Finite Group Databasebeta

← #10823639

Subgroups — #10823639

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85811C211yesmaximal, characteristic
2C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#942C212yesmaximal, complemented
4#71795C212yesmaximal, complemented
5C4×#836C211yesmaximal, characteristic, complemented
6#69695C212yesmaximal, complemented
7#69694C211yesmaximal, characteristic
8#69673C211yesmaximal, characteristic, complemented
9#69593C211yesmaximal, characteristic
10#69573C211yesmaximal, characteristic, complemented
11C4×#805C212yesmaximal, complemented
12#23590—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×#389—22no—
30C3×#384C2211yescharacteristic
31C3×C4×C4⋊C4—22no—
32C3×C43C2211yescharacteristic
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—
50C4×C4⋊Q8D311yescharacteristic, complemented
51#2060—31no—
52#2039—31no—
53#2038—32no—
54#2028—31no—
55#2007—31no—
56#2005—32no—
57C4×#104—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—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4D412yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42—21no—
77C2×C3×C42D412yes—
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81#947—22no—
82#946—22no—
83#946—22no—
84#944—22no—
85C2×#216—22no—
86#942—24no—
87#942—24no—
88C2×#215—22no—
89#930—22no—
90#929—24no—
91C4×#67—22no—
92C4×#67—22no—
93#926C2312yes—
94#924—22no—
95C2×#208C2312yes—
96C2×#208C2312yes—
97#898—24no—
98#897—22no—
99#896—24no—
100#895—22no—
101#894—24no—
102#890—22no—
103#889—24no—
104#888—22no—
105#876—24no—
106#875—22no—
107C4×D12—22no—
108C4×D12D412yes—
109C4×D12D412yes—
110#872—24no—
111#872—22no—
112C42×D3—22no—
113#870—22no—
114C4×Dic6—22no—
115C4×Dic6D412yes—
116C4×Dic6D412yes—
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#389—22no—
158#384C2×D311yescharacteristic
159C4×C4⋊C4—22no—
160C43C2×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×D411yescharacteristic
179C22×C3×C4C2×D412yes—
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—44no—
199C3×C4⋊C4—42no—
200C3×C42—24no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—21no—
204C3×C42—22no—
205C3×C42—42no—
206C3×C42—41no—
207C3×C42—41no—
208C3×C42C2×D412yes—
209C3×C42C2×D412yes—
210C3×C42#4112yes—
211#217—22no—
212#216—28no—
213#215—28no—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3C2×D412yes—
222C4⋊Dic3C2×D412yes—
223Dic3⋊C4—24no—
224Dic3⋊C4—24no—
225Dic3⋊C4—24no—
226Dic3⋊C4—22no—
227Dic3⋊C4—22no—
228C4×Dic3—22no—
229C4×Dic3—22no—
230C4×Dic3—22no—
231#208—22no—
232#208—22no—
233#208—22no—
234#208#4114yes—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×#41—32no—
238C22×Q8C22×D311yescharacteristic
239C22×Q8C22×D311yescharacteristic
240#137—64no—
241#137—64no—
242C2×Q16—62no—
243C2×SD16—62no—
244C2×M16—32no—
245C2×M16—32no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—62no—
249C4⋊Q8—62no—
250#126—64no—
251#126—64no—
252#125—62no—
253#125—62no—
254#125—64no—
255#124—62no—
256#124—62no—
257#123—64no—
258#123—64no—
259C22⋊Q8—62no—
260C22⋊Q8—64no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264#121—62no—
265#121—62no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—32no—
271C4×Q8—32no—
272C4×Q8—62no—
273C4×D4—32no—
274C4×D4—32no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—62no—
278#117—32no—
279#117—64no—
280#117—62no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4#6712yes—
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—62no—
291C2×C42#6712yes—
292C2×C42C22×D311yescharacteristic
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#106—62no—
297#104—64no—
298#104—64no—
299#103—62no—
300#103—62no—
301#103—62no—
302#102—62no—
303#101—64no—
304#97—62no—
305C4×C8—62no—
306#95—22no—
307#95—22no—
308C23×C3C22×D411yescharacteristic
309C22×D3—22no—
310C3×Q8—24no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×Q8—42no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4#12012yes—
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4C22×D412yes—
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4112yes—
339#67—42no—
340#67—42no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344D12—22no—
345D12—22no—
346D12—82no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—44no—
350C4×D3—82no—
351Dic6—22no—
352Dic6—22no—
353Dic6—82no—
354C3⋊C8—24no—
355C3⋊C8—42no—
356#41—62no—
357#41—62no—
358#41—62no—
359#41—62no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—32no—
368C2×Q8—62no—
369C2×Q8#23512yes—
370C2×D4—32no—
371C2×D4—62no—
372C2×D4—62no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4C2×#6712yes—
381C22×C4C2×#6711yescharacteristic
382C22×C4C2×#6712yes—
383C22×C4C2×#6711yescharacteristic
384C22×C4C23×D311yescharacteristic
385Q16—68no—
386SD16—68no—
387M16—34no—
388M16—62no—
389M16—62no—
390M16—62no—
391C2×C8—62no—
392C2×C8—62no—
393C4⋊C4—28no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—32no—
398C4⋊C4—32no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—64no—
402C4⋊C4—64no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—64no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C22⋊C4—64no—
409C22⋊C4—64no—
410C22⋊C4—64no—
411C22⋊C4—62no—
412C22⋊C4—62no—
413C22⋊C4—62no—
414C42—24no—
415C42—22no—
416C42—22no—
417C42—21no—
418C42—22no—
419C42—42no—
420C42—41no—
421C42—41no—
422C42—62no—
423C42—62no—
424C42—62no—
425C42#23512yes—
426C42D3×D412yes—
427C42D3×D412yes—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3C2×#12011yescharacteristic
431C22×C3#52511yescharacteristic
432C22×C3#53211yescharacteristic
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C3×C4—24no—
437C3×C4—21no—
438C3×C4—22no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—81no—
444C3×C4#45612yes—
445C3×C4#52512yes—
446C3×C4#53212yes—
447Dic3—42no—
448Dic3—42no—
449Dic3—42no—
450C23—62no—
451C23C22×#6711yescharacteristic
452Q8—24no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—62no—
457Q8—62no—
458Q8—242no—
459D4—62no—
460D4—62no—
461D4—122no—
462D4—122no—
463D4—242no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—124no—
488C2×C4—242no—
489C2×C4#93711yescharacteristic
490C2×C4#93712yes—
491C2×C4#93711yescharacteristic
492C2×C4C2×#23511yescharacteristic
493C2×C4C2×#23511yescharacteristic
494C2×C4C2×D3×D412yes—
495C2×C4D3×#4112yes—
496C8—64no—
497C8—122no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3#238811yescharacteristic
501C2×C3C2×#45611yescharacteristic
502C2×C3#301811yescharacteristic
503D3—82no—
504C22—21no—
505C22—21no—
506C22—122no—
507C22—122no—
508C22—122no—
509C22#579111yescharacteristic
510C22C2×#93711yescharacteristic
511C22#583211yescharacteristic
512C4—24no—
513C4—21no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4—122no—
521C4—122no—
522C4—122no—
523C4#509512yes—
524C4#554712yes—
525C4#554812yes—
526C3#2359011yescharacteristic, complemented, Sylow
527C2—21no—
528C2—21no—
529C2—242no—
530C2#8240011yescharacteristic
531C2C2×#509511yescharacteristic
532C2#8332211yescharacteristic
subgroupG/H
centerC2×C4#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊Q8C2
socleC22×C3C2×#120
prime radical#5510C22
hypercenterC4×C4⋊Q8D3
hypocenterC3#23590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×C4×C4⋊Q8C2semidirect
3. C4×#942C2semidirect
4. #71795C2semidirect
5. C4×#836C2semidirect
6. #69695C2semidirect
8. #69673C2semidirect
10. #69573C2semidirect
11. C4×#805C2semidirect
50. C4×C4⋊Q8D3semidirect
526. C3#23590semidirect

Maximal subgroups

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

subgroupindex
#858112
C3×C4×C4⋊Q82
C4×#9422
#717952
C4×#8362
#696952
#696942
#696732
#695932
#695732
C4×#8052
#235903