Finite Group Databasebeta

← #10824896

Subgroups — #10824896

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2923C211yesmaximal, characteristic, complemented
2C4×#949C212yesmaximal, complemented
3C2×#4711C211yesmaximal, characteristic, complemented
4#70542C212yesmaximal, complemented
5#70540C211yesmaximal, characteristic, complemented
6#70533C211yesmaximal, characteristic, complemented
7#70006C211yesmaximal, characteristic
8#70003C212yesmaximal, complemented
9#69995C211yesmaximal, characteristic, complemented
10C4×#823C212yesmaximal, complemented
11#69840C211yesmaximal, characteristic, complemented
12#23719—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#949C2212yes—
15C22×D24C2211yescharacteristic
16C2×C4×D12C2212yes—
17C3×#433—22no—
18C3×#432—22no—
19C3×#424C2211yescharacteristic
20C3×#424C2211yescharacteristic
21C3×#408C2211yescharacteristic
22C3×#408C2211yescharacteristic
23C3×C4×M16C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25#5232C2212yes—
26C2×#823C2212yes—
27C2×#823C2212yes—
28C2×#821C2212yes—
29C2×#819C2212yes—
30C2×#817C2211yescharacteristic
31#5026—22no—
32C4×#215—22no—
33#5014—22no—
34C4×#213—22no—
35#5001C2212yes—
36#4995C2212yes—
37C4×#208C2212yes—
38#4711—22no—
39#4711—22no—
40#4554C2212yes—
41#4554C2212yes—
42#4552C2211yescharacteristic
43#4552C2211yescharacteristic
44C4×AGL(1,ℤ/8)—32no—
45C2×#442—31no—
46#2923D311yescharacteristic, complemented
47#2090—31no—
48#2079—31no—
49#2063—32no—
50#2051—32no—
51#2045—31no—
52#2018—31no—
53#1778—31no—
54C4×#100—32no—
55C2×C3×#41C2311yescharacteristic
56C22×D12C2312yes—
57C22×C4×D3—22no—
58C2×C3×M16C2311yescharacteristic
59C2×C3×M16C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×C4×Q8D412yes—
63C3×C4×D4—22no—
64C3×C4×D4D412yes—
65C3×#117—21no—
66C3×#117C2311yescharacteristic
67C2×C3×C42—21no—
68C2×C3×C42C2311yescharacteristic
69#949—24no—
70#949—24no—
71C2×#215—22no—
72C2×#213—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×C4⋊Dic3C2312yes—
75C2×#208C2312yes—
76C2×#208C2312yes—
77C2×D24—24no—
78C2×D24—22no—
79C4×D12—22no—
80C4×D12—22no—
81C4×D12—22no—
82C4×D12—22no—
83C3×C4⋊C8—22no—
84C3×C4⋊C8—22no—
85C3×C22⋊C8—22no—
86C3×C22⋊C8—22no—
87C3×#97—21no—
88C3×#97D412yes—
89C3×#97D412yes—
90C3×C4×C8—21no—
91#835—22no—
92#832—22no—
93#823—24no—
94#823—24no—
95#821—24no—
96#819—24no—
97#817—22no—
98#809—22no—
99#809—22no—
100#808—22no—
101#807—22no—
102#803—22no—
103C4×C3⋊C8—22no—
104C2×AGL(1,ℤ/8)—32no—
105C22×D8—31no—
106C4×#41C2×D311yescharacteristic
107C2×C4×D4—32no—
108#442—62no—
109#442—62no—
110#442—62no—
111#440—62no—
112C4×SD16—62no—
113C4×D8—62no—
114#433—22no—
115#432—22no—
116C2×#108—32no—
117#428—32no—
118C2×#106—31no—
119#424C2×D311yescharacteristic
120#424C2×D311yescharacteristic
121#418—32no—
122#417—32no—
123C2×#102—32no—
124C2×#100—32no—
125C2×#100—32no—
126#408C2×D311yescharacteristic
127#408C2×D311yescharacteristic
128C4×M16—32no—
129C4×M16C2×D311yescharacteristic
130C2×#97C2×D311yescharacteristic
131#343—32no—
132#343—32no—
133#341—31no—
134#341—31no—
135C23×D3—22no—
136C3×#41—22no—
137C3×#41—22no—
138C2×C3×Q8C2×D411yescharacteristic
139C2×C3×D4—21no—
140C2×C3×D4C2×D411yescharacteristic
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4#4112yes—
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—22no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12#4114yes—
152C2×C4×D3—42no—
153C2×C4×D3—42no—
154C2×C4×D3—84no—
155C3×M16—22no—
156C3×M16—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8C2×D412yes—
162C2×C3×C8C2×D412yes—
163C3×C4⋊C4—22no—
164C3×C4⋊C4C2×D412yes—
165C3×C22⋊C4—21no—
166C3×C22⋊C4—22no—
167C3×C42—22no—
168C3×C42—21no—
169C3×C42C2×D412yes—
170C3×C42C2×D411yescharacteristic
171C3×C42C2×D411yescharacteristic
172#215—28no—
173#213—28no—
174(C2×D3)⋊C4—42no—
175(C2×D3)⋊C4—42no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178#208—22no—
179#208—22no—
180#208—22no—
181#208#4114yes—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184D24—28no—
185C2×#41C22×D311yescharacteristic
186C22×D4—32no—
187C23×C4—62no—
188AGL(1,ℤ/8)—64no—
189AGL(1,ℤ/8)—64no—
190C2×SD16—62no—
191C2×D8—64no—
192C2×D8—62no—
193C2×D8—62no—
194C2×M16—32no—
195C2×M16—32no—
196C2×M16C22×D311yescharacteristic
197C2×M16C22×D311yescharacteristic
198C22×C8C22×D311yescharacteristic
199C22×C8C22×D311yescharacteristic
200C4×Q8D1212yes—
201C4×D4—22no—
202C4×D4—62no—
203C4×D4—62no—
204C4×D4—62no—
205C4×D4—62no—
206C4×D4D1212yes—
207#117—21no—
208#117C22×D311yescharacteristic
209C2×C4⋊C4—32no—
210C2×C22⋊C4—62no—
211C2×C42—21no—
212C2×C42C22×D311yescharacteristic
213#108—64no—
214#107—62no—
215#106—62no—
216#106—62no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219#103—62no—
220#102—62no—
221#102—64no—
222#102—62no—
223#102—62no—
224#100—64no—
225#100—64no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228#97—21no—
229#97—62no—
230#97#6712yes—
231#97#6712yes—
232C4×C8—21no—
233C4×C8—62no—
234C23×C3—21no—
235C23×C3C2×#4111yescharacteristic
236C22×D3—24no—
237C22×D3—42no—
238C22×D3—42no—
239C22×D3—82no—
240C22×D3—82no—
241C3×Q8—22no—
242C3×D4—22no—
243C3×D4—42no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—82no—
250C2×C3×C4#12112yes—
251C2×C3×C4#12112yes—
252C2×C3×C4C22×D411yescharacteristic
253C2×C3×C4C22×D411yescharacteristic
254C2×C3×C4C2×#4112yes—
255C2×C3×C4C2×#4112yes—
256C2×Dic3—42no—
257C2×Dic3—82no—
258D12—24no—
259D12—24no—
260D12—42no—
261D12—42no—
262C4×D3—84no—
263C4×D3—84no—
264C3×C8—22no—
265C3×C8—22no—
266C3×C8—41no—
267C3×C8—41no—
268C3⋊C8—24no—
269C3⋊C8—42no—
270C24—62no—
271#41—22no—
272#41—22no—
273C2×Q8C2×D1211yescharacteristic
274C2×D4—21no—
275C2×D4—34no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4C2×D1211yescharacteristic
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—62no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4—244no—
288C22×C4#23512yes—
289C22×C4C23×D311yescharacteristic
290SD16—68no—
291D8—68no—
292D8—68no—
293M16—22no—
294M16—22no—
295M16—34no—
296M16—62no—
297M16—62no—
298M16—62no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—62no—
304C2×C8—62no—
305C2×C8C2×#6712yes—
306C2×C8C2×#6712yes—
307C4⋊C4—22no—
308C4⋊C4—62no—
309C4⋊C4—62no—
310C4⋊C4C2×D1212yes—
311C22⋊C4—21no—
312C22⋊C4—22no—
313C22⋊C4—122no—
314C22⋊C4—122no—
315C42—22no—
316C42—21no—
317C42C2×D1212yes—
318C42C2×#6711yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—22no—
321C22×C3—41no—
322C22×C3—42no—
323C22×C3C2×#12111yescharacteristic
324C22×C3#52912yes—
325C2×D3—44no—
326C2×D3—84no—
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—84no—
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—41no—
333C3×C4—42no—
334C3×C4#52512yes—
335C3×C4#52511yescharacteristic
336C3×C4#53211yescharacteristic
337Dic3—82no—
338C23—21no—
339C23—64no—
340C23—122no—
341C23—122no—
342C23—242no—
343C23—242no—
344C23C2×#23511yescharacteristic
345Q8—22no—
346D4—22no—
347D4—42no—
348D4—64no—
349D4—64no—
350D4—122no—
351D4—122no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—122no—
359C2×C4—244no—
360C2×C4—244no—
361C2×C4—242no—
362C2×C4#93012yes—
363C2×C4#93012yes—
364C2×C4C22×D1211yescharacteristic
365C2×C4D3×#4112yes—
366C2×C4D3×#4112yes—
367C2×C4C22×#6711yescharacteristic
368C8—22no—
369C8—22no—
370C8—41no—
371C8—41no—
372C8—64no—
373C8—122no—
374C2×C3—22no—
375C2×C3—81no—
376C2×C3#239211yescharacteristic
377C2×C3#306812yes—
378D3—84no—
379C22—22no—
380C22—41no—
381C22—42no—
382C22—124no—
383C22—244no—
384C22—244no—
385C22—242no—
386C22—242no—
387C22#552812yes—
388C22C2×#93011yescharacteristic
389C4—22no—
390C4—22no—
391C4—41no—
392C4—42no—
393C4—242no—
394C4#552412yes—
395C4#582811yescharacteristic
396C4#583211yescharacteristic
397C3#2371911yescharacteristic, complemented, Sylow
398C2—22no—
399C2—81no—
400C2—244no—
401C2#8227311yescharacteristic
402C2#8381412yes—
subgroupG/H
centerC22C2×#930
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2923C2
socleC22×C3C2×#121
prime radical#70006C2
hypercenter#2923D3
hypocenterC3#23719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2923

Complements

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

NH ≅ G/Ntype
1. C3×#2923C2semidirect
2. C4×#949C2semidirect
3. C2×#4711C2semidirect
4. #70542C2semidirect
5. #70540C2semidirect
6. #70533C2semidirect
8. #70003C2semidirect
9. #69995C2semidirect
10. C4×#823C2semidirect
11. #69840C2semidirect
46. #2923D3semidirect
397. C3#23719semidirect

Maximal subgroups

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

subgroupindex
C3×#29232
C4×#9492
C2×#47112
#705422
#705402
#705332
#700062
#700032
#699952
C4×#8232
#698402
#237193