Finite Group Databasebeta

← #10802137

Subgroups — #10802137

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×#2875C211yesmaximal, characteristic, complemented
2#83805C211yesmaximal, characteristic
3#83792C212yesmaximal, complemented
4C2×#4717C211yesmaximal, characteristic, complemented
5#72013C211yesmaximal, characteristic, complemented
6#71645C212yesmaximal, complemented
7#69901C211yesmaximal, characteristic, complemented
8#69898C212yesmaximal, complemented
9#69897C211yesmaximal, characteristic, complemented
10#69874C212yesmaximal, complemented
11#69829C211yesmaximal, characteristic, complemented
12#21559—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×#97—21no—
26C2×C3×#97C2211yescharacteristic
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#4717—21no—
43#4717—21no—
44#4717—21no—
45#4717—21no—
46#4559C2212yes—
47#4533C2212yes—
48#4523C2212yes—
49#3067—31no—
50#2955—32no—
51C2×#444—31no—
52#2875D311yescharacteristic, complemented
53#1976—31no—
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1773—31no—
59#1757—32no—
60C2×C3×#41C2311yescharacteristic
61C2×#235C2312yes—
62C3×#131—21no—
63C3×#131—21no—
64C3×#131—22no—
65C3×#131C2×C412yes—
66C3×#131C2×C412yes—
67C3×#131D414yes—
68C2×C3×M16—21no—
69C2×C3×M16—21no—
70C2×C3×M16C2311yescharacteristic
71C2×C3×M16C2311yescharacteristic
72C22×C3×C8—21no—
73C22×C3×C8—21no—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C3×C4×Q8—21no—
77C3×C4×D4—21no—
78C3×C4×D4—22no—
79C3×#117—21no—
80C3×#117C2311yescharacteristic
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83#951C2×C414yes—
84#950—24no—
85#949C2×C414yes—
86#942—22no—
87#932—22no—
88#926C2312yes—
89C2×#208C2312yes—
90C2×#208C2312yes—
91C2×Dic12C2×C411yescharacteristic, complemented
92#904—21no—
93#904—21no—
94#904—21no—
95#904—21no—
96C2×D24C2×C411yescharacteristic, complemented
97C2×#204C2×C412yescomplemented
98#901—22no—
99#901—22no—
100#901—22no—
101#901—22no—
102C2×ΓL(1,𝔽25)—22no—
103C2×D3×C8—22no—
104C3×#97—21no—
105C3×#97—21no—
106C3×#97—21no—
107C3×#97D411yescharacteristic
108C3×#97D411yescharacteristic
109C3×#97D411yescharacteristic
110C3×#97D411yescharacteristic
111C3×C4×C8—21no—
112C3×C4×C8—21no—
113C3×C4×C8—21no—
114#837—24no—
115#825—24no—
116#824C2×C414yescomplemented
117#823C2×C414yescomplemented
118#822—24no—
119#819—21no—
120#819—21no—
121#819C2×C412yescomplemented
122#819C2×C412yescomplemented
123#818—21no—
124#817—21no—
125#815—22no—
126C8×Dic3—22no—
127#806—24no—
128#805—22no—
129#805—22no—
130#575—32no—
131C2×#135—31no—
132C2×#131—32no—
133C2×#131C2×D311yescharacteristic
134C2×#131C2×D311yescharacteristic
135C4×#41C2×D311yescharacteristic
136#473—32no—
137#472—32no—
138#471—34no—
139#444—64no—
140#444—61no—
141#444—61no—
142#444—61no—
143#444—61no—
144#443—64no—
145#430—32no—
146C2×#108—31no—
147C2×#108—31no—
148#427—31no—
149#421—32no—
150#421—32no—
151C2×#104—32no—
152#413—32no—
153#410—32no—
154#405—21no—
155#405—21no—
156#405—32no—
157#405C2×D311yescharacteristic
158#405C2×D311yescharacteristic
159C4×M16—21no—
160C4×M16C2×D311yescharacteristic
161C2×#97—21no—
162C2×#97C2×D311yescharacteristic
163#344—32no—
164#344—32no—
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×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4C2411yescharacteristic
177C2×#67—22no—
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—21no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—24no—
190C3×M16C22×C412yes—
191C3×M16C2×D412yes—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—24no—
197C2×C3×C8C22×C411yescharacteristic
198C2×C3×C8C22×C411yescharacteristic
199C2×C3×C8C2×D412yes—
200C2×C3×C8C2×D411yescharacteristic
201C2×C3×C8C2×D411yescharacteristic
202C2×C3×C8C2×D411yescharacteristic
203C2×C3×C8C2×D411yescharacteristic
204C2×C3×C8#4111yescharacteristic
205C2×C3×C8#4111yescharacteristic
206C2×C3×C8#4112yes—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4#4112yes—
211C3×C42—22no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42C2×D411yescharacteristic
215C3×C42C2×D411yescharacteristic
216#217—22no—
217#216—24no—
218#215—24no—
219#214—24no—
220#213—24no—
221C4⋊Dic3—22no—
222C4×Dic3—22no—
223#208—22no—
224#208—22no—
225#208—22no—
226#208C22×C414yes—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
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—22no—
256#131—62no—
257#131—62no—
258#131—62no—
259#131—62no—
260#131C4×D312yes—
261#131C4×D312yes—
262#131D1214yes—
263C2×M16—21no—
264C2×M16—21no—
265C2×M16—32no—
266C2×M16—32no—
267C2×M16—62no—
268C2×M16C22×D311yescharacteristic
269C2×M16C22×D311yescharacteristic
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—62no—
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C4×Q8—21no—
276C4×D4—21no—
277C4×D4—22no—
278#117—21no—
279#117—32no—
280#117C22×D311yescharacteristic
281C2×C42—21no—
282C2×C42C22×D311yescharacteristic
283#108—32no—
284#108—32no—
285#108—64no—
286#108—61no—
287#108—61no—
288#107—61no—
289#106—61no—
290#104—64no—
291#104—64no—
292#104—62no—
293#104—62no—
294#101—34no—
295#100—34no—
296#99—64no—
297#97—21no—
298#97—21no—
299#97—21no—
300#97—62no—
301#97#6711yescharacteristic
302#97#6711yescharacteristic
303#97#6711yescharacteristic
304#97#6711yescharacteristic
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—62no—
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—21no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—81no—
325C2×C3×C4—82no—
326C2×C3×C4C23×C411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332#67—42no—
333#67—42no—
334C2×Dic3—22no—
335C2×Dic3—42no—
336D12—22no—
337D12—22no—
338C4×D3—42no—
339C4×D3—42no—
340Dic6—22no—
341Dic6—22no—
342C3×C8—22no—
343C3×C8—22no—
344C3×C8C4×D411yescharacteristic
345C3×C8C4×D411yescharacteristic
346C3×C8C4×D411yescharacteristic
347C3×C8C4×D411yescharacteristic
348C3×C8#12112yes—
349C3×C8#12112yes—
350C3⋊C8—24no—
351C3⋊C8—42no—
352#41—21no—
353#41—21no—
354#41—62no—
355#41—62no—
356#41—62no—
357#41—62no—
358#41C2×C4×D312yes—
359#41C2×D1212yes—
360C2×Q8—32no—
361C2×Q8#23511yescharacteristic
362C2×D4—21no—
363C2×D4—32no—
364C2×D4—62no—
365C2×D4#23511yescharacteristic
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
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—21no—
383M16—22no—
384M16—22no—
385M16—24no—
386M16—34no—
387M16—62no—
388M16—62no—
389M16—62no—
390M16—122no—
391M16—122no—
392M16C2×C4×D312yes—
393M16C2×D1212yes—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—62no—
400C2×C8—62no—
401C2×C8—122no—
402C2×C8—122no—
403C2×C8C2×C4×D311yescharacteristic
404C2×C8C2×C4×D311yescharacteristic
405C2×C8C2×D1212yes—
406C2×C8#23511yescharacteristic
407C2×C8#23511yescharacteristic
408C2×C8#23512yes—
409C2×C8C2×#6711yescharacteristic
410C2×C8C2×#6711yescharacteristic
411C2×C8C2×#6711yescharacteristic
412C2×C8C2×#6711yescharacteristic
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—62no—
416C22⋊C4—22no—
417C22⋊C4—62no—
418C22⋊C4#23512yes—
419C42—22no—
420C42—21no—
421C42—21no—
422C42—62no—
423C42C2×#6711yescharacteristic
424C42C2×#6711yescharacteristic
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—42no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4C2×C4×D411yescharacteristic
438C3×C4C2×C4×D411yescharacteristic
439C3×C4C2×#12111yescharacteristic
440C3×C4#52911yescharacteristic
441Dic3—42no—
442Dic3—82no—
443C23—21no—
444C23—62no—
445C23C2×#23511yescharacteristic
446Q8—21no—
447Q8—62no—
448Q8—62no—
449Q8C4×D1212yes—
450D4—22no—
451D4—22no—
452D4—21no—
453D4—62no—
454D4—62no—
455D4—122no—
456D4—122no—
457D4C4×D1212yes—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—62no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4C22×C4×D311yescharacteristic
471C2×C4C22×D1211yescharacteristic
472C2×C4C2×#23511yescharacteristic
473C2×C4C2×#23511yescharacteristic
474C2×C4C2×#23511yescharacteristic
475C2×C4C22×#6711yescharacteristic
476C8—22no—
477C8—22no—
478C8—64no—
479C8—122no—
480C8C4×#6711yescharacteristic
481C8C4×#6711yescharacteristic
482C8C4×#6711yescharacteristic
483C8C4×#6711yescharacteristic
484C8#93012yes—
485C8#93012yes—
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—
497C22C4×#23511yescharacteristic
498C22#547711yescharacteristic
499C22#579111yescharacteristic
500C4—22no—
501C4—41no—
502C4—42no—
503C4—41no—
504C4—122no—
505C4—242no—
506C4C2×C4×D1211yescharacteristic
507C4#547911yescharacteristic
508C4C2×C4×#6711yescharacteristic
509C4C2×#93011yescharacteristic
510C3#2155911yescharacteristic, complemented, Sylow
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—242no—
516C2C4×#93011yescharacteristic
subgroupG/H
centerC4C2×#930
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2875C2
socleC2×C3C4×#121
prime radical#83805C2
cosocleC22×C3×C4C24
hypercenter#2875D3
hypocenterC3#21559
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2875

Complements

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

NH ≅ G/Ntype
1. C3×#2875C2semidirect
3. #83792C2semidirect
4. C2×#4717C2semidirect
5. #72013C2semidirect
6. #71645C2semidirect
7. #69901C2semidirect
8. #69898C2semidirect
9. #69897C2semidirect
10. #69874C2semidirect
11. #69829C2semidirect
30. #5143C4semidirect
31. #5142C4semidirect
32. #5141C4semidirect
52. #2875D3semidirect
91. C2×Dic12C2×C4semidirect
96. C2×D24C2×C4semidirect
97. C2×#204C2×C4semidirect
116. #824C2×C4semidirect
117. #823C2×C4semidirect
121. #819C2×C4semidirect
122. #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
C3×#28752
#838052
#837922
C2×#47172
#720132
#716452
#699012
#698982
#698972
#698742
#698292
#215593