Finite Group Databasebeta

← #10802138

Subgroups — #10802138

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3067C211yesmaximal, characteristic
2#83999C212yesmaximal, complemented
3C2×#5197C211yesmaximal, characteristic, complemented
4#83782C211yesmaximal, characteristic, complemented
5#72114C212yesmaximal, complemented
6#72086C211yesmaximal, characteristic, complemented
7#72013C211yesmaximal, characteristic, complemented
8#70707C211yesmaximal, characteristic, complemented
9#70703C211yesmaximal, characteristic, complemented
10#70133C212yesmaximal, complemented
11#70130C212yesmaximal, complemented
12#21559—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#5197—21no—
30#5197—21no—
31#5197—21no—
32#5197—21no—
33#5196—24no—
34#5125C2212yes—
35#5122—21no—
36#5122C2212yes—
37#5122C2211yescharacteristic
38Dic3×M16—21no—
39Dic3×M16C2211yescharacteristic
40C2×#819C2211yescharacteristic
41#5106C2211yescharacteristic
42#5101—21no—
43#5101C2211yescharacteristic
44C2×#815—21no—
45C2×#815C2211yescharacteristic
46#4559C2212yes—
47#4559C2212yes—
48#4555C2212yes—
49#3067D311yescharacteristic
50#2955—32no—
51C2×#444—31no—
52#2875—31no—
53#1976—31no—
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1773—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—22no—
84#948—22no—
85#948—22no—
86#948—22no—
87#948—22no—
88#948D414yes—
89Q8×Dic3—21no—
90D4×Dic3—21no—
91D4×Dic3—22no—
92#926—21no—
93#926C2312yes—
94#926C2311yescharacteristic
95C2×C4×Dic3—21no—
96C2×C4×Dic3C2311yescharacteristic
97C2×#208—21no—
98C2×#208—22no—
99C2×#208C2312yes—
100C2×#208C2311yescharacteristic
101C22×C3⋊C8—21no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8C2311yescharacteristic
104C3×#108C2×C412yescomplemented
105C3×#108C2×C412yescomplemented
106C3×#101C2×C414yescomplemented
107C3×#100C2×C414yescomplemented
108#837—24no—
109#837—22no—
110#837—22no—
111#837—24no—
112#834—24no—
113#819—21no—
114#819—21no—
115#819—24no—
116#818—21no—
117#817—21no—
118#815—21no—
119#815—21no—
120#815—21no—
121#815—22no—
122#815D411yescharacteristic
123#815D411yescharacteristic
124#815D411yescharacteristic
125#815D411yescharacteristic
126C8×Dic3—21no—
127C8×Dic3—21no—
128C8×Dic3—22no—
129C8×Dic3—21no—
130#575C2×D312yes—
131C2×#135C2×D311yescharacteristic
132C2×#131—31no—
133C2×#131—32no—
134C2×#131C2×D311yescharacteristic
135C4×#41—31no—
136#473Dic312yescomplemented
137#472Dic312yescomplemented
138#471Dic314yescomplemented
139#444—61no—
140#444—61no—
141#444—61no—
142#444—61no—
143#444—64no—
144#443—64no—
145#430—32no—
146C2×#108—31no—
147C2×#108C2×D311yescharacteristic
148#427—31no—
149#421—32no—
150#421—32no—
151C2×#104—32no—
152#413C2×D312yes—
153#410—32no—
154#405—31no—
155#405—32no—
156#405—31no—
157#405—61no—
158#405—61no—
159C4×M16—31no—
160C4×M16—61no—
161C2×#97—31no—
162C2×#97—61no—
163#344—32no—
164#344—32no—
165#339—32no—
166C3×#41—21no—
167C3×#41—21no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—22no—
172C3×#41C22×C412yes—
173C3×#41C2×D412yes—
174C2×C3×Q8C22×C412yes—
175C2×C3×Q8#4111yescharacteristic
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4C22×C412yes—
179C2×C3×D4#4111yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4C2411yescharacteristic
183C22×Dic3—22no—
184C22×Dic3—21no—
185C3×Q16—24no—
186C3×Q16—21no—
187C3×Q16—21no—
188C3×SD16—24no—
189C3×SD16—24no—
190C3×SD16—22no—
191C3×SD16—22no—
192C3×D8—24no—
193C3×D8—21no—
194C3×D8—21no—
195C3×M16—21no—
196C3×M16—22no—
197C3×M16—22no—
198C3×M16—22no—
199C3×M16C22×C412yes—
200C3×M16C22×C414yes—
201C2×C3×C8—21no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8C22×C411yescharacteristic
206C2×C3×C8C22×C411yescharacteristic
207C2×C3×C8C2×D411yescharacteristic
208C2×C3×C8C2×D411yescharacteristic
209C2×C3×C8#4111yescharacteristic
210C2×C3×C8#4111yescharacteristic
211#217—22no—
212#217—22no—
213#217#4112yes—
214C4⋊Dic3—22no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—21no—
219C4×Dic3—21no—
220C4×Dic3—22no—
221C4×Dic3C2×D411yescharacteristic
222C4×Dic3C2×D411yescharacteristic
223#208—21no—
224#208—24no—
225#208—22no—
226#208—22no—
227#208—42no—
228#208—42no—
229#208C2×D412yes—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—24no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—42no—
234C2×C3⋊C8—42no—
235C2×C3⋊C8C2×D412yes—
236C2×C3⋊C8C2×D411yescharacteristic
237C2×C3⋊C8C2×D411yescharacteristic
238C2×C3⋊C8#4112yes—
239C2×#41C22×D311yescharacteristic
240C2×#41C22×D312yes—
241#137—22no—
242#137C2×Dic314yes—
243AGL(1,ℤ/8)—22no—
244AGL(1,ℤ/8)C2×Dic314yes—
245#135—24no—
246#135—21no—
247#135—21no—
248#135—21no—
249#135—21no—
250C2×Q16C2×Dic311yescharacteristic, complemented
251C2×SD16C2×Dic312yescomplemented
252C2×D8C2×Dic311yescharacteristic, complemented
253#131—21no—
254#131—21no—
255#131—34no—
256#131—62no—
257#131—62no—
258#131—62no—
259#131—62no—
260#131—62no—
261#131C2×Dic312yes—
262#131C2×Dic312yes—
263C2×M16—21no—
264C2×M16—31no—
265C2×M16—32no—
266C2×M16—61no—
267C2×M16—62no—
268C2×M16C22×D311yescharacteristic
269C2×M16C22×D312yes—
270C22×C8—21no—
271C22×C8—31no—
272C22×C8—61no—
273C22×C8—62no—
274C22×C8C22×D311yescharacteristic
275C4×Q8—61no—
276C4×D4—62no—
277C4×D4—61no—
278#117—31no—
279#117—32no—
280#117—61no—
281C2×C42—31no—
282C2×C42—61no—
283#108—64no—
284#108—61no—
285#108—61no—
286#108C2×Dic312yescomplemented
287#108C2×Dic312yescomplemented
288#107—61no—
289#106—61no—
290#104—62no—
291#104—62no—
292#104—64no—
293#104—64no—
294#101C2×Dic314yescomplemented
295#100C2×Dic314yescomplemented
296#99—64no—
297#97—31no—
298#97—31no—
299#97—31no—
300#97—31no—
301#97—61no—
302#97—61no—
303#97—62no—
304#97—61no—
305C4×C8—61no—
306C4×C8—61no—
307C4×C8—62no—
308C4×C8—61no—
309C23×C3—21no—
310C23×C3—22no—
311C23×C3C2×#4111yescharacteristic
312C3×Q8—21no—
313C3×Q8—22no—
314C3×Q8—22no—
315C3×Q8C4×D412yes—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—21no—
321C3×D4—42no—
322C3×D4—42no—
323C3×D4C4×D412yes—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4C23×C411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×Dic3—21no—
337C2×Dic3—22no—
338C2×Dic3—21no—
339C2×Dic3—42no—
340C2×Dic3—81no—
341C2×Dic3—82no—
342C3×C8—22no—
343C3×C8—24no—
344C3×C8C4×D411yescharacteristic
345C3×C8C4×D411yescharacteristic
346C3×C8C4×D411yescharacteristic
347C3×C8C4×D411yescharacteristic
348C3⋊C8—22no—
349C3⋊C8—42no—
350C3⋊C8#12112yes—
351C3⋊C8#12112yes—
352#41—21no—
353#41—21no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41D3×D412yes—
359#41C22×Dic312yes—
360C2×Q8#23711yescharacteristic
361C2×Q8C22×Dic312yes—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4#23711yescharacteristic
365C2×D4C22×Dic312yes—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4C23×D311yescharacteristic
371Q16—24no—
372Q16—21no—
373Q16—21no—
374SD16—24no—
375SD16—24no—
376SD16—22no—
377SD16—22no—
378D8—24no—
379D8—21no—
380D8—21no—
381M16—21no—
382M16—22no—
383M16—22no—
384M16—22no—
385M16—32no—
386M16—64no—
387M16—61no—
388M16—62no—
389M16—62no—
390M16—122no—
391M16—122no—
392M16C22×Dic312yes—
393M16C22×Dic314yes—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—31no—
399C2×C8—32no—
400C2×C8—31no—
401C2×C8—32no—
402C2×C8—64no—
403C2×C8—61no—
404C2×C8—62no—
405C2×C8—122no—
406C2×C8—122no—
407C2×C8D3×D411yescharacteristic
408C2×C8D3×D411yescharacteristic
409C2×C8#23711yescharacteristic
410C2×C8#23711yescharacteristic
411C2×C8C22×Dic311yescharacteristic
412C2×C8C22×Dic311yescharacteristic
413C4⋊C4—62no—
414C4⋊C4—61no—
415C4⋊C4—62no—
416C22⋊C4—32no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C42—31no—
420C42—31no—
421C42—62no—
422C42—61no—
423C42—61no—
424C42—62no—
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—82no—
430C22×C3C4×#4111yescharacteristic
431C22×C3#52511yescharacteristic
432C22×C3#53211yescharacteristic
433C3×C4—22no—
434C3×C4—42no—
435C3×C4C2×C4×D411yescharacteristic
436C3×C4C2×C4×D411yescharacteristic
437C3×C4C2×#12111yescharacteristic
438C3×C4#52911yescharacteristic
439Dic3—42no—
440Dic3—41no—
441Dic3—41no—
442Dic3—82no—
443C23—21no—
444C23—22no—
445C23D3×#4111yescharacteristic
446Q8—21no—
447Q8—22no—
448Q8—22no—
449Q8D4×Dic312yes—
450D4—22no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—21no—
455D4—42no—
456D4—42no—
457D4D4×Dic312yes—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—62no—
467C2×C4—122no—
468C2×C4—241no—
469C2×C4—242no—
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×#23711yescharacteristic
473C2×C4C2×#23711yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4C23×Dic311yescharacteristic
476C8—22no—
477C8—24no—
478C8—32no—
479C8—32no—
480C8—62no—
481C8—122no—
482C8D4×Dic311yescharacteristic
483C8D4×Dic311yescharacteristic
484C8D4×Dic311yescharacteristic
485C8D4×Dic311yescharacteristic
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—42no—
490C2×C3—82no—
491C2×C3C4×#12111yescharacteristic
492C22—22no—
493C22—41no—
494C22—42no—
495C22—42no—
496C22—82no—
497C22#559611yescharacteristic
498C22#560611yescharacteristic
499C22Dic3×#4111yescharacteristic
500C4—22no—
501C4—42no—
502C4—121no—
503C4—121no—
504C4—122no—
505C4—242no—
506C4#560011yescharacteristic
507C4D3×#12111yescharacteristic
508C4C2×D4×Dic311yescharacteristic
509C4C2×D4×Dic311yescharacteristic
510C3#2155911yescharacteristic, complemented, Sylow
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—82no—
516C2Dic3×#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#21559
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×#5197C2semidirect
4. #83782C2semidirect
5. #72114C2semidirect
6. #72086C2semidirect
7. #72013C2semidirect
8. #70707C2semidirect
9. #70703C2semidirect
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
104. C3×#108C2×C4semidirect
105. C3×#108C2×C4semidirect
106. C3×#101C2×C4semidirect
107. C3×#100C2×C4semidirect
136. #473Dic3semidirect
137. #472Dic3semidirect
138. #471Dic3semidirect
250. C2×Q16C2×Dic3semidirect
251. C2×SD16C2×Dic3semidirect
252. C2×D8C2×Dic3semidirect
286. #108C2×Dic3semidirect
287. #108C2×Dic3semidirect
294. #101C2×Dic3semidirect
295. #100C2×Dic3semidirect
510. C3#21559semidirect

Maximal subgroups

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

subgroupindex
C3×#30672
#839992
C2×#51972
#837822
#721142
#720862
#720132
#707072
#707032
#701332
#701302
#215593