Finite Group Databasebeta

← #10909550

Subgroups — #10909550

Order 768.

920 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84102C212yesmaximal, complemented
2#78890C212yesmaximal, complemented
3C3×#1792C211yesmaximal, characteristic, complemented
4#70390C212yesmaximal, complemented
5#70361C212yesmaximal, complemented
6#70302C212yesmaximal, complemented
7#70088C212yesmaximal, complemented
8#69664C211yesmaximal, characteristic, complemented
9#69656C211yesmaximal, characteristic, complemented
10#30429—31nomaximal, Sylow
11C2×C3×H(5,𝔽2)C2211yescharacteristic
12#5833C2212yes—
13C2×#950C2212yes—
14C2×#932C2212yes—
15C2×#907C2212yescomplemented
16#5510C2211yescharacteristic, complemented
17C2×C3×#104C2212yes—
18C3×#417C2212yes—
19C2×C3×#100C2211yescharacteristic
20C3×#390C2211yescharacteristic
21#5239—22no—
22#5238—22no—
23#5234C2212yes—
24#5232C2212yes—
25#5218—22no—
26#5217—22no—
27C2×#832C2212yes—
28#5133C2212yescomplemented
29C2×#808C2212yescomplemented
30#4935C2211yescharacteristic, complemented
31C2×#805C2212yescomplemented
32#4816—22no—
33#4807—22no—
34#4782—22no—
35#4778—22no—
36#4558C2212yescomplemented
37#4554C2212yescomplemented
38#4530C2211yescharacteristic, complemented
39#4524C2211yescharacteristic, complemented
40#3060—32no—
41#3012—32no—
42#2094—31no—
43#2079—32no—
44#2039—32no—
45#2006—31no—
46#1907—32no—
47#1876—32no—
48#1792D311yescharacteristic, complemented
49C3×H(5,𝔽2)—22no—
50C3×H(5,𝔽2)—42no—
51C3×H(5,𝔽2)—41no—
52C2×C3×#41—22no—
53C2×C3×#41C2312yes—
54C22×C3×D4—22no—
55C22×C3×D4—41no—
56C22×C3×D4C2311yescharacteristic
57C2×#235C2312yes—
58C2×C3×M16C2312yes—
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C22⋊C4—41no—
61C2×C3×C42C2311yescharacteristic
62#950—42no—
63#950—42no—
64#946—22no—
65C2×#216D412yes—
66C2×#215D412yes—
67#939—42no—
68#938—22no—
69C2×#214—22no—
70C2×#214D412yes—
71#932—24no—
72#932—42no—
73C2×#213—22no—
74C2×#213D412yes—
75#930—22no—
76#930—41no—
77C2×C4⋊Dic3—41no—
78C2×C4⋊Dic3C2312yes—
79#924—22no—
80C2×#208C2312yes—
81C22×C3⋊C8C2312yes—
82#907—42no—
83#907—42no—
84C2×ΓL(1,𝔽25)—22no—
85C2×D3×C8—22no—
86#897—22no—
87#895—22no—
88C3×#104—42no—
89C3×#103—22no—
90C3×#102—22no—
91C3×#100—41no—
92#835—22no—
93#832—22no—
94#832—22no—
95#832—22no—
96#830—22no—
97#830—22no—
98#821—22no—
99#816—22no—
100#808—22no—
101#808—22no—
102#805—42no—
103C2×H(5,𝔽2)C2×D311yescharacteristic
104C2×AGL(1,ℤ/8)—32no—
105C2×#135—32no—
106C2×#131—32no—
107#537—32no—
108#537—31no—
109#470—62no—
110#469—62no—
111#465—62no—
112#465—62no—
113#452—62no—
114#450—62no—
115#449—62no—
116#447—62no—
117C2×#106—32no—
118C2×#104—32no—
119C2×#104C2×D312yes—
120#417—32no—
121#417—32no—
122#417C2×D312yes—
123C2×#102—32no—
124C2×#100C2×D311yescharacteristic
125#408—32no—
126#390C2×D311yescharacteristic
127#384—31no—
128#343—32no—
129#343—31no—
130#341—32no—
131#336—31no—
132C24×C3—21no—
133C24×C3—41no—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—44no—
139C3×#41—44no—
140C2×C3×Q8C2×D412yes—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—21no—
145C2×C3×D4—42no—
146C2×C3×D4—44no—
147C2×C3×D4—44no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C2×C3×D4C2×D412yes—
151C2×C3×D4#4112yes—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—41no—
156C22×C3×C4C2411yescharacteristic
157C2×#67—22no—
158C2×#67—42no—
159C22×Dic3—22no—
160#235—22no—
161#235—22no—
162#235—22no—
163#235—22no—
164C2×D12—41no—
165C2×D12C2×D412yes—
166C2×C4×D3—22no—
167C2×Dic6C2×D412yes—
168C3×M16—22no—
169C3×M16—22no—
170C2×C3×C8—22no—
171C3×C4⋊C4—22no—
172C3×C22⋊C4—82no—
173C3×C42—22no—
174#217—42no—
175#216—42no—
176#215—42no—
177#214—42no—
178#214—44no—
179#213—42no—
180#213—44no—
181(C2×D3)⋊C4—42no—
182(C2×D3)⋊C4—42no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—41no—
186C4⋊Dic3—82no—
187Dic3⋊C4—42no—
188#208—22no—
189#208—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8C2×D412yes—
194C2×C3⋊C8C2×D412yes—
195ΓL(1,𝔽25)—44no—
196D3×C8—44no—
197H(5,𝔽2)—22no—
198H(5,𝔽2)—42no—
199H(5,𝔽2)—41no—
200C2×#41—22no—
201C2×#41—32no—
202C2×#41C22×D312yes—
203C22×D4—22no—
204C22×D4—41no—
205C22×D4C22×D311yescharacteristic
206AGL(1,ℤ/8)—64no—
207AGL(1,ℤ/8)—122no—
208#135—122no—
209#135—122no—
210C2×Q16—32no—
211C2×SD16—32no—
212C2×SD16—32no—
213C2×SD16—62no—
214C2×D8—32no—
215C2×D8—62no—
216#131—122no—
217#131—122no—
218C2×M16—32no—
219C2×M16—62no—
220C2×M16C22×D312yes—
221C22×C8—32no—
222C22×C8—62no—
223C22⋊Q8—62no—
224C22⋊Q8—62no—
225C22⋊Q8—62no—
226#121—62no—
227#121—62no—
228#121—62no—
229#121—121no—
230#121—122no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—121no—
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C22⋊C4—41no—
235C2×C42C22×D311yescharacteristic
236#106—62no—
237#106—62no—
238#104—42no—
239#104—122no—
240#103—22no—
241#103—62no—
242#103—62no—
243#102—22no—
244#102—62no—
245#102—62no—
246#102—62no—
247#102—62no—
248#100—41no—
249C22⋊C8—62no—
250C22⋊C8—62no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—44no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—81no—
257C23×C3—81no—
258C23×C3—161no—
259C23×C3—161no—
260C23×C3H(5,𝔽2)11yescharacteristic
261C22×D3—22no—
262C3×Q8—22no—
263C3×Q8—22no—
264C3×D4—22no—
265C3×D4—24no—
266C3×D4—22no—
267C3×D4—48no—
268C3×D4—42no—
269C3×D4—42no—
270C3×D4—42no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—41no—
274C2×C3×C4—44no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—81no—
280C2×C3×C4C22×D412yes—
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4H(5,𝔽2)11yescharacteristic
283C2×C3×C4H(5,𝔽2)12yes—
284#67—42no—
285#67—42no—
286#67—162no—
287C2×Dic3—22no—
288C2×Dic3—42no—
289C2×Dic3—82no—
290D12—22no—
291D12—22no—
292D12—161no—
293C4×D3—42no—
294C4×D3—42no—
295Dic6—22no—
296Dic6—22no—
297C3×C8—42no—
298C3⋊C8—22no—
299C3⋊C8—22no—
300C3⋊C8—42no—
301C24—21no—
302C24—41no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—44no—
308#41—44no—
309#41—62no—
310#41—62no—
311#41—62no—
312#41—62no—
313C2×Q8—32no—
314C2×Q8D3×D412yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—32no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—62no—
326C2×D4—121no—
327C2×D4—122no—
328C2×D4#23512yes—
329C2×D4D3×D412yes—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4C23×D311yescharacteristic
337Q16—122no—
338SD16—122no—
339SD16—122no—
340SD16—124no—
341D8—122no—
342D8—124no—
343M16—22no—
344M16—22no—
345M16—62no—
346M16—62no—
347M16—124no—
348C2×C8—22no—
349C2×C8—32no—
350C2×C8—32no—
351C2×C8—62no—
352C2×C8—62no—
353C2×C8—62no—
354C2×C8—124no—
355C4⋊C4—22no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—122no—
359C4⋊C4—121no—
360C4⋊C4—242no—
361C22⋊C4—82no—
362C22⋊C4—122no—
363C22⋊C4—122no—
364C22⋊C4—122no—
365C42—22no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—82no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—161no—
376C22×C3—164no—
377C22×C3#53811yescharacteristic
378C22×C3D4211yescharacteristic
379C22×C3#56011yescharacteristic
380C2×D3—42no—
381C2×D3—42no—
382C2×D3—42no—
383C3×C4—42no—
384C3×C4—42no—
385C3×C4—81no—
386C3×C4#53412yes—
387C3×C4#54612yes—
388Dic3—42no—
389Dic3—82no—
390C23—22no—
391C23—22no—
392C23—44no—
393C23—41no—
394C23—41no—
395C23—62no—
396C23—81no—
397C23—81no—
398C23—161no—
399C23—161no—
400C23#100411yescharacteristic
401Q8—22no—
402Q8—22no—
403Q8—62no—
404Q8—62no—
405D4—24no—
406D4—22no—
407D4—22no—
408D4—48no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—62no—
413D4—62no—
414D4—122no—
415D4—122no—
416D4—481no—
417D4—482no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—62no—
427C2×C4—81no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—242no—
432C2×C4C2×#23511yescharacteristic
433C2×C4C2×D3×D412yes—
434C2×C4#100412yes—
435C2×C4#100411yescharacteristic
436C8—42no—
437C8—62no—
438C8—62no—
439C8—122no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—82no—
443C2×C3—82no—
444C2×C3#266811yescharacteristic
445C2×C3#328811yescharacteristic
446C2×C3#329111yescharacteristic
447D3—82no—
448C22—21no—
449C22—21no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—82no—
455C22—81no—
456C22—81no—
457C22—122no—
458C22—122no—
459C22—122no—
460C22—161no—
461C22—164no—
462C22#549411yescharacteristic
463C22#559011yescharacteristic
464C22#559511yescharacteristic
465C4—42no—
466C4—42no—
467C4—81no—
468C4—122no—
469C4—242no—
470C4#549012yes—
471C4#558912yes—
472C3#3042911yescharacteristic, complemented, Sylow
473C2—21no—
474C2—21no—
475C2—82no—
476C2—82no—
477C2—242no—
478C2#7385711yescharacteristic
479C2#8028111yescharacteristic
480C2#8029411yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1792C2
socleC22×C3D42
hypercenter#1792D3
hypocenterC3#30429
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1792

Complements

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

NH ≅ G/Ntype
1. #84102C2semidirect
2. #78890C2semidirect
3. C3×#1792C2semidirect
4. #70390C2semidirect
5. #70361C2semidirect
6. #70302C2semidirect
7. #70088C2semidirect
8. #69664C2semidirect
9. #69656C2semidirect
15. C2×#907C22semidirect
16. #5510C22semidirect
28. #5133C22semidirect
29. C2×#808C22semidirect
30. #4935C22semidirect
31. C2×#805C22semidirect
36. #4558C22semidirect
37. #4554C22semidirect
38. #4530C22semidirect
39. #4524C22semidirect
48. #1792D3semidirect
472. C3#30429semidirect

Maximal subgroups

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

subgroupindex
#841022
#788902
C3×#17922
#703902
#703612
#703022
#700882
#696642
#696562
#304293