Finite Group Databasebeta

← #10802127

Subgroups — #10802127

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×#3067C211yesmaximal, characteristic
2#83999C212yesmaximal, complemented
3C2×#5196C211yesmaximal, characteristic, complemented
4Dic3×#131C211yesmaximal, characteristic, complemented
5#72114C212yesmaximal, complemented
6#72086C211yesmaximal, characteristic, complemented
7#72013C211yesmaximal, characteristic, complemented
8#70707C211yesmaximal, characteristic, complemented
9#70702C211yesmaximal, characteristic, complemented
10#70133C212yesmaximal, complemented
11#70130C212yesmaximal, complemented
12#21558—31nomaximal, Sylow
13C3×#575C2212yes—
14C2×C3×#135C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16Dic3×#41C2211yescharacteristic
17C2×#948C2212yes—
18C2×#948C2211yescharacteristic
19C3×#473C412yescomplemented
20C3×#472C412yescomplemented
21C3×#471C414yescomplemented
22C2×C3×#108C2211yescharacteristic
23C3×#413C2212yes—
24#5240C2212yes—
25#5236C2212yes—
26#5236C2212yes—
27C2×#837C2212yes—
28#5197—24no—
29#5196—24no—
30#5196—21no—
31#5196—21no—
32#5196—21no—
33#5196—21no—
34#5125C2212yes—
35#5122—21no—
36#5122C2212yes—
37#5122C2211yescharacteristic
38Dic3×M16—21no—
39Dic3×M16C2211yescharacteristic
40C2×#819C2211yescharacteristic
41#5106C2211yescharacteristic
42#5101—21no—
43#5101C2211yescharacteristic
44C2×C8×Dic3—21no—
45C2×C8×Dic3C2211yescharacteristic
46#4559C2212yes—
47#4559C2212yes—
48#4555C2212yes—
49#3067D311yescharacteristic
50#2955—32no—
51C2×#443—31no—
52C4×#131—31no—
53#1976—31no—
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1772—31no—
59#1757—32no—
60C2×C3×#41C2312yes—
61C2×C3×#41C2311yescharacteristic
62C3×#137—22no—
63C3×#137C2×C414yes—
64C3×AGL(1,ℤ/8)—22no—
65C3×AGL(1,ℤ/8)C2×C414yes—
66C3×#135—24no—
67C3×#135—21no—
68C3×#135—21no—
69C3×#135—21no—
70C3×#135—21no—
71C2×C3×Q16C2×C411yescharacteristic, complemented
72C2×C3×SD16C2×C412yescomplemented
73C2×C3×D8C2×C411yescharacteristic, complemented
74C3×#131—21no—
75C3×#131—21no—
76C3×#131C2×C412yes—
77C3×#131C2×C412yes—
78C2×C3×M16—21no—
79C2×C3×M16C2312yes—
80C2×C3×M16C2311yescharacteristic
81C22×C3×C8—21no—
82C22×C3×C8C2311yescharacteristic
83#948—21no—
84#948—21no—
85#948—22no—
86#948—22no—
87#948—22no—
88#948—22no—
89#948D412yes—
90#948D412yes—
91Q8×Dic3—21no—
92D4×Dic3—21no—
93D4×Dic3—22no—
94#926—21no—
95#926C2312yes—
96#926C2311yescharacteristic
97C2×C4×Dic3—21no—
98C2×C4×Dic3C2311yescharacteristic
99C2×#208—21no—
100C2×#208—22no—
101C2×#208C2312yes—
102C2×#208C2311yescharacteristic
103C22×C3⋊C8—21no—
104C22×C3⋊C8—22no—
105C22×C3⋊C8C2311yescharacteristic
106C3×#108C2×C412yescomplemented
107C3×#108C2×C412yescomplemented
108C3×#101C2×C414yescomplemented
109C3×#100C2×C414yescomplemented
110#837—24no—
111#837—22no—
112#837—22no—
113#837—24no—
114#834—24no—
115#819—21no—
116#819—21no—
117#819—24no—
118#818—21no—
119#817—21no—
120#815—21no—
121#815—21no—
122#815—22no—
123#815—21no—
124C8×Dic3—21no—
125C8×Dic3—21no—
126C8×Dic3—21no—
127C8×Dic3—22no—
128C8×Dic3D411yescharacteristic
129C8×Dic3D411yescharacteristic
130C8×Dic3D411yescharacteristic
131C8×Dic3D411yescharacteristic
132#575C2×D312yes—
133C2×#135C2×D311yescharacteristic
134C2×#131—31no—
135C2×#131—32no—
136C2×#131C2×D311yescharacteristic
137C4×#41—31no—
138#473Dic312yescomplemented
139#472Dic312yescomplemented
140#471Dic314yescomplemented
141#444—64no—
142#443—61no—
143#443—61no—
144#443—61no—
145#443—61no—
146#443—64no—
147#430—32no—
148C2×#108—31no—
149C2×#108C2×D311yescharacteristic
150#427—31no—
151#421—32no—
152#421—32no—
153C2×#104—32no—
154#413C2×D312yes—
155#410—32no—
156#405—31no—
157#405—32no—
158#405—31no—
159#405—61no—
160#405—61no—
161C4×M16—31no—
162C4×M16—61no—
163C2×C4×C8—31no—
164C2×C4×C8—61no—
165#344—32no—
166#344—32no—
167#339—32no—
168C3×#41—21no—
169C3×#41—21no—
170C3×#41—22no—
171C3×#41—22no—
172C3×#41—22no—
173C3×#41—22no—
174C3×#41C22×C412yes—
175C3×#41C2×D412yes—
176C2×C3×Q8C22×C412yes—
177C2×C3×Q8#4111yescharacteristic
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4C22×C412yes—
181C2×C3×D4#4111yescharacteristic
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4C2411yescharacteristic
185C22×Dic3—22no—
186C22×Dic3—21no—
187C3×Q16—24no—
188C3×Q16—21no—
189C3×Q16—21no—
190C3×SD16—24no—
191C3×SD16—24no—
192C3×SD16—22no—
193C3×SD16—22no—
194C3×D8—24no—
195C3×D8—21no—
196C3×D8—21no—
197C3×M16—21no—
198C3×M16—22no—
199C3×M16—22no—
200C3×M16—22no—
201C3×M16C22×C412yes—
202C3×M16C22×C414yes—
203C2×C3×C8—21no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8C22×C411yescharacteristic
208C2×C3×C8C22×C411yescharacteristic
209C2×C3×C8C2×D411yescharacteristic
210C2×C3×C8C2×D411yescharacteristic
211C2×C3×C8#4111yescharacteristic
212C2×C3×C8#4111yescharacteristic
213#217—22no—
214#217—22no—
215#217#4112yes—
216C4⋊Dic3—22no—
217C4⋊Dic3—21no—
218C4⋊Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—22no—
223C4×Dic3C2×D411yescharacteristic
224C4×Dic3C2×D411yescharacteristic
225#208—21no—
226#208—22no—
227#208—22no—
228#208—22no—
229#208—22no—
230#208—42no—
231#208—42no—
232#208C2×D412yes—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—42no—
238C2×C3⋊C8—42no—
239C2×C3⋊C8C2×D411yescharacteristic
240C2×C3⋊C8C2×D411yescharacteristic
241C2×C3⋊C8C2×D411yescharacteristic
242C2×C3⋊C8C2×D411yescharacteristic
243C2×C3⋊C8#4111yescharacteristic
244C2×C3⋊C8#4111yescharacteristic
245C2×#41C22×D311yescharacteristic
246C2×#41C22×D312yes—
247#137—22no—
248#137C2×Dic314yes—
249AGL(1,ℤ/8)—22no—
250AGL(1,ℤ/8)C2×Dic314yes—
251#135—24no—
252#135—21no—
253#135—21no—
254#135—21no—
255#135—21no—
256C2×Q16C2×Dic311yescharacteristic, complemented
257C2×SD16C2×Dic312yescomplemented
258C2×D8C2×Dic311yescharacteristic, complemented
259#131—21no—
260#131—21no—
261#131—32no—
262#131—32no—
263#131—61no—
264#131—61no—
265#131—62no—
266#131—62no—
267#131—62no—
268#131—62no—
269#131C2×Dic312yes—
270#131C2×Dic312yes—
271C2×M16—21no—
272C2×M16—31no—
273C2×M16—32no—
274C2×M16—61no—
275C2×M16—62no—
276C2×M16C22×D311yescharacteristic
277C2×M16C22×D312yes—
278C22×C8—21no—
279C22×C8—31no—
280C22×C8—61no—
281C22×C8—62no—
282C22×C8C22×D311yescharacteristic
283C4×Q8—61no—
284C4×D4—62no—
285C4×D4—61no—
286#117—31no—
287#117—32no—
288#117—61no—
289C2×C42—31no—
290C2×C42—61no—
291#108—64no—
292#108—61no—
293#108—61no—
294#108C2×Dic312yescomplemented
295#108C2×Dic312yescomplemented
296#107—61no—
297#106—61no—
298#104—62no—
299#104—62no—
300#104—64no—
301#104—64no—
302#101C2×Dic314yescomplemented
303#100C2×Dic314yescomplemented
304#99—64no—
305#97—61no—
306#97—61no—
307#97—62no—
308#97—61no—
309C4×C8—31no—
310C4×C8—31no—
311C4×C8—31no—
312C4×C8—31no—
313C4×C8—61no—
314C4×C8—61no—
315C4×C8—62no—
316C4×C8—61no—
317C23×C3—21no—
318C23×C3—22no—
319C23×C3C2×#4111yescharacteristic
320C3×Q8—21no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8C4×D412yes—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—21no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4C4×D412yes—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4C23×C411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—22no—
346C2×Dic3—21no—
347C2×Dic3—42no—
348C2×Dic3—44no—
349C2×Dic3—42no—
350C3×C8—22no—
351C3×C8—24no—
352C3×C8C4×D411yescharacteristic
353C3×C8C4×D411yescharacteristic
354C3×C8C4×D411yescharacteristic
355C3×C8C4×D411yescharacteristic
356C3⋊C8—22no—
357C3⋊C8—42no—
358C3⋊C8#12111yescharacteristic
359C3⋊C8#12111yescharacteristic
360C3⋊C8#12111yescharacteristic
361C3⋊C8#12111yescharacteristic
362#41—21no—
363#41—21no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41D3×D412yes—
369#41C22×Dic312yes—
370C2×Q8#23711yescharacteristic
371C2×Q8C22×Dic312yes—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4#23711yescharacteristic
375C2×D4C22×Dic312yes—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4C23×D311yescharacteristic
381Q16—24no—
382Q16—21no—
383Q16—21no—
384SD16—24no—
385SD16—24no—
386SD16—22no—
387SD16—22no—
388D8—24no—
389D8—21no—
390D8—21no—
391M16—21no—
392M16—22no—
393M16—22no—
394M16—22no—
395M16—32no—
396M16—62no—
397M16—62no—
398M16—61no—
399M16—62no—
400M16—62no—
401M16—122no—
402M16—122no—
403M16C22×Dic312yes—
404M16C22×Dic314yes—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—31no—
410C2×C8—31no—
411C2×C8—31no—
412C2×C8—31no—
413C2×C8—31no—
414C2×C8—31no—
415C2×C8—62no—
416C2×C8—62no—
417C2×C8—61no—
418C2×C8—62no—
419C2×C8—122no—
420C2×C8—122no—
421C2×C8D3×D411yescharacteristic
422C2×C8D3×D411yescharacteristic
423C2×C8#23711yescharacteristic
424C2×C8#23711yescharacteristic
425C2×C8C22×Dic311yescharacteristic
426C2×C8C22×Dic311yescharacteristic
427C4⋊C4—62no—
428C4⋊C4—61no—
429C4⋊C4—62no—
430C22⋊C4—32no—
431C22⋊C4—62no—
432C22⋊C4—62no—
433C42—31no—
434C42—31no—
435C42—62no—
436C42—61no—
437C42—61no—
438C42—62no—
439C22×C3—22no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—82no—
444C22×C3C4×#4111yescharacteristic
445C22×C3#52511yescharacteristic
446C22×C3#53211yescharacteristic
447C3×C4—22no—
448C3×C4—42no—
449C3×C4C2×C4×D411yescharacteristic
450C3×C4C2×C4×D411yescharacteristic
451C3×C4C2×#12111yescharacteristic
452C3×C4#52911yescharacteristic
453Dic3—22no—
454Dic3—42no—
455Dic3—41no—
456Dic3—82no—
457C23—21no—
458C23—22no—
459C23D3×#4111yescharacteristic
460Q8—21no—
461Q8—22no—
462Q8—22no—
463Q8D4×Dic312yes—
464D4—22no—
465D4—22no—
466D4—22no—
467D4—22no—
468D4—21no—
469D4—42no—
470D4—42no—
471D4D4×Dic312yes—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—62no—
481C2×C4—122no—
482C2×C4—124no—
483C2×C4—122no—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×#23711yescharacteristic
487C2×C4C2×#23711yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4C23×Dic311yescharacteristic
490C8—22no—
491C8—24no—
492C8—31no—
493C8—31no—
494C8—31no—
495C8—31no—
496C8—62no—
497C8—122no—
498C8D4×Dic311yescharacteristic
499C8D4×Dic311yescharacteristic
500C8D4×Dic311yescharacteristic
501C8D4×Dic311yescharacteristic
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—42no—
506C2×C3—82no—
507C2×C3C4×#12111yescharacteristic
508C22—22no—
509C22—41no—
510C22—42no—
511C22—42no—
512C22—82no—
513C22#559611yescharacteristic
514C22#560611yescharacteristic
515C22Dic3×#4111yescharacteristic
516C4—22no—
517C4—42no—
518C4—62no—
519C4—121no—
520C4—122no—
521C4—242no—
522C4#560011yescharacteristic
523C4D3×#12111yescharacteristic
524C4C2×D4×Dic311yescharacteristic
525C4C2×D4×Dic311yescharacteristic
526C3#2155811yescharacteristic, complemented, Sylow
527C2—21no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—82no—
532C2Dic3×#12111yescharacteristic
subgroupG/H
centerC4D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3067C2
socleC2×C3C4×#121
cosocleC22×C3×C4C24
hypercenter#3067D3
hypocenterC3#21558
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3067

Complements

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

NH ≅ G/Ntype
2. #83999C2semidirect
3. C2×#5196C2semidirect
4. Dic3×#131C2semidirect
5. #72114C2semidirect
6. #72086C2semidirect
7. #72013C2semidirect
8. #70707C2semidirect
9. #70702C2semidirect
10. #70133C2semidirect
11. #70130C2semidirect
19. C3×#473C4semidirect
20. C3×#472C4semidirect
21. C3×#471C4semidirect
71. C2×C3×Q16C2×C4semidirect
72. C2×C3×SD16C2×C4semidirect
73. C2×C3×D8C2×C4semidirect
106. C3×#108C2×C4semidirect
107. C3×#108C2×C4semidirect
108. C3×#101C2×C4semidirect
109. C3×#100C2×C4semidirect
138. #473Dic3semidirect
139. #472Dic3semidirect
140. #471Dic3semidirect
256. C2×Q16C2×Dic3semidirect
257. C2×SD16C2×Dic3semidirect
258. C2×D8C2×Dic3semidirect
294. #108C2×Dic3semidirect
295. #108C2×Dic3semidirect
302. #101C2×Dic3semidirect
303. #100C2×Dic3semidirect
526. C3#21558semidirect

Maximal subgroups

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

subgroupindex
C3×#30672
#839992
C2×#51962
Dic3×#1312
#721142
#720862
#720132
#707072
#707022
#701332
#701302
#215583