Finite Group Databasebeta

← #10799207

Subgroups — #10799207

Order 768.

1434 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#942C211yesmaximal, characteristic
2C22×C3×#101C211yesmaximal, characteristic, complemented
3C2×#5225C211yesmaximal, characteristic, complemented
4#83768C211yesmaximal, characteristic
5C2×#5132C211yesmaximal, characteristic
6C2×#5124C211yesmaximal, characteristic, complemented
7#83747C211yesmaximal, characteristic, complemented
8#70108C214yesmaximal, complemented
9#70107C214yesmaximal, complemented
10#21250—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×#216C2212yes—
16C2×#942—24no—
17C2×#942—22no—
18C2×#942—22no—
19C2×#942C418yescomplemented
20C22×#215C2212yes—
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C2×C3×#101—24no—
25C2×C3×#101C2214yes—
26#5225—22no—
27#5225C2214yes—
28C2×#835—22no—
29C2×#824C2212yes—
30#5133—22no—
31#5132—24no—
32#5131C2212yes—
33C2×#823C2212yes—
34#5124—24no—
35C2×#822C2214yes—
36#5122C2214yes—
37#5120C2212yes—
38Dic3×M16—22no—
39#5113—22no—
40#5101—22no—
41C22×#137—31no—
42C2×#430—31no—
43#2896—31no—
44C2×#419—31no—
45C2×#413—31no—
46C22×#101D311yescharacteristic, complemented
47#2873—31no—
48#1979—34no—
49#1977—34no—
50C22×C3×Q8—22no—
51C22×C3×Q8—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—21no—
54C22×C3×Q8—21no—
55C22×C3×Q8—21no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8C2×C414yes—
59C22×C3×Q8C2312yes—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C2×#235—22no—
64C2×#235—21no—
65C2×#235—21no—
66C2×#235—21no—
67C2×#235—22no—
68C2×#235—21no—
69C2×#235—22no—
70C2×#235C2×C414yes—
71C22×D12C2311yescharacteristic
72C22×C4×D3—21no—
73C22×Dic6C2311yescharacteristic
74C2×C3×M16—21no—
75C2×C3×M16—22no—
76C2×C3×M16—21no—
77C2×C3×M16—22no—
78C2×C3×M16—22no—
79C2×C3×M16D414yes—
80C2×C3×M16C2312yes—
81C2×C3×M16C2312yes—
82C22×C3×C8—21no—
83C22×C3×C8—21no—
84C2×#217—21no—
85C2×#216—24no—
86C2×#216—24no—
87C2×#216C2×C418yescomplemented
88#942—216no—
89#942—216no—
90C2×#215—24no—
91C2×#215—24no—
92C2×#215C2×C418yescomplemented
93#926—21no—
94#926—21no—
95#926D412yes—
96#926C2312yes—
97C2×C4⋊Dic3D412yes—
98C2×C4×Dic3—21no—
99C2×#208—21no—
100C2×#208—22no—
101C2×#208—22no—
102C2×#208C2×C414yes—
103C2×#208D414yes—
104C2×#208D412yes—
105C2×#208D412yes—
106C2×#208C2312yes—
107C22×C3⋊C8—21no—
108C22×C3⋊C8C2312yes—
109C3×#101—28no—
110#835—28no—
111#824—24no—
112#823—24no—
113#822—28no—
114#821—24no—
115#816—24no—
116#815—24no—
117C8×Dic3—24no—
118C22×#41—31no—
119C23×Q8C2×D311yescharacteristic
120C2×#137—38no—
121C2×#137—62no—
122C2×#137—62no—
123C2×#137—64no—
124C22×Q16—32no—
125C22×SD16—32no—
126C22×M16—31no—
127C22×M16—31no—
128C22×M16C2×D311yescharacteristic
129C22×M16C2×D311yescharacteristic
130C2×#117—31no—
131#430—64no—
132C2×#108—34no—
133#419—34no—
134#419—32no—
135#419—62no—
136#418—32no—
137#417—62no—
138#416—62no—
139C2×#103—62no—
140#413—64no—
141C2×#101—24no—
142C2×#101—32no—
143C2×#101C2×D314yes—
144C2×#100—32no—
145#405—34no—
146#405—62no—
147C4×M16—62no—
148C24×C3#4111yescharacteristic
149C23×D3—21no—
150C2×C3×Q8—24no—
151C2×C3×Q8—22no—
152C2×C3×Q8—24no—
153C2×C3×Q8—24no—
154C2×C3×Q8—24no—
155C2×C3×Q8—24no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—41no—
161C2×C3×Q8—41no—
162C2×C3×Q8—42no—
163C2×C3×Q8—42no—
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8—42no—
167C2×C3×Q8C22×C414yes—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—41no—
171C22×C3×C4—41no—
172C22×C3×C4—42no—
173C22×C3×C4C22×C412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4#4112yes—
182C22×C3×C4C2411yescharacteristic
183C2×#67—24no—
184C2×#67—41no—
185C2×#67—41no—
186C2×#67—42no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189#235—24no—
190#235—24no—
191#235—24no—
192#235—24no—
193#235—42no—
194#235—42no—
195#235—42no—
196#235—42no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12C22×C412yes—
203C2×C4×D3—24no—
204C2×C4×D3—41no—
205C2×C4×D3—41no—
206C2×C4×D3—42no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C22×C412yes—
213C3×M16—24no—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—22no—
217C3×M16—42no—
218C3×M16C2×D414yes—
219C2×C3×C8—24no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—42no—
223#217—22no—
224#217—22no—
225#216—216no—
226#215—216no—
227C4⋊Dic3—22no—
228C4⋊Dic3C2×D414yes—
229C4×Dic3—22no—
230C4×Dic3—22no—
231#208—22no—
232#208—22no—
233#208—42no—
234#208C22⋊C418yes—
235#208C22⋊C418yes—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—42no—
238C2×C3⋊C8C22×C414yes—
239C2×C3⋊C8C2×D414yes—
240C2×C3⋊C8C2×D414yes—
241C2×#41—34no—
242C2×#41—62no—
243C2×#41—61no—
244C2×#41—61no—
245C2×#41—62no—
246C2×#41—61no—
247C2×#41—61no—
248C2×#41—62no—
249C22×Q8—21no—
250C22×Q8—22no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8—22no—
257C22×Q8—31no—
258C22×Q8C4×D314yes—
259C22×Q8C22×D312yes—
260C22×D4—31no—
261C22×D4—61no—
262C23×C4—21no—
263C23×C4—61no—
264C23×C4C22×D311yescharacteristic
265#137—616no—
266#137—616no—
267C2×Q16—38no—
268C2×Q16—64no—
269C2×Q16—64no—
270C2×SD16—38no—
271C2×SD16—64no—
272C2×SD16—64no—
273C2×M16—21no—
274C2×M16—22no—
275C2×M16—21no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—34no—
279C2×M16—32no—
280C2×M16—32no—
281C2×M16—34no—
282C2×M16—32no—
283C2×M16—61no—
284C2×M16—62no—
285C2×M16—62no—
286C2×M16#6714yes—
287C2×M16C22×D312yes—
288C2×M16C22×D312yes—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—32no—
292C22×C8—61no—
293#117—32no—
294#117—32no—
295#117—61no—
296#117—61no—
297C2×C4⋊C4—32no—
298C2×C22⋊C4—61no—
299C2×C42—61no—
300#108—68no—
301#103—64no—
302#103—68no—
303#102—64no—
304#101—28no—
305#101—64no—
306#100—64no—
307#97—64no—
308C4×C8—64no—
309C23×C3—42no—
310C23×C3C4×D412yes—
311C23×C3#12111yescharacteristic
312C23×C3#12111yescharacteristic
313C23×C3#12412yes—
314C23×C3C2×#4111yescharacteristic
315C22×D3—22no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—82no—
319C3×Q8—24no—
320C3×Q8—24no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×Q8—44no—
324C3×Q8—44no—
325C2×C3×C4—24no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—44no—
329C2×C3×C4—44no—
330C2×C3×C4C2×C22⋊C412yes—
331C2×C3×C4C2×C22⋊C412yes—
332C2×C3×C4C4×D412yes—
333C2×C3×C4C4×D412yes—
334C2×C3×C4C4×D412yes—
335C2×C3×C4#12111yescharacteristic
336C2×C3×C4#12111yescharacteristic
337C2×C3×C4#12112yes—
338C2×C3×C4#12112yes—
339C2×C3×C4#12412yes—
340C2×C3×C4#12712yes—
341C2×C3×C4#12712yes—
342C2×C3×C4C23×C411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348#67—44no—
349#67—44no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—82no—
355D12—22no—
356D12—22no—
357D12—41no—
358D12—41no—
359C4×D3—44no—
360C4×D3—44no—
361Dic6—22no—
362Dic6—22no—
363Dic6—41no—
364Dic6—41no—
365C3×C8—24no—
366C3×C8—42no—
367C3⋊C8—42no—
368C3⋊C8C2×C22⋊C418yes—
369C24—61no—
370C24#23511yescharacteristic
371#41—64no—
372#41—64no—
373#41—64no—
374#41—64no—
375#41—122no—
376#41—122no—
377#41—122no—
378#41—122no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—32no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8C2×C4×D314yes—
403C2×D4—32no—
404C2×D4—64no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—121no—
411C2×D4—121no—
412C2×D4—122no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4—64no—
421C22×C4—121no—
422C22×C4—121no—
423C22×C4—122no—
424C22×C4C2×C4×D312yes—
425C22×C4#23511yescharacteristic
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D412yes—
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4#23712yes—
432C22×C4C2×#6712yes—
433C22×C4C23×D311yescharacteristic
434Q16—616no—
435SD16—616no—
436M16—24no—
437M16—22no—
438M16—22no—
439M16—22no—
440M16—38no—
441M16—38no—
442M16—42no—
443M16—62no—
444M16—62no—
445M16—122no—
446M16C2×#6714yes—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—34no—
451C2×C8—34no—
452C2×C8—34no—
453C2×C8—42no—
454C2×C8—62no—
455C2×C8—122no—
456C4⋊C4—34no—
457C4⋊C4—62no—
458C22⋊C4—62no—
459C22⋊C4—62no—
460C42—62no—
461C42—62no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3#39012yes—
468C22×C3#39012yes—
469C22×C3C2×C4×D411yescharacteristic
470C22×C3C2×#12111yescharacteristic
471C22×C3C2×#12411yescharacteristic
472C2×D3—42no—
473C2×D3—82no—
474C2×D3—82no—
475C2×D3—82no—
476C3×C4—44no—
477C3×C4#39012yes—
478C3×C4#39012yes—
479C3×C4C22×C22⋊C411yescharacteristic
480C3×C4C2×C4×D411yescharacteristic
481C3×C4C2×#12111yescharacteristic
482C3×C4C2×#12711yescharacteristic
483Dic3—42no—
484Dic3—44no—
485C23—42no—
486C23—62no—
487C23—121no—
488C23—121no—
489C23—242no—
490C23#88411yescharacteristic
491C23#88411yescharacteristic
492C23#88512yes—
493C23C4×#6712yes—
494C23C2×#23511yescharacteristic
495Q8—24no—
496Q8—24no—
497Q8—44no—
498Q8—44no—
499Q8—42no—
500Q8—42no—
501Q8—62no—
502Q8—62no—
503Q8—121no—
504Q8—121no—
505D4—62no—
506D4—62no—
507D4—124no—
508D4—124no—
509D4—121no—
510D4—121no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—124no—
521C2×C4—124no—
522C2×C4—242no—
523C2×C4D3×C22⋊C412yes—
524C2×C4D3×C22⋊C412yes—
525C2×C4#67⋊C412yes—
526C2×C4#67⋊C412yes—
527C2×C4#88411yescharacteristic
528C2×C4#88411yescharacteristic
529C2×C4#88512yes—
530C2×C4C4×#6712yes—
531C2×C4#93912yes—
532C2×C4#93912yes—
533C2×C4#94012yes—
534C2×C4#94012yes—
535C2×C4C22×C4×D311yescharacteristic
536C2×C4C2×D3×D411yescharacteristic
537C2×C4C2×D3×D411yescharacteristic
538C2×C4C2×D3×D411yescharacteristic
539C2×C4C2×#23711yescharacteristic
540C2×C4C22×#6711yescharacteristic
541C8—24no—
542C8—38no—
543C8—42no—
544C8—122no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—22no—
548C2×C3—22no—
549C2×C3C2×#39011yescharacteristic
550C2×C3#306912yes—
551D3—82no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—42no—
556C22—42no—
557C22—122no—
558C22—242no—
559C22—242no—
560C22—242no—
561C22#495712yes—
562C22#495712yes—
563C22C2×#88411yescharacteristic
564C22C2×#88511yescharacteristic
565C22C2×C4×#6711yescharacteristic
566C4—44no—
567C4—124no—
568C4—122no—
569C4#495712yes—
570C4#495712yes—
571C4C2×D3×C22⋊C411yescharacteristic
572C4C2×#67⋊C411yescharacteristic
573C4C2×#93911yescharacteristic
574C4C2×#94011yescharacteristic
575C3#2125011yescharacteristic, complemented, Sylow
576C2—21no—
577C2—21no—
578C2—22no—
579C2—22no—
580C2—242no—
581C2#7871212yes—
582C2C2×#495711yescharacteristic
subgroupG/H
centerC22C2×#884
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC22×C3×#101C2
socleC22×C3C2×#121
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC22×#101D3
hypocenterC3#21250
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
2. C22×C3×#101C2semidirect
3. C2×#5225C2semidirect
6. C2×#5124C2semidirect
7. #83747C2semidirect
8. #70108C2semidirect
9. #70107C2semidirect
19. C2×#942C4semidirect
46. C22×#101D3semidirect
87. C2×#216C2×C4semidirect
92. C2×#215C2×C4semidirect
575. C3#21250semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C22×C3×#1012
C2×#52252
#837682
C2×#51322
C2×#51242
#837472
#701082
#701072
#212503