Finite Group Databasebeta

← #10802126

Subgroups — #10802126

Order 768.

876 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
1C3×C4×#131C211yesmaximal, characteristic, complemented
2#83805C211yesmaximal, characteristic
3#83792C212yesmaximal, complemented
4C2×#4700C211yesmaximal, characteristic, complemented
5#72013C211yesmaximal, characteristic, complemented
6#71645C212yesmaximal, complemented
7#69901C211yesmaximal, characteristic, complemented
8#69898C212yesmaximal, complemented
9#69897C211yesmaximal, characteristic, complemented
10#69874C212yesmaximal, complemented
11#69779C211yesmaximal, characteristic, complemented
12#21558—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14C2×C3×#131C2211yescharacteristic
15C3×C4×#41C2211yescharacteristic
16#5822C2212yes—
17C2×#904C2211yescharacteristic
18C2×#901C2212yes—
19C3×#405—21no—
20C3×#405—21no—
21C3×#405C2211yescharacteristic
22C3×#405C2211yescharacteristic
23C3×C4×M16—21no—
24C3×C4×M16C2211yescharacteristic
25C2×C3×C4×C8—21no—
26C2×C3×C4×C8C2211yescharacteristic
27#5236C2212yes—
28#5145—24no—
29#5144—24no—
30#5143C412yescomplemented
31#5142C412yescomplemented
32#5141C414yescomplemented
33#5139C2212yes—
34#5132C2212yes—
35#5124C2212yes—
36C2×#819C2211yescharacteristic
37C2×#819C2211yescharacteristic
38#5106C2211yescharacteristic
39#5101C2212yes—
40#5006C2212yes—
41C2×#805C2212yes—
42#4700—21no—
43#4700—21no—
44#4700—21no—
45#4700—21no—
46#4559C2212yes—
47#4533C2212yes—
48#4523C2212yes—
49#3067—31no—
50#2955—32no—
51C2×#443—31no—
52C4×#131D311yescharacteristic, complemented
53#1976—31no—
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1772—31no—
59#1757—32no—
60C2×C3×#41C2311yescharacteristic
61C2×#235C2312yes—
62C3×#131—21no—
63C3×#131—21no—
64C3×#131—21no—
65C3×#131—21no—
66C3×#131C2×C412yes—
67C3×#131C2×C412yes—
68C3×#131D412yes—
69C3×#131D412yes—
70C2×C3×M16—21no—
71C2×C3×M16—21no—
72C2×C3×M16C2311yescharacteristic
73C2×C3×M16C2311yescharacteristic
74C22×C3×C8—21no—
75C22×C3×C8—21no—
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C3×C4×Q8—21no—
79C3×C4×D4—21no—
80C3×C4×D4—22no—
81C3×#117—21no—
82C3×#117C2311yescharacteristic
83C2×C3×C42—21no—
84C2×C3×C42C2311yescharacteristic
85#951C2×C414yes—
86#950—24no—
87#949C2×C414yes—
88#942—22no—
89#932—22no—
90#926C2312yes—
91C2×#208C2312yes—
92C2×#208C2312yes—
93C2×Dic12C2×C411yescharacteristic, complemented
94#904—21no—
95#904—21no—
96#904—21no—
97#904—21no—
98C2×D24C2×C411yescharacteristic, complemented
99C2×#204C2×C412yescomplemented
100#901—22no—
101#901—22no—
102#901—22no—
103#901—22no—
104C2×ΓL(1,𝔽25)—22no—
105C2×D3×C8—22no—
106C3×#97—21no—
107C3×#97—21no—
108C3×#97—21no—
109C3×C4×C8—21no—
110C3×C4×C8—21no—
111C3×C4×C8—21no—
112C3×C4×C8D411yescharacteristic
113C3×C4×C8D411yescharacteristic
114C3×C4×C8D411yescharacteristic
115C3×C4×C8D411yescharacteristic
116#837—24no—
117#825—24no—
118#824C2×C414yescomplemented
119#823C2×C414yescomplemented
120#822—24no—
121#819—21no—
122#819—21no—
123#819C2×C412yescomplemented
124#819C2×C412yescomplemented
125#818—21no—
126#817—21no—
127#815—22no—
128C8×Dic3—22no—
129#806—24no—
130#805—22no—
131#805—22no—
132#575—32no—
133C2×#135—31no—
134C2×#131—32no—
135C2×#131C2×D311yescharacteristic
136C2×#131C2×D311yescharacteristic
137C4×#41C2×D311yescharacteristic
138#473—32no—
139#472—32no—
140#471—34no—
141#444—64no—
142#443—64no—
143#443—61no—
144#443—61no—
145#443—61no—
146#443—61no—
147#430—32no—
148C2×#108—31no—
149C2×#108—31no—
150#427—31no—
151#421—32no—
152#421—32no—
153C2×#104—32no—
154#413—32no—
155#410—32no—
156#405—21no—
157#405—21no—
158#405—32no—
159#405C2×D311yescharacteristic
160#405C2×D311yescharacteristic
161C4×M16—21no—
162C4×M16C2×D311yescharacteristic
163C2×C4×C8—21no—
164C2×C4×C8C2×D311yescharacteristic
165#344—32no—
166#344—32no—
167#339—32no—
168C3×#41—21no—
169C3×#41—21no—
170C3×#41C22×C412yes—
171C3×#41C2×D412yes—
172C2×C3×Q8#4111yescharacteristic
173C2×C3×D4—21no—
174C2×C3×D4#4111yescharacteristic
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4C2411yescharacteristic
179C2×#67—22no—
180#235—22no—
181#235—22no—
182#235—22no—
183#235—22no—
184C2×D12C22×C412yes—
185C2×C4×D3—22no—
186C2×Dic6C22×C412yes—
187C3×M16—21no—
188C3×M16—21no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—22no—
192C3×M16—22no—
193C3×M16C22×C412yes—
194C3×M16C2×D412yes—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8C22×C411yescharacteristic
202C2×C3×C8C22×C411yescharacteristic
203C2×C3×C8C2×D411yescharacteristic
204C2×C3×C8C2×D411yescharacteristic
205C2×C3×C8C2×D411yescharacteristic
206C2×C3×C8C2×D411yescharacteristic
207C2×C3×C8C2×D411yescharacteristic
208C2×C3×C8C2×D411yescharacteristic
209C2×C3×C8#4111yescharacteristic
210C2×C3×C8#4111yescharacteristic
211C2×C3×C8#4111yescharacteristic
212C2×C3×C8#4111yescharacteristic
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4#4112yes—
217C3×C42—22no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222#217—22no—
223#216—24no—
224#215—24no—
225#214—24no—
226#213—24no—
227C4⋊Dic3—22no—
228C4×Dic3—22no—
229#208—22no—
230#208—22no—
231#208—22no—
232#208C22×C414yes—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235Dic12—21no—
236Dic12—21no—
237D24—21no—
238D24—21no—
239#204—22no—
240#204—22no—
241ΓL(1,𝔽25)—42no—
242ΓL(1,𝔽25)—42no—
243D3×C8—42no—
244D3×C8—42no—
245C2×#41—32no—
246C2×#41C22×D311yescharacteristic
247#137—34no—
248#137—62no—
249AGL(1,ℤ/8)—34no—
250AGL(1,ℤ/8)—62no—
251#135—64no—
252#135—61no—
253#135—61no—
254#135—61no—
255#135—61no—
256C2×Q16—31no—
257C2×SD16—32no—
258C2×D8—31no—
259#131—21no—
260#131—21no—
261#131—21no—
262#131—21no—
263#131—62no—
264#131—62no—
265#131—62no—
266#131—62no—
267#131C4×D312yes—
268#131C4×D312yes—
269#131D1212yes—
270#131D1212yes—
271C2×M16—21no—
272C2×M16—21no—
273C2×M16—32no—
274C2×M16—32no—
275C2×M16—62no—
276C2×M16C22×D311yescharacteristic
277C2×M16C22×D311yescharacteristic
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—62no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C4×Q8—21no—
284C4×D4—21no—
285C4×D4—22no—
286#117—21no—
287#117—32no—
288#117C22×D311yescharacteristic
289C2×C42—21no—
290C2×C42C22×D311yescharacteristic
291#108—32no—
292#108—32no—
293#108—64no—
294#108—61no—
295#108—61no—
296#107—61no—
297#106—61no—
298#104—64no—
299#104—64no—
300#104—62no—
301#104—62no—
302#101—34no—
303#100—34no—
304#99—64no—
305#97—21no—
306#97—21no—
307#97—21no—
308#97—62no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—62no—
313C4×C8#6711yescharacteristic
314C4×C8#6711yescharacteristic
315C4×C8#6711yescharacteristic
316C4×C8#6711yescharacteristic
317C23×C3—21no—
318C23×C3C2×#4111yescharacteristic
319C22×D3—22no—
320C3×Q8—21no—
321C3×Q8C4×D412yes—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—21no—
325C3×D4C4×D412yes—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4C23×C411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340#67—42no—
341#67—42no—
342C2×Dic3—22no—
343C2×Dic3—42no—
344D12—22no—
345D12—22no—
346C4×D3—42no—
347C4×D3—42no—
348Dic6—22no—
349Dic6—22no—
350C3×C8—22no—
351C3×C8—22no—
352C3×C8C4×D411yescharacteristic
353C3×C8C4×D411yescharacteristic
354C3×C8C4×D411yescharacteristic
355C3×C8C4×D411yescharacteristic
356C3×C8#12111yescharacteristic
357C3×C8#12111yescharacteristic
358C3×C8#12111yescharacteristic
359C3×C8#12111yescharacteristic
360C3⋊C8—24no—
361C3⋊C8—42no—
362#41—21no—
363#41—21no—
364#41—62no—
365#41—62no—
366#41—62no—
367#41—62no—
368#41C2×C4×D312yes—
369#41C2×D1212yes—
370C2×Q8—32no—
371C2×Q8#23511yescharacteristic
372C2×D4—21no—
373C2×D4—32no—
374C2×D4—62no—
375C2×D4#23511yescharacteristic
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—62no—
380C22×C4C23×D311yescharacteristic
381Q16—64no—
382Q16—61no—
383Q16—61no—
384SD16—64no—
385SD16—64no—
386SD16—62no—
387SD16—62no—
388D8—64no—
389D8—61no—
390D8—61no—
391M16—21no—
392M16—21no—
393M16—22no—
394M16—22no—
395M16—22no—
396M16—22no—
397M16—34no—
398M16—62no—
399M16—62no—
400M16—62no—
401M16—122no—
402M16—122no—
403M16C2×C4×D312yes—
404M16C2×D1212yes—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8—122no—
414C2×C8—122no—
415C2×C8C2×C4×D311yescharacteristic
416C2×C8C2×C4×D311yescharacteristic
417C2×C8C2×D1211yescharacteristic
418C2×C8C2×D1211yescharacteristic
419C2×C8#23511yescharacteristic
420C2×C8#23511yescharacteristic
421C2×C8#23511yescharacteristic
422C2×C8#23511yescharacteristic
423C2×C8C2×#6711yescharacteristic
424C2×C8C2×#6711yescharacteristic
425C2×C8C2×#6711yescharacteristic
426C2×C8C2×#6711yescharacteristic
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—62no—
430C22⋊C4—22no—
431C22⋊C4—62no—
432C22⋊C4#23512yes—
433C42—22no—
434C42—21no—
435C42—21no—
436C42—62no—
437C42C2×#6711yescharacteristic
438C42C2×#6711yescharacteristic
439C22×C3—22no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3C4×#4111yescharacteristic
443C22×C3#52511yescharacteristic
444C22×C3#53211yescharacteristic
445C2×D3—42no—
446C2×D3—82no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—42no—
450C3×C4—41no—
451C3×C4C2×C4×D411yescharacteristic
452C3×C4C2×C4×D411yescharacteristic
453C3×C4C2×#12111yescharacteristic
454C3×C4#52911yescharacteristic
455Dic3—42no—
456Dic3—82no—
457C23—21no—
458C23—62no—
459C23C2×#23511yescharacteristic
460Q8—21no—
461Q8—62no—
462Q8—62no—
463Q8C4×D1212yes—
464D4—22no—
465D4—22no—
466D4—21no—
467D4—62no—
468D4—62no—
469D4—122no—
470D4—122no—
471D4C4×D1212yes—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—62no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4C22×C4×D311yescharacteristic
485C2×C4C22×D1211yescharacteristic
486C2×C4C2×#23511yescharacteristic
487C2×C4C2×#23511yescharacteristic
488C2×C4C2×#23511yescharacteristic
489C2×C4C22×#6711yescharacteristic
490C8—22no—
491C8—22no—
492C8—64no—
493C8—122no—
494C8C4×#6711yescharacteristic
495C8C4×#6711yescharacteristic
496C8C4×#6711yescharacteristic
497C8C4×#6711yescharacteristic
498C8#93011yescharacteristic
499C8#93011yescharacteristic
500C8#93011yescharacteristic
501C8#93011yescharacteristic
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—42no—
506C2×C3C4×#12111yescharacteristic
507D3—82no—
508C22—22no—
509C22—41no—
510C22—42no—
511C22—122no—
512C22—242no—
513C22C4×#23511yescharacteristic
514C22#547711yescharacteristic
515C22#579111yescharacteristic
516C4—22no—
517C4—22no—
518C4—42no—
519C4—41no—
520C4—122no—
521C4—242no—
522C4C2×C4×D1211yescharacteristic
523C4#547911yescharacteristic
524C4C2×C4×#6711yescharacteristic
525C4C2×#93011yescharacteristic
526C3#2155811yescharacteristic, complemented, Sylow
527C2—21no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—242no—
532C2C4×#93011yescharacteristic
subgroupG/H
centerC8#930
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#131C2
socleC2×C3C4×#121
prime radical#83805C2
cosocleC22×C3×C4C24
hypercenterC4×#131D3
hypocenterC3#21558
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ C4×#131

Complements

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

NH ≅ G/Ntype
1. C3×C4×#131C2semidirect
3. #83792C2semidirect
4. C2×#4700C2semidirect
5. #72013C2semidirect
6. #71645C2semidirect
7. #69901C2semidirect
8. #69898C2semidirect
9. #69897C2semidirect
10. #69874C2semidirect
11. #69779C2semidirect
30. #5143C4semidirect
31. #5142C4semidirect
32. #5141C4semidirect
52. C4×#131D3semidirect
93. C2×Dic12C2×C4semidirect
98. C2×D24C2×C4semidirect
99. C2×#204C2×C4semidirect
118. #824C2×C4semidirect
119. #823C2×C4semidirect
123. #819C2×C4semidirect
124. #819C2×C4semidirect
526. C3#21558semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1312
#838052
#837922
C2×#47002
#720132
#716452
#699012
#698982
#698972
#698742
#697792
#215583