Finite Group Databasebeta

← #10802130

Subgroups — #10802130

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
1#83805C211yesmaximal, characteristic
2#83782C211yesmaximal, characteristic, complemented
3C2×#4894C211yesmaximal, characteristic, complemented
4#78989C212yesmaximal, complemented
5C3×#1814C211yesmaximal, characteristic, complemented
6#70706C211yesmaximal, characteristic, complemented
7#70660C211yesmaximal, characteristic, complemented
8#70658C211yesmaximal, characteristic, complemented
9#70395C212yesmaximal, complemented
10#69014C212yesmaximal, complemented
11#69012C212yesmaximal, complemented
12#21559—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14Dic3×#41C2211yescharacteristic
15#5822C2212yes—
16C2×#948C2211yescharacteristic
17C2×#907C2212yes—
18C2×#904C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C3×#427C2211yescharacteristic
21C3×#421C2212yes—
22#5143C412yescomplemented
23#5142C412yescomplemented
24#5141C414yescomplemented
25C2×#825C2212yes—
26#5132C2212yes—
27#5124C2212yes—
28#5122—21no—
29#5122C2211yescharacteristic
30Dic3×M16—21no—
31Dic3×M16C2211yescharacteristic
32C2×#819C2211yescharacteristic
33#5101—21no—
34#5101C2211yescharacteristic
35C2×#815—21no—
36C2×#815C2211yescharacteristic
37#5006C2212yes—
38#5005C2212yes—
39#4997C2212yes—
40#4894—21no—
41#4894—21no—
42#4894—21no—
43#4894—21no—
44#4824—24no—
45#4823—24no—
46C3×#344C2212yes—
47#4556C2212yes—
48#4532C2212yes—
49#3067—31no—
50#2955—32no—
51C2×#444—31no—
52#2875—31no—
53#1976—31no—
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814D311yescharacteristic, complemented
58#1773—31no—
59#1757—32no—
60C2×C3×#41C2311yescharacteristic
61C2×#235C2312yes—
62C3×#131—21no—
63C3×#131—21no—
64C3×#131C2×C412yes—
65C3×#131C2×C412yes—
66C2×C3×M16—21no—
67C2×C3×M16C2312yes—
68C2×C3×M16C2311yescharacteristic
69C22×C3×C8—21no—
70C22×C3×C8C2311yescharacteristic
71C3×#117C2312yes—
72#951C2×C414yes—
73#950—24no—
74#949C2×C414yes—
75#948—22no—
76#948D414yes—
77Q8×Dic3—21no—
78#942—22no—
79D4×Dic3—21no—
80D4×Dic3—22no—
81#932—22no—
82#926—21no—
83#926C2311yescharacteristic
84C2×C4×Dic3—21no—
85C2×C4×Dic3C2311yescharacteristic
86C2×#208—21no—
87C2×#208C2312yes—
88C2×#208C2311yescharacteristic
89C22×C3⋊C8—21no—
90C22×C3⋊C8C2311yescharacteristic
91#907—22no—
92#907—22no—
93#907—22no—
94#907—22no—
95C2×Dic12C2×C411yescharacteristic, complemented
96#904—21no—
97#904—21no—
98#904—21no—
99#904—21no—
100C2×D24C2×C411yescharacteristic, complemented
101C2×#204C2×C412yescomplemented
102C2×ΓL(1,𝔽25)—22no—
103C2×D3×C8—22no—
104C3×#108—21no—
105C3×#108—21no—
106C3×#107—21no—
107C3×#106—21no—
108C3×#104—24no—
109#825—22no—
110#825—22no—
111#824C2×C414yescomplemented
112#823C2×C414yescomplemented
113#822—24no—
114#819C2×C412yescomplemented
115#819C2×C412yescomplemented
116#815—21no—
117#815—21no—
118#815—21no—
119#815D411yescharacteristic
120#815D411yescharacteristic
121#815D411yescharacteristic
122#815D411yescharacteristic
123C8×Dic3—21no—
124C8×Dic3—21no—
125C8×Dic3—21no—
126#806—24no—
127#805—24no—
128#803—22no—
129C4×C3⋊C8—22no—
130#575—32no—
131C2×#135—31no—
132C2×#131—31no—
133C2×#131—32no—
134C2×#131C2×D311yescharacteristic
135C4×#41—31no—
136#473—32no—
137#472—32no—
138#471—34no—
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#427C2×D311yescharacteristic
149#421—32no—
150#421C2×D312yes—
151C2×#104—32no—
152#413—32no—
153#410—32no—
154#405—31no—
155#405—31no—
156#405—32no—
157#405—61no—
158#405—61no—
159C4×M16—31no—
160C4×M16—61no—
161C2×#97—31no—
162C2×#97—61no—
163#344—32no—
164#344C2×D312yes—
165#339—32no—
166C3×#41—21no—
167C3×#41—21no—
168C3×#41C22×C412yes—
169C3×#41C2×D412yes—
170C2×C3×Q8#4111yescharacteristic
171C2×C3×D4—21no—
172C2×C3×D4#4111yescharacteristic
173C22×C3×C4—21no—
174C22×C3×C4C2411yescharacteristic
175C2×#67—22no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178#235—22no—
179#235—22no—
180#235—22no—
181#235—22no—
182C2×D12C22×C412yes—
183C2×C4×D3—22no—
184C2×Dic6C22×C412yes—
185C3×M16—21no—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16C22×C412yes—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8C22×C411yescharacteristic
196C2×C3×C8C22×C411yescharacteristic
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8#4111yescharacteristic
200C2×C3×C8#4111yescharacteristic
201C3×C4⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C42—22no—
204#217—22no—
205#217#4112yes—
206#216—24no—
207#215—24no—
208#214—24no—
209#213—24no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4×Dic3—22no—
213C4×Dic3—21no—
214C4×Dic3—21no—
215C4×Dic3C2×D411yescharacteristic
216C4×Dic3C2×D411yescharacteristic
217#208—22no—
218#208—21no—
219#208—24no—
220#208C22×C414yes—
221#208C2×D412yes—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—24no—
225C2×C3⋊C8C2×D412yes—
226C2×C3⋊C8C2×D411yescharacteristic
227C2×C3⋊C8C2×D411yescharacteristic
228C2×C3⋊C8#4112yes—
229Dic12—21no—
230Dic12—21no—
231D24—21no—
232D24—21no—
233#204—22no—
234#204—22no—
235ΓL(1,𝔽25)—42no—
236ΓL(1,𝔽25)—42no—
237D3×C8—42no—
238D3×C8—42no—
239C2×#41—32no—
240C2×#41C22×D311yescharacteristic
241#137—34no—
242#137—62no—
243AGL(1,ℤ/8)—34no—
244AGL(1,ℤ/8)—62no—
245#135—64no—
246#135—61no—
247#135—61no—
248#135—61no—
249#135—61no—
250C2×Q16—31no—
251C2×SD16—32no—
252C2×D8—31no—
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#131C4×D312yes—
262#131C4×D312yes—
263C2×M16—21no—
264C2×M16—31no—
265C2×M16—32no—
266C2×M16—62no—
267C2×M16—61no—
268C2×M16C22×D312yes—
269C2×M16C22×D311yescharacteristic
270C22×C8—21no—
271C22×C8—31no—
272C22×C8—62no—
273C22×C8—61no—
274C22×C8C22×D311yescharacteristic
275C4×Q8—61no—
276C4×D4—62no—
277C4×D4—61no—
278#117—31no—
279#117—61no—
280#117C22×D312yes—
281C2×C42—31no—
282C2×C42—61no—
283#108—21no—
284#108—21no—
285#108—32no—
286#108—32no—
287#108—64no—
288#107—21no—
289#106—21no—
290#104—24no—
291#104—64no—
292#104—62no—
293#104—62no—
294#101—34no—
295#100—34no—
296#99—64no—
297#97—31no—
298#97—31no—
299#97—31no—
300#97—31no—
301#97—62no—
302#97—61no—
303#97—61no—
304#97—61no—
305C4×C8—62no—
306C4×C8—61no—
307C4×C8—61no—
308C4×C8—61no—
309C23×C3—21no—
310C23×C3C2×#4111yescharacteristic
311C22×D3—22no—
312C3×Q8—21no—
313C3×Q8C4×D412yes—
314C3×D4—22no—
315C3×D4—22no—
316C3×D4—21no—
317C3×D4C4×D412yes—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4C23×C411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328#67—42no—
329#67—42no—
330C2×Dic3—22no—
331C2×Dic3—21no—
332C2×Dic3—22no—
333C2×Dic3—21no—
334C2×Dic3—81no—
335C2×Dic3—82no—
336D12—22no—
337D12—22no—
338C4×D3—42no—
339C4×D3—42no—
340Dic6—22no—
341Dic6—22no—
342C3×C8—22no—
343C3×C8—42no—
344C3×C8C4×D411yescharacteristic
345C3×C8C4×D411yescharacteristic
346C3×C8C4×D411yescharacteristic
347C3×C8C4×D411yescharacteristic
348C3⋊C8—24no—
349C3⋊C8—22no—
350C3⋊C8#12112yes—
351C3⋊C8#12112yes—
352#41—21no—
353#41—21no—
354#41—62no—
355#41—62no—
356#41—62no—
357#41—62no—
358#41C2×C4×D312yes—
359#41D3×D412yes—
360C2×Q8—32no—
361C2×Q8#23911yescharacteristic
362C2×D4—21no—
363C2×D4—32no—
364C2×D4—62no—
365C2×D4#23911yescharacteristic
366C22×C4—21no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—62no—
370C22×C4C23×D311yescharacteristic
371Q16—64no—
372Q16—61no—
373Q16—61no—
374SD16—64no—
375SD16—64no—
376SD16—62no—
377SD16—62no—
378D8—64no—
379D8—61no—
380D8—61no—
381M16—21no—
382M16—22no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16—34no—
387M16—32no—
388M16—64no—
389M16—62no—
390M16—61no—
391M16—122no—
392M16—122no—
393M16C2×C4×D312yes—
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×C8C2×C4×D311yescharacteristic
408C2×C8C2×C4×D311yescharacteristic
409C2×C8D3×D411yescharacteristic
410C2×C8D3×D411yescharacteristic
411C2×C8#23711yescharacteristic
412C2×C8#23711yescharacteristic
413C4⋊C4—22no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C22⋊C4—22no—
417C22⋊C4—32no—
418C22⋊C4—62no—
419C42—22no—
420C42—31no—
421C42—31no—
422C42—62no—
423C42—61no—
424C42—61no—
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—42no—
428C22×C3C4×#4111yescharacteristic
429C22×C3#52511yescharacteristic
430C22×C3#53211yescharacteristic
431C2×D3—42no—
432C2×D3—82no—
433C3×C4—22no—
434C3×C4—82no—
435C3×C4C2×C4×D411yescharacteristic
436C3×C4C2×C4×D411yescharacteristic
437C3×C4C2×#12111yescharacteristic
438C3×C4#52911yescharacteristic
439Dic3—42no—
440Dic3—41no—
441Dic3—41no—
442Dic3—42no—
443C23—21no—
444C23—62no—
445C23D3×#4111yescharacteristic
446Q8—21no—
447Q8—62no—
448Q8—62no—
449Q8#89312yes—
450D4—22no—
451D4—22no—
452D4—21no—
453D4—62no—
454D4—62no—
455D4—122no—
456D4—122no—
457D4#89312yes—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—62no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—62no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—241no—
469C2×C4—242no—
470C2×C4C22×C4×D311yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×#23711yescharacteristic
474C2×C4C2×#23911yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C8—22no—
477C8—32no—
478C8—32no—
479C8—42no—
480C8—64no—
481C8—62no—
482C8#67⋊C411yescharacteristic
483C8#67⋊C411yescharacteristic
484C8#67⋊C411yescharacteristic
485C8#67⋊C411yescharacteristic
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—42no—
490C2×C3C4×#12111yescharacteristic
491D3—82no—
492C22—22no—
493C22—41no—
494C22—42no—
495C22—122no—
496C22—242no—
497C22#552211yescharacteristic
498C22#559611yescharacteristic
499C22#562711yescharacteristic
500C4—22no—
501C4—82no—
502C4—121no—
503C4—121no—
504C4—122no—
505C4—122no—
506C4C2×#67⋊C411yescharacteristic
507C4C2×#89311yescharacteristic
508C4D3×#12111yescharacteristic
509C4#562411yescharacteristic
510C3#2155911yescharacteristic, complemented, Sylow
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—242no—
516C2#7386511yescharacteristic
subgroupG/H
centerC4D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1814C2
socleC2×C3C4×#121
prime radical#83805C2
cosocleC22×C3×C4C24
hypercenter#1814D3
hypocenterC3#21559
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1814

Complements

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

NH ≅ G/Ntype
2. #83782C2semidirect
3. C2×#4894C2semidirect
4. #78989C2semidirect
5. C3×#1814C2semidirect
6. #70706C2semidirect
7. #70660C2semidirect
8. #70658C2semidirect
9. #70395C2semidirect
10. #69014C2semidirect
11. #69012C2semidirect
22. #5143C4semidirect
23. #5142C4semidirect
24. #5141C4semidirect
57. #1814D3semidirect
95. C2×Dic12C2×C4semidirect
100. C2×D24C2×C4semidirect
101. C2×#204C2×C4semidirect
111. #824C2×C4semidirect
112. #823C2×C4semidirect
114. #819C2×C4semidirect
115. #819C2×C4semidirect
510. C3#21559semidirect

Maximal subgroups

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

subgroupindex
#838052
#837822
C2×#48942
#789892
C3×#18142
#707062
#706602
#706582
#703952
#690142
#690122
#215593