characteristic, complemented
6#69695C212yesmaximal, complemented
7#69694C211yesmaximal, characteristic
8#69673C211yesmaximal, characteristic, complemented
9#69563C211yesmaximal, characteristic
10#69543C211yesmaximal, characteristic, complemented
11#69430C212yesmaximal, complemented
12#23594—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16#5816C2212yes—
17C2×#942C2212yes—
18#5583—22no—
19#5582—22no—
20#5580—22no—
21#5578—22no—
22#5515—22no—
23#5510C2211yescharacteristic
24#5500C2211yescharacteristic
25#5479—22no—
26C4×#235C2212yes—
27C3×#391C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#387—22no—
30C3×#387—22no—
31C3×#382C2211yescharacteristic
32C3×#376C2211yescharacteristic
33C2×#836C2212yes—
34#5225C2212yes—
35#5030—22no—
36C4×#216—22no—
37#5026—22no—
38C4×#215—22no—
39#4966C2212yes—
40#4964C2212yes—
41C2×#805C2212yes—
42C2×#805C2212yes—
43C4×#208C2212yes—
44#4530C2211yescharacteristic
45#4530C2211yescharacteristic
46#4523C2212yes—
47#4523C2212yes—
48#3461—31no—
49C4×#137—32no—
50#2327D311yescharacteristic, complemented
51#2061—31no—
52#2040—31no—
53#2038—32no—
54#2028—31no—
55#2007—31no—
56#2005—32no—
57#1759—32no—
58C4×#101—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C2×#235C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42D412yes—
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78#947—22no—
79#946—22no—
80#946—22no—
81#944—22no—
82C2×#216—22no—
83#942—24no—
84#942—24no—
85C2×#215—22no—
86#930—22no—
87#929—24no—
88C4×#67—22no—
89C4×#67—22no—
90#926C2312yes—
91#924—22no—
92C2×#208C2312yes—
93C2×#208C2312yes—
94#898—24no—
95#897—22no—
96#896—24no—
97#895—22no—
98#894—24no—
99#890—22no—
100#889—24no—
101#888—22no—
102#876—24no—
103#875—22no—
104C4×D12—22no—
105C4×D12D412yes—
106C4×D12D412yes—
107#872—24no—
108#872—22no—
109C42×D3—22no—
110#870—22no—
111C4×Dic6—22no—
112C4×Dic6D412yes—
113C4×Dic6D412yes—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—22no—
117C3×#95—22no—
118C3×#95—22no—
119#836—24no—
120#835—22no—
121#835—22no—
122#810—22no—
123#809—22no—
124#808—22no—
125#807—22no—
126#805—24no—
127#805—24no—
128#803—22no—
129C4×C3⋊C8—22no—
130C2×#137—32no—
131#544—31no—
132#542—62no—
133#541—62no—
134#539—62no—
135#537—31no—
136#536—32no—
137#533—62no—
138C2×C4⋊Q8C2×D311yescharacteristic
139#529—62no—
140#529—62no—
141C4×#41—32no—
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×Q8C2×D311yescharacteristic
144#441—62no—
145#440—62no—
146C4×Q16—62no—
147C4×SD16—62no—
148#428—32no—
149C2×#104—32no—
150C2×#104—32no—
151#419—32no—
152#417—32no—
153C2×#101—32no—
154C4×M16—32no—
155#391C2×D311yescharacteristic
156#391C2×D311yescharacteristic
157#387—22no—
158#387—22no—
159#382C2×D311yescharacteristic
160#376C2×D311yescharacteristic
161#343—31no—
162#343—31no—
163#339—32no—
164#339—32no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8#4112yes—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C2×#67—22no—
183#235—22no—
184#235—22no—
185#235—22no—
186#235—22no—
187C2×D12—22no—
188C2×D12C2×D412yes—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×Dic6—22no—
193C2×Dic6C2×D412yes—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—21no—
203C3×C42—22no—
204C3×C42—42no—
205C3×C42C2×D412yes—
206C3×C42C2×D412yes—
207C3×C42#4112yes—
208#217—22no—
209#216—28no—
210#215—28no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3C2×D412yes—
219C4⋊Dic3C2×D412yes—
220Dic3⋊C4—24no—
221Dic3⋊C4—24no—
222Dic3⋊C4—24no—
223Dic3⋊C4—22no—
224Dic3⋊C4—22no—
225C4×Dic3—22no—
226C4×Dic3—22no—
227C4×Dic3—22no—
228#208—22no—
229#208—22no—
230#208—22no—
231#208#4114yes—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×#41—32no—
235C22×Q8C22×D311yescharacteristic
236C22×Q8C22×D311yescharacteristic
237#137—64no—
238#137—64no—
239C2×Q16—62no—
240C2×SD16—62no—
241C2×M16—32no—
242C2×M16—32no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—62no—
246C4⋊Q8—62no—
247#126—64no—
248#126—64no—
249#125—62no—
250#125—62no—
251#125—64no—
252#124—62no—
253#124—62no—
254#123—64no—
255#123—64no—
256C22⋊Q8—62no—
257C22⋊Q8—64no—
258C22⋊Q8—62no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261#121—62no—
262#121—62no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—32no—
268C4×Q8—32no—
269C4×Q8—62no—
270C4×D4—32no—
271C4×D4—32no—
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—62no—
275#117—32no—
276#117—64no—
277#117—62no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4#6712yes—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C42—21no—
284C2×C42—62no—
285C2×C42#6712yes—
286C2×C42C22×D311yescharacteristic
287C2×C42C22×D311yescharacteristic
288C2×C42C22×D311yescharacteristic
289#107—62no—
290#106—62no—
291#104—64no—
292#104—64no—
293#103—62no—
294#103—62no—
295#103—62no—
296#102—62no—
297#101—64no—
298#97—62no—
299C4×C8—62no—
300#95—21no—
301#95—21no—
302#95—22no—
303#95—22no—
304#95—22no—
305C23×C3C22×D411yescharacteristic
306C22×D3—22no—
307C3×Q8—24no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×Q8—42no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4#12012yes—
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4C22×D412yes—
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4112yes—
336#67—42no—
337#67—42no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341D12—22no—
342D12—22no—
343D12—82no—
344C4×D3—44no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—82no—
348Dic6—22no—
349Dic6—22no—
350Dic6—82no—
351C3⋊C8—24no—
352C3⋊C8—42no—
353#41—62no—
354#41—62no—
355#41—62no—
356#41—62no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—32no—
365C2×Q8—62no—
366C2×Q8#23512yes—
367C2×D4—32no—
368C2×D4—62no—
369C2×D4—62no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4C2×#6712yes—
378C22×C4C2×#6712yes—
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6711yescharacteristic
381C22×C4C23×D311yescharacteristic
382Q16—68no—
383SD16—68no—
384M16—34no—
385M16—62no—
386M16—62no—
387M16—62no—
388C2×C8—62no—
389C2×C8—62no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—32no—
395C4⋊C4—32no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—64no—
399C4⋊C4—64no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—64no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C22⋊C4—64no—
406C22⋊C4—64no—
407C22⋊C4—64no—
408C22⋊C4—62no—
409C22⋊C4—62no—
410C22⋊C4—62no—
411C42—22no—
412C42—22no—
413C42—21no—
414C42—22no—
415C42—42no—
416C42—62no—
417C42—62no—
418C42—62no—
419C42#23512yes—
420C42D3×D412yes—
421C42D3×D412yes—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3C2×#12011yescharacteristic
425C22×C3#52511yescharacteristic
426C22×C3#53211yescharacteristic
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—81no—
438C3×C4—81no—
439C3×C4#52512yes—
440C3×C4#53212yes—
441Dic3—42no—
442Dic3—42no—
443Dic3—42no—
444C23—62no—
445C23C22×#6711yescharacteristic
446Q8—24no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—62no—
451Q8—62no—
452Q8—242no—
453D4—62no—
454D4—62no—
455D4—122no—
456D4—122no—
457D4—242no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—124no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—242no—
483C2×C4#93711yescharacteristic
484C2×C4#93711yescharacteristic
485C2×C4#93712yes—
486C2×C4C2×#23511yescharacteristic
487C2×C4C2×#23511yescharacteristic
488C2×C4C2×D3×D412yes—
489C2×C4D3×#4112yes—
490C8—64no—
491C8—122no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3#238811yescharacteristic
495C2×C3#302011yescharacteristic
496C2×C3#302211yescharacteristic
497D3—82no—
498C22—21no—
499C22—21no—
500C22—122no—
501C22—122no—
502C22—122no—
503C22#579111yescharacteristic
504C22C2×#93711yescharacteristic
505C22#583211yescharacteristic
506C4—21no—
507C4—21no—
508C4—22no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—122no—
516C4—122no—
517C4—122no—
518C4#554712yes—
519C4#554812yes—
520C3#2359411yescharacteristic, complemented, Sylow
521C2—21no—
522C2—21no—
523C2—242no—
524C2#8240011yescharacteristic
525C2#8341511yescharacteristic
526C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2327C2
socleC22×C3C2×#120
prime radical#5510C22
hypercenter#2327D3
hypocenterC3#23594
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
2. C3×#2327C2semidirect
3. C4×#942C2semidirect
4. #71795C2semidirect
5. C4×#836C2semidirect
6. #69695C2semidirect
8. #69673C2semidirect
10. #69543C2semidirect
11. #69430C2semidirect
50. #2327D3semidirect
520. C3#23594semidirect

Maximal subgroups

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

subgroupindex
#858112
C3×#23272
C4×#9422
#717952
C4×#8362
#696952
#696942
#696732
#695632
#695432
#694302
#235943