Finite Group Databasebeta

← #10767978

Subgroups — #10767978

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86280C211yesmaximal, characteristic
2C3×#3113C211yesmaximal, characteristic, complemented
3#82492C211yesmaximal, characteristic
4#82448C211yesmaximal, characteristic, complemented
5#71836C211yesmaximal, characteristic, complemented
6#71172C214yesmaximal, complemented
7#70980C214yesmaximal, complemented
8#70836C211yesmaximal, characteristic, complemented
9#70827C211yesmaximal, characteristic
10#18181—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13#5730—21no—
14#5727—21no—
15#5690—22no—
16#5634C2214yes—
17D3×C22⋊Q8—22no—
18#5579—22no—
19#5579—21no—
20Q8×D12—21no—
21#5521—21no—
22#5519—21no—
23C2×C4×D12C2211yescharacteristic
24#5470C2211yescharacteristic
25C3×#483C2212yes—
26C3×#475—21no—
27C3×#462—21no—
28C3×#451C2212yes—
29C3×#441—22no—
30C3×#432C2211yescharacteristic
31#5231C2212yes—
32#5228C2212yes—
33#5031—21no—
34#5030—22no—
35#5020—21no—
36#4992C2212yes—
37#4991—21no—
38#4986—21no—
39#4971C2212yes—
40C4×(C2×D3)⋊C4—22no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4538C2214yes—
44#4468C2214yes—
45#4467C2214yes—
46#3501—31no—
47#3113—31no—
48#3113D311yescharacteristic, complemented
49#2398—31no—
50#1608—34no—
51#1602—34no—
52#1506—31no—
53#1506—31no—
54#1466—31no—
55C22×C3×Q8C2311yescharacteristic
56C2×D3×Q8—22no—
57C22×D12D411yescharacteristic
58C22×C4×D3D412yes—
59C2×C3×Q16—22no—
60C3×C4⋊Q8D411yescharacteristic
61C3×C4⋊Q8D411yescharacteristic
62C3×#125—21no—
63C3×C22⋊Q8—21no—
64C3×C22⋊Q8C2312yes—
65C3×C4×Q8—21no—
66C3×C4×Q8—21no—
67C3×C4×Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C3×C4×Q8D412yes—
71C3×#117—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76#946—22no—
77#946—24no—
78C2×#216—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4C2312yes—
82#926—21no—
83C2×C4⋊Dic3D411yescharacteristic
84#924—21no—
85#924C2312yes—
86C2×C4×Dic3—21no—
87#897—22no—
88#897—24no—
89#896—24no—
90#896—24no—
91#896—24no—
92#893—22no—
93#893—22no—
94D3×C4⋊C4—24no—
95D3×C4⋊C4—22no—
96#890—22no—
97#888—22no—
98D3×C22⋊C4—24no—
99#878—24no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—21no—
103C4×D12—21no—
104C4×D12—21no—
105C4×D12—21no—
106#870—22no—
107#870—21no—
108#869—22no—
109#869D411yescharacteristic
110#869D411yescharacteristic
111C4×Dic6—21no—
112C4×Dic6D412yes—
113C3×#106—22no—
114C3×C4⋊C8—21no—
115C3×#103—22no—
116C3×#103—22no—
117C3×C22⋊C8C2312yes—
118C3×#97—21no—
119#835—22no—
120C3⋊#95—21no—
121C3⋊#95—22no—
122C3⋊#95—22no—
123C3⋊#95C2312yes—
124#830C2312yes—
125#810—22no—
126#808—22no—
127C4⋊(C3⋊C8)—21no—
128#803—21no—
129#554—61no—
130#554—62no—
131#554—61no—
132#554—62no—
133D4×Q8—61no—
134D4×Q8—61no—
135#544—61no—
136#543—34no—
137#533—31no—
138#533C2×D311yescharacteristic
139C2×C22⋊Q8—62no—
140#518—61no—
141C2×C4×Q8C2×D311yescharacteristic
142C2×C4×D4—31no—
143#483—32no—
144#483C2×D312yes—
145#475—21no—
146#475—61no—
147#462—21no—
148#462—61no—
149#451—32no—
150#451C2×D312yes—
151#441—22no—
152#441—62no—
153#432—31no—
154#432C2×D311yescharacteristic
155#400—61no—
156#397—32no—
157#394—61no—
158#393—32no—
159C4×C22⋊C4—62no—
160#376—31no—
161#328—34no—
162#328—34no—
163#323—34no—
164C23×D3C2×D412yes—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—42no—
169C2×C3×Q8C2×D412yes—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—22no—
176C22×Dic3C2×D412yes—
177D3×Q8—84no—
178C2×D12—24no—
179C2×D12—21no—
180C2×D12—21no—
181C2×C4×D3—22no—
182C2×C4×D3—24no—
183C2×C4×D3—24no—
184C2×C4×D3—22no—
185C2×C4×D3—44no—
186C2×Dic6—21no—
187C2×Dic6—24no—
188C2×Dic6C2×D412yes—
189C3×Q16—44no—
190C2×C3×C8—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4C2×D412yes—
201C3×C22⋊C4—22no—
202C3×C42—22no—
203C3×C42—41no—
204C3×C42—41no—
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209#217—22no—
210#216—44no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—22no—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—44no—
216(C2×D3)⋊C4—44no—
217C4⋊Dic3—24no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3C2×D412yes—
224Dic3⋊C4—24no—
225Dic3⋊C4—24no—
226Dic3⋊C4—22no—
227Dic3⋊C4—22no—
228C4×Dic3—22no—
229C4×Dic3—42no—
230C2×C3⋊C8—22no—
231C22×Q8—62no—
232C22×Q8C22×D311yescharacteristic
233C22×D4—31no—
234C23×C4—32no—
235C2×Q16—22no—
236C2×Q16—62no—
237C4⋊Q8—31no—
238C4⋊Q8—31no—
239C4⋊Q8—62no—
240C4⋊Q8—62no—
241C4⋊Q8#6711yescharacteristic
242C4⋊Q8#6711yescharacteristic
243#125—21no—
244#125—62no—
245#125—62no—
246#125—61no—
247C22⋊Q8—21no—
248C22⋊Q8—32no—
249C22⋊Q8—64no—
250C22⋊Q8—64no—
251C22⋊Q8—64no—
252C22⋊Q8—62no—
253C22⋊Q8—64no—
254C22⋊Q8—62no—
255C22⋊Q8—64no—
256C22⋊Q8—61no—
257C22⋊Q8—64no—
258C22⋊Q8C22×D312yes—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—32no—
265C4×Q8—61no—
266C4×Q8#6712yes—
267C4×D4—62no—
268C4×D4—62no—
269C4×D4—62no—
270C4×D4—62no—
271C4×D4—61no—
272C4×D4—61no—
273C4×D4—61no—
274C4×D4—61no—
275#117—21no—
276#117—61no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—64no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C22⋊C4—32no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C42—21no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289#106—22no—
290#106—62no—
291C4⋊C8—21no—
292C4⋊C8—61no—
293#103—22no—
294#103—22no—
295#103—62no—
296#103—62no—
297C22⋊C8—32no—
298C22⋊C8C22×D312yes—
299#97—21no—
300#97—61no—
301#95—32no—
302#95—62no—
303#95—62no—
304#95—61no—
305C23×C3C22×D411yescharacteristic
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3#13714yes—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—42no—
314C3×Q8—41no—
315C3×Q8—41no—
316C3×Q8—42no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340D12—24no—
341D12—24no—
342C4×D3—44no—
343C4×D3—82no—
344C4×D3—84no—
345C4×D3—84no—
346Dic6—42no—
347Dic6—42no—
348Dic6—84no—
349C3×C8—42no—
350C3⋊C8—42no—
351C24—32no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—32no—
356C2×Q8—42no—
357C2×Q8—64no—
358C2×Q8—61no—
359C2×Q8—244no—
360C2×Q8C2×#6712yes—
361C2×D4—64no—
362C2×D4—61no—
363C2×D4—61no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—32no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—124no—
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D411yescharacteristic
375C22×C4C23×D311yescharacteristic
376Q16—44no—
377Q16—124no—
378C2×C8—22no—
379C2×C8—62no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—32no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—64no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—62no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—62no—
400C4⋊C4C2×#6712yes—
401C22⋊C4—22no—
402C22⋊C4—64no—
403C22⋊C4—64no—
404C22⋊C4—62no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C42—22no—
410C42—41no—
411C42—41no—
412C42—62no—
413C42—122no—
414C42D3×D411yescharacteristic
415C42D3×D411yescharacteristic
416C42C2×#6711yescharacteristic
417C42C2×#6711yescharacteristic
418C22×C3—21no—
419C22×C3—21no—
420C22×C3C2×#12011yescharacteristic
421C22×C3C2×#13712yes—
422C2×D3—24no—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—44no—
426C2×D3—82no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—81no—
436C3×C4#45512yes—
437Dic3—42no—
438Dic3—42no—
439Dic3—42no—
440Dic3—82no—
441C23—34no—
442C23—62no—
443C23—62no—
444C23—122no—
445C23—122no—
446C23C2×D3×D411yescharacteristic
447Q8—22no—
448Q8—22no—
449Q8—42no—
450Q8—41no—
451Q8—41no—
452Q8—42no—
453Q8—122no—
454Q8—122no—
455Q8—244no—
456D4—64no—
457D4—64no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—124no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—242no—
477C2×C4—244no—
478C2×C4—244no—
479C2×C4#93711yescharacteristic
480C2×C4#93711yescharacteristic
481C2×C4#93711yescharacteristic
482C2×C4#93711yescharacteristic
483C2×C4C2×D3×D411yescharacteristic
484C2×C4C22×#6711yescharacteristic
485C8—42no—
486C8—122no—
487C2×C3—22no—
488C2×C3#300411yescharacteristic
489C2×C3C2×#45511yescharacteristic
490C2×C3#302511yescharacteristic
491D3—44no—
492C22—21no—
493C22—21no—
494C22—64no—
495C22—124no—
496C22—122no—
497C22—122no—
498C22—242no—
499C22D3×#13712yes—
500C22C2×#93711yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—81no—
510C4—122no—
511C4—122no—
512C4—122no—
513C4—242no—
514C4#520212yes—
515C3#1818111yescharacteristic, complemented, Sylow
516C2—22no—
517C2—124no—
518C2#8293611yescharacteristic
519C2#8397911yescharacteristic
520C2C2×#520211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3113C2
socleC22×C3C2×#120
prime radicalC22×D12D4
hypercenter#3113D3
hypocenterC3#18181
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
2. C3×#3113C2semidirect
4. #82448C2semidirect
5. #71836C2semidirect
6. #71172C2semidirect
7. #70980C2semidirect
8. #70836C2semidirect
48. #3113D3semidirect
515. C3#18181semidirect

Maximal subgroups

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

subgroupindex
#862802
C3×#31132
#824922
#824482
#718362
#711722
#709802
#708362
#708272
#181813