Finite Group Databasebeta

← #10824859

Subgroups — #10824859

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
1C3×#2922C211yesmaximal, characteristic, complemented
2C4×#949C212yesmaximal, complemented
3C2×C4×D24C211yesmaximal, characteristic, complemented
4#70542C212yesmaximal, complemented
5#70540C211yesmaximal, characteristic, complemented
6#70532C211yesmaximal, characteristic, complemented
7#70006C211yesmaximal, characteristic
8#70003C212yesmaximal, complemented
9#69994C211yesmaximal, characteristic, complemented
10C4×#823C212yesmaximal, complemented
11C4×#819C211yesmaximal, characteristic, complemented
12#23716—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#949C2212yes—
15C22×D24C2211yescharacteristic
16C2×C4×D12C2212yes—
17C3×#433—22no—
18C3×#431—22no—
19C3×#424C2211yescharacteristic
20C3×#424C2211yescharacteristic
21C3×#408C2211yescharacteristic
22C3×#408C2211yescharacteristic
23C3×C4×M16C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25#5232C2212yes—
26C2×#823C2212yes—
27C2×#823C2212yes—
28C2×#821C2212yes—
29C2×#819C2212yes—
30C2×#818C2211yescharacteristic
31#5026—22no—
32C4×#215—22no—
33#5014—22no—
34C4×#213—22no—
35#5001C2212yes—
36#4995C2212yes—
37C4×#208C2212yes—
38C4×D24—22no—
39C4×D24—22no—
40#4554C2212yes—
41#4554C2212yes—
42#4552C2211yescharacteristic
43#4552C2211yescharacteristic
44C4×AGL(1,ℤ/8)—32no—
45C2×C4×D8—31no—
46#2922D311yescharacteristic, complemented
47#2089—31no—
48#2078—31no—
49#2063—32no—
50#2051—32no—
51#2045—31no—
52#2018—31no—
53C4×#108—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×Q8D411yescharacteristic
63C3×C4×Q8D411yescharacteristic
64C3×C4×D4—21no—
65C3×C4×D4—21no—
66C3×C4×D4D411yescharacteristic
67C3×C4×D4D411yescharacteristic
68C3×#117—21no—
69C3×#117C2311yescharacteristic
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72#949—24no—
73#949—24no—
74C2×#215—22no—
75C2×#213—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3C2312yes—
78C2×#208C2312yes—
79C2×#208C2312yes—
80C2×D24—24no—
81C2×D24—22no—
82C4×D12—22no—
83C4×D12—22no—
84C4×D12—22no—
85C4×D12—22no—
86C3×C4⋊C8—22no—
87C3×C4⋊C8—22no—
88C3×C22⋊C8—22no—
89C3×C22⋊C8—22no—
90C3×#97—21no—
91C3×C4×C8—21no—
92C3×C4×C8D412yes—
93C3×C4×C8D412yes—
94#835—22no—
95#832—22no—
96#823—24no—
97#823—24no—
98#821—24no—
99#819—24no—
100#818—22no—
101#809—22no—
102#809—22no—
103#808—22no—
104#807—22no—
105#803—22no—
106C4×C3⋊C8—22no—
107C2×AGL(1,ℤ/8)—32no—
108C22×D8—31no—
109C4×#41C2×D311yescharacteristic
110C2×C4×D4—32no—
111#442—62no—
112#440—62no—
113C4×SD16—62no—
114C4×D8—62no—
115C4×D8—62no—
116C4×D8—62no—
117#433—22no—
118#431—22no—
119C2×#108—32no—
120#428—32no—
121C2×#107—31no—
122#424C2×D311yescharacteristic
123#424C2×D311yescharacteristic
124#418—32no—
125#417—32no—
126C2×#102—32no—
127C2×#100—32no—
128C2×#100—32no—
129#408C2×D311yescharacteristic
130#408C2×D311yescharacteristic
131C4×M16—32no—
132C4×M16C2×D311yescharacteristic
133C2×C4×C8C2×D311yescharacteristic
134#343—32no—
135#343—32no—
136#341—31no—
137#341—31no—
138C23×D3—22no—
139C3×#41—22no—
140C3×#41—22no—
141C2×C3×Q8C2×D411yescharacteristic
142C2×C3×D4—21no—
143C2×C3×D4C2×D411yescharacteristic
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×C3×C4C2411yescharacteristic
150C22×Dic3—22no—
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12#4114yes—
157C2×C4×D3—42no—
158C2×C4×D3—42no—
159C2×C4×D3—84no—
160C3×M16—22no—
161C3×M16—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8C2×D412yes—
167C2×C3×C8C2×D412yes—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4C2×D411yescharacteristic
171C3×C4⋊C4C2×D411yescharacteristic
172C3×C22⋊C4—21no—
173C3×C22⋊C4—21no—
174C3×C22⋊C4—21no—
175C3×C42—21no—
176C3×C42—21no—
177C3×C42—21no—
178C3×C42C2×D411yescharacteristic
179C3×C42C2×D411yescharacteristic
180C3×C42C2×D411yescharacteristic
181C3×C42C2×D411yescharacteristic
182#215—28no—
183#213—28no—
184(C2×D3)⋊C4—42no—
185(C2×D3)⋊C4—42no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188#208—22no—
189#208—22no—
190#208—22no—
191#208#4114yes—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194D24—28no—
195C2×#41C22×D311yescharacteristic
196C22×D4—32no—
197C23×C4—62no—
198AGL(1,ℤ/8)—64no—
199AGL(1,ℤ/8)—64no—
200C2×SD16—62no—
201C2×D8—64no—
202C2×D8—62no—
203C2×D8—62no—
204C2×M16—32no—
205C2×M16—32no—
206C2×M16C22×D311yescharacteristic
207C2×M16C22×D311yescharacteristic
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C4×Q8D1211yescharacteristic
211C4×Q8D1211yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—62no—
215C4×D4—62no—
216C4×D4—62no—
217C4×D4—62no—
218C4×D4D1211yescharacteristic
219C4×D4D1211yescharacteristic
220#117—21no—
221#117C22×D311yescharacteristic
222C2×C4⋊C4—32no—
223C2×C22⋊C4—62no—
224C2×C42—21no—
225C2×C42C22×D311yescharacteristic
226#108—64no—
227#107—62no—
228#107—62no—
229#106—62no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232#103—62no—
233#102—62no—
234#102—64no—
235#102—62no—
236#102—62no—
237#100—64no—
238#100—64no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241#97—21no—
242#97—62no—
243C4×C8—21no—
244C4×C8—62no—
245C4×C8#6712yes—
246C4×C8#6712yes—
247C23×C3—21no—
248C23×C3C2×#4111yescharacteristic
249C22×D3—24no—
250C22×D3—42no—
251C22×D3—42no—
252C22×D3—82no—
253C22×D3—82no—
254C3×Q8—22no—
255C3×D4—22no—
256C3×D4—42no—
257C2×C3×C4—22no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C2×#4112yes—
273C2×C3×C4C2×#4112yes—
274C2×Dic3—42no—
275C2×Dic3—82no—
276D12—24no—
277D12—24no—
278D12—42no—
279D12—42no—
280C4×D3—84no—
281C4×D3—84no—
282C3×C8—22no—
283C3×C8—22no—
284C3×C8—41no—
285C3×C8—41no—
286C3⋊C8—24no—
287C3⋊C8—42no—
288C24—62no—
289#41—22no—
290#41—22no—
291C2×Q8C2×D1211yescharacteristic
292C2×D4—21no—
293C2×D4—34no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4C2×D1211yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—62no—
304C22×C4—122no—
305C22×C4—122no—
306C22×C4—244no—
307C22×C4#23511yescharacteristic
308C22×C4#23511yescharacteristic
309C22×C4C23×D311yescharacteristic
310SD16—68no—
311D8—68no—
312D8—68no—
313M16—22no—
314M16—22no—
315M16—34no—
316M16—62no—
317M16—62no—
318M16—62no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8C2×#6712yes—
326C2×C8C2×#6712yes—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—62no—
330C4⋊C4—62no—
331C4⋊C4C2×D1211yescharacteristic
332C4⋊C4C2×D1211yescharacteristic
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—122no—
337C22⋊C4—122no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42C2×D1211yescharacteristic
342C42C2×D1211yescharacteristic
343C42C2×#6711yescharacteristic
344C42C2×#6711yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3C2×#12111yescharacteristic
351C22×C3#52912yes—
352C2×D3—44no—
353C2×D3—84no—
354C2×D3—82no—
355C2×D3—82no—
356C2×D3—84no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4#47212yes—
364C3×C4#52512yes—
365C3×C4#52511yescharacteristic
366C3×C4#53211yescharacteristic
367Dic3—82no—
368C23—21no—
369C23—64no—
370C23—122no—
371C23—122no—
372C23—242no—
373C23—242no—
374C23C2×#23511yescharacteristic
375Q8—22no—
376D4—22no—
377D4—42no—
378D4—64no—
379D4—64no—
380D4—122no—
381D4—122no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—122no—
392C2×C4—244no—
393C2×C4—244no—
394C2×C4—242no—
395C2×C4#93011yescharacteristic
396C2×C4#93011yescharacteristic
397C2×C4#93011yescharacteristic
398C2×C4#93011yescharacteristic
399C2×C4C22×D1211yescharacteristic
400C2×C4D3×#4112yes—
401C2×C4D3×#4112yes—
402C2×C4C22×#6711yescharacteristic
403C8—22no—
404C8—22no—
405C8—41no—
406C8—41no—
407C8—64no—
408C8—122no—
409C2×C3—22no—
410C2×C3—81no—
411C2×C3#239211yescharacteristic
412C2×C3C2×#47211yescharacteristic
413C2×C3#306711yescharacteristic
414D3—84no—
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—124no—
421C22—244no—
422C22—244no—
423C22—242no—
424C22—242no—
425C22#552812yes—
426C22C2×#93011yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—242no—
434C4#514212yes—
435C4#552412yes—
436C4#582811yescharacteristic
437C4#583211yescharacteristic
438C3#2371611yescharacteristic, complemented, Sylow
439C2—22no—
440C2—81no—
441C2—244no—
442C2#8227311yescharacteristic
443C2C2×#514211yescharacteristic
444C2#8380511yescharacteristic
subgroupG/H
centerC2×C4#930
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2922C2
socleC22×C3C2×#121
prime radical#70006C2
hypercenter#2922D3
hypocenterC3#23716
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2922

Complements

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

NH ≅ G/Ntype
1. C3×#2922C2semidirect
2. C4×#949C2semidirect
3. C2×C4×D24C2semidirect
4. #70542C2semidirect
5. #70540C2semidirect
6. #70532C2semidirect
8. #70003C2semidirect
9. #69994C2semidirect
10. C4×#823C2semidirect
11. C4×#819C2semidirect
46. #2922D3semidirect
438. C3#23716semidirect

Maximal subgroups

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

subgroupindex
C3×#29222
C4×#9492
C2×C4×D242
#705422
#705402
#705322
#700062
#700032
#699942
C4×#8232
C4×#8192
#237163