Finite Group Databasebeta

← #10824861

Subgroups — #10824861

Order 768.

816 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D8C211yesmaximal, characteristic, complemented
2C4×#932C212yesmaximal, complemented
3#78455C211yesmaximal, characteristic, complemented
4#72046C211yesmaximal, characteristic
5#72042C211yesmaximal, characteristic, complemented
6C4×#833C212yesmaximal, complemented
7#70647C211yesmaximal, characteristic, complemented
8#70626C211yesmaximal, characteristic, complemented
9#70197C212yesmaximal, complemented
10#70196C212yesmaximal, complemented
11C4×#819C211yesmaximal, characteristic, complemented
12#23716—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×#932C2212yes—
16C4×#235C2211yescharacteristic
17C3×C4×D8—22no—
18C3×C4×D8—22no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#102C2212yes—
21C2×C3×C4×C8C2211yescharacteristic
22C2×#833C2212yes—
23C2×#833C2212yes—
24#5215C2212yes—
25#5110C2211yescharacteristic
26#5110C2211yescharacteristic
27C2×#819C2212yes—
28#5102C2211yescharacteristic
29#5102C2211yescharacteristic
30#5018—22no—
31C4×#214—22no—
32#5014—22no—
33C4×#213—22no—
34#4966C2212yes—
35#4964C2212yes—
36C4×#208C2212yes—
37C4×#208C2211yescharacteristic
38#4690—22no—
39#4685—22no—
40#4552C2211yescharacteristic
41#4552C2211yescharacteristic
42#4530C2212yes—
43#4530C2212yes—
44C4×AGL(1,ℤ/8)—32no—
45C2×C4×D8D311yescharacteristic, complemented
46#2922—31no—
47#2089—31no—
48#2078—31no—
49#2063—32no—
50#2051—32no—
51#2045—31no—
52#2018—31no—
53C4×#108—31no—
54C4×#100—32no—
55C22×C3×D4C2312yes—
56C23×C3×C4—22no—
57C2×#235C2311yescharacteristic
58C2×C3×D8—24no—
59C2×C3×D8—22no—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C42C2311yescharacteristic
69C2×#214—22no—
70#932—24no—
71#932—24no—
72C2×#213—22no—
73C4×#67—21no—
74C4×#67—21no—
75#926C2311yescharacteristic
76C2×#208C2312yes—
77C2×#208C2311yescharacteristic
78C2×#208C2311yescharacteristic
79C2×#208C2312yes—
80C4×D12D411yescharacteristic
81C4×D12D411yescharacteristic
82#872—21no—
83C42×D3—21no—
84C4×Dic6D411yescharacteristic
85C4×Dic6D411yescharacteristic
86C3×#107—22no—
87C3×#102—24no—
88C3×C4×C8D412yes—
89C3×C4×C8D412yes—
90#833—24no—
91#833—24no—
92#832—22no—
93#832—22no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—22no—
96#819—24no—
97Dic3⋊C8—22no—
98Dic3⋊C8—22no—
99#810—22no—
100#809—22no—
101#808—22no—
102#807—22no—
103#803—21no—
104#803—22no—
105C4×C3⋊C8—21no—
106C4×C3⋊C8—22no—
107C2×AGL(1,ℤ/8)—32no—
108C22×D8C2×D311yescharacteristic
109C4×#41—31no—
110C2×C4×D4C2×D312yes—
111#442—62no—
112#440—62no—
113C4×SD16—62no—
114C4×D8—22no—
115C4×D8—22no—
116C4×D8—62no—
117#433—62no—
118#431—62no—
119C2×#108—32no—
120#428—32no—
121C2×#107C2×D311yescharacteristic
122#424—31no—
123#424—31no—
124#418—32no—
125#417—32no—
126C2×#102C2×D312yes—
127C2×#100—32no—
128C2×#100—32no—
129#408—31no—
130#408—31no—
131C4×M16—31no—
132C4×M16—32no—
133C2×C4×C8C2×D311yescharacteristic
134#343—32no—
135#343—32no—
136#341—31no—
137#341—31no—
138C24×C3—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4#4114yes—
145C22×C3×C4—22no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—84no—
149C22×C3×C4#4111yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C2×#67—21no—
153#235—22no—
154#235—22no—
155C2×D12C2×D411yescharacteristic
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×Dic6C2×D411yescharacteristic
160C3×D8—28no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8C2×D412yes—
164C2×C3×C8C2×D412yes—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C22⋊C4—42no—
168C3×C22⋊C4—42no—
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171C3×C42C2×D411yescharacteristic
172C3×C42C2×D411yescharacteristic
173#217—21no—
174#214—28no—
175#213—28no—
176(C2×D3)⋊C4—21no—
177(C2×D3)⋊C4—21no—
178C4⋊Dic3C2×D411yescharacteristic
179C4⋊Dic3C2×D411yescharacteristic
180Dic3⋊C4—21no—
181Dic3⋊C4—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185#208—22no—
186#208—22no—
187#208—22no—
188#208—22no—
189#208—22no—
190#208#4114yes—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×#41—31no—
196C22×D4C22×D312yes—
197C23×C4—22no—
198AGL(1,ℤ/8)—64no—
199AGL(1,ℤ/8)—64no—
200C2×SD16—62no—
201C2×D8—24no—
202C2×D8—22no—
203C2×D8—62no—
204C2×M16—32no—
205C2×M16—32no—
206C2×M16—31no—
207C2×M16—31no—
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C4×Q8—31no—
211C4×Q8—31no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—31no—
217C4×D4—31no—
218C4×D4—61no—
219C4×D4—61no—
220#117—31no—
221#117—61no—
222C2×C4⋊C4C22×D312yes—
223C2×C22⋊C4—22no—
224C2×C42—61no—
225C2×C42C22×D311yescharacteristic
226#108—64no—
227#107—22no—
228#107—62no—
229#106—62no—
230C4⋊C8—62no—
231C4⋊C8—62no—
232#103—62no—
233#102—24no—
234#102—62no—
235#102—62no—
236#102—62no—
237#100—64no—
238#100—64no—
239C22⋊C8—62no—
240C22⋊C8—62no—
241#97—62no—
242#97—61no—
243C4×C8—62no—
244C4×C8—61no—
245C4×C8#6712yes—
246C4×C8#6712yes—
247C23×C3—24no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—82no—
251C23×C3—82no—
252C23×C3C2×#4111yescharacteristic
253C22×D3—21no—
254C3×D4—24no—
255C3×D4—24no—
256C3×D4—42no—
257C3×D4—42no—
258C2×C3×C4—22no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—84no—
262C2×C3×C4—84no—
263C2×C3×C4—82no—
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C2×#4112yes—
271C2×C3×C4C2×#4112yes—
272#67—42no—
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—21no—
276D12—22no—
277C4×D3—42no—
278C4×D3—41no—
279C4×D3—41no—
280C4×D3—42no—
281Dic6—22no—
282C3×C8—22no—
283C3×C8—22no—
284C3⋊C8—24no—
285C3⋊C8—41no—
286C3⋊C8—42no—
287C3⋊C8—41no—
288C24—22no—
289#41—62no—
290#41—62no—
291C2×Q8—31no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—31no—
298C2×D4—61no—
299C2×D4#23514yes—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—84no—
307C22×C4#23711yescharacteristic
308C22×C4#23711yescharacteristic
309C22×C4C23×D311yescharacteristic
310SD16—68no—
311D8—28no—
312D8—68no—
313M16—34no—
314M16—62no—
315M16—62no—
316M16—62no—
317M16—62no—
318M16—62no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—61no—
324C2×C8—61no—
325C2×C8C2×#6712yes—
326C2×C8C2×#6712yes—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—31no—
330C4⋊C4—31no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C42—61no—
339C42—61no—
340C42—61no—
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C42C2×#6711yescharacteristic
344C42C2×#6711yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—84no—
349C22×C3—84no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3C2×#12111yescharacteristic
353C22×C3#52912yes—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—82no—
361C3×C4#47212yes—
362C3×C4#52511yescharacteristic
363C3×C4#52512yes—
364C3×C4#53211yescharacteristic
365Dic3—41no—
366Dic3—41no—
367Dic3—41no—
368C23—24no—
369C23—42no—
370C23—42no—
371C23—61no—
372C23—82no—
373C23—82no—
374C23C2×#23711yescharacteristic
375Q8—62no—
376D4—24no—
377D4—24no—
378D4—42no—
379D4—42no—
380D4—62no—
381D4—122no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4#93811yescharacteristic
396C2×C4#93811yescharacteristic
397C2×C4#93811yescharacteristic
398C2×C4#93811yescharacteristic
399C2×C4C2×#23512yes—
400C2×C4C2×D3×D411yescharacteristic
401C2×C4D3×#4112yes—
402C2×C4C22×#6711yescharacteristic
403C8—22no—
404C8—22no—
405C8—64no—
406C8—121no—
407C8—122no—
408C8—121no—
409C2×C3—22no—
410C2×C3—84no—
411C2×C3#239211yescharacteristic
412C2×C3C2×#47211yescharacteristic
413C2×C3#306711yescharacteristic
414D3—81no—
415C22—21no—
416C22—21no—
417C22—44no—
418C22—84no—
419C22—84no—
420C22—82no—
421C22—82no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22#553812yes—
426C22C2×#93811yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—121no—
432C4—121no—
433C4—121no—
434C4#516012yes—
435C4#554712yes—
436C4#579111yescharacteristic
437C4#583211yescharacteristic
438C3#2371611yescharacteristic, complemented, Sylow
439C2—22no—
440C2—84no—
441C2—241no—
442C2#8240111yescharacteristic
443C2C2×#516011yescharacteristic
444C2#8392811yescharacteristic
subgroupG/H
centerC2×C4#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC22×C3C2×#121
prime radical#72046C2
hypercenterC2×C4×D8D3
hypocenterC3#23716
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D8C2semidirect
2. C4×#932C2semidirect
3. #78455C2semidirect
5. #72042C2semidirect
6. C4×#833C2semidirect
7. #70647C2semidirect
8. #70626C2semidirect
9. #70197C2semidirect
10. #70196C2semidirect
11. C4×#819C2semidirect
45. C2×C4×D8D3semidirect
438. C3#23716semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
C4×#9322
#784552
#720462
#720422
C4×#8332
#706472
#706262
#701972
#701962
C4×#8192
#237163