Finite Group Databasebeta

← #10824899

Subgroups — #10824899

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
1C2×#5152C211yesmaximal, characteristic, complemented
2#79071C211yesmaximal, characteristic, complemented
3#78914C212yesmaximal, complemented
4C3×#2045C211yesmaximal, characteristic, complemented
5#72046C211yesmaximal, characteristic
6#70533C211yesmaximal, characteristic, complemented
7#70317C212yesmaximal, complemented
8#70092C212yesmaximal, complemented
9Dic3×#100C212yesmaximal, complemented
10#68922C211yesmaximal, characteristic, complemented
11#68917C211yesmaximal, characteristic, complemented
12#23719—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×D4×Dic3C2212yes—
15C2×#932C2212yes—
16#5522C2211yescharacteristic
17C3×#424C2211yescharacteristic
18C3×#418C2212yes—
19C2×C3×#100C2212yes—
20C2×#833C2212yes—
21C2×#832C2212yes—
22#5152—22no—
23#5152—22no—
24#5133C2212yes—
25#5129C2211yescharacteristic
26C2×#822C2212yes—
27Dic3×M16C2212yes—
28Dic3×M16C2211yescharacteristic
29#5110C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#815C2211yescharacteristic
32#4999C2211yescharacteristic
33#4995C2212yes—
34#4840—22no—
35#4835—22no—
36#4785—22no—
37#4758—22no—
38#4757—22no—
39#4756—22no—
40C3×#341C2211yescharacteristic
41#4554C2212yes—
42#4554C2212yes—
43#4552C2211yescharacteristic
44C4×AGL(1,ℤ/8)—32no—
45C2×#442—31no—
46#2923—31no—
47#2090—31no—
48#2079—31no—
49#2063—32no—
50#2051—32no—
51#2045D311yescharacteristic, complemented
52#2018—31no—
53#1778—31no—
54C4×#100—32no—
55C22×C3×D4C2312yes—
56C23×Dic3—22no—
57C2×#235C2311yescharacteristic
58C2×C3×D8—24no—
59C2×C3×D8—22no—
60C2×C3×M16C2312yes—
61C2×C3×M16C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#117C2311yescharacteristic
64C2×#217—22no—
65D4×Dic3—22no—
66D4×Dic3—22no—
67D4×Dic3—22no—
68D4×Dic3—22no—
69C2×#214—22no—
70#932—24no—
71#932—24no—
72C2×#213—22no—
73C2×C4⋊Dic3C2312yes—
74C2×C4×Dic3C2311yescharacteristic
75C2×#208C2311yescharacteristic
76C2×#208C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78#893D412yes—
79#887D412yes—
80#67⋊C4—22no—
81#872—21no—
82C42×D3—21no—
83C3×C4⋊C8—22no—
84C3×#102—22no—
85C3×#102—22no—
86C3×#100—24no—
87#833—24no—
88#832—24no—
89#822—24no—
90#821—22no—
91(C2×D3)⋊C8—22no—
92(C2×D3)⋊C8—22no—
93#817—22no—
94#816—22no—
95#815—22no—
96#815—21no—
97#815D412yes—
98#815D412yes—
99C8×Dic3—22no—
100C8×Dic3—21no—
101#808—22no—
102#807—22no—
103C4⋊(C3⋊C8)—22no—
104C2×AGL(1,ℤ/8)—32no—
105C22×D8C2×D311yescharacteristic
106C4×#41—31no—
107C2×C4×D4—32no—
108#442—62no—
109#442—62no—
110#442—62no—
111#440—62no—
112C4×SD16—62no—
113C4×D8—62no—
114#433—62no—
115#432—62no—
116C2×#108—32no—
117#428—32no—
118C2×#106—31no—
119#424—31no—
120#424C2×D311yescharacteristic
121#418C2×D312yes—
122#417—32no—
123C2×#102—32no—
124C2×#100—32no—
125C2×#100C2×D312yes—
126#408—31no—
127#408—31no—
128C4×M16—31no—
129C4×M16—32no—
130C2×#97—31no—
131#343—32no—
132#343—32no—
133#341—31no—
134#341C2×D311yescharacteristic
135C24×C3—22no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4#4114yes—
142C22×C3×C4C2411yescharacteristic
143C2×#67—21no—
144C22×Dic3—22no—
145C22×Dic3—42no—
146C22×Dic3—42no—
147C22×Dic3—84no—
148C22×Dic3#4112yes—
149#235—22no—
150#235—22no—
151C2×D12C2×D411yescharacteristic
152C2×C4×D3—21no—
153C2×C4×D3—22no—
154C2×Dic6C2×D411yescharacteristic
155C3×D8—28no—
156C3×M16—22no—
157C3×M16—22no—
158C3×M16—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—21no—
162C2×C3×C8C2×D412yes—
163C3×C4⋊C4C2×D412yes—
164C3×C22⋊C4—21no—
165C3×C42—21no—
166#217—42no—
167#217—42no—
168#214—28no—
169#213—28no—
170(C2×D3)⋊C4—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—22no—
173Dic3⋊C4—22no—
174C4×Dic3—22no—
175C4×Dic3C2×D412yes—
176C4×Dic3C2×D411yescharacteristic
177C4×Dic3C2×D411yescharacteristic
178#208—22no—
179#208—22no—
180#208#4114yes—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8C2×D412yes—
185C2×#41—31no—
186C22×D4C22×D312yes—
187C23×C4—62no—
188AGL(1,ℤ/8)—64no—
189AGL(1,ℤ/8)—64no—
190C2×SD16—62no—
191C2×D8—24no—
192C2×D8—22no—
193C2×D8—62no—
194C2×M16—32no—
195C2×M16—31no—
196C2×M16C22×D312yes—
197C2×M16C22×D311yescharacteristic
198C22×C8—31no—
199C22×C8C22×D311yescharacteristic
200C4×Q8—32no—
201C4×D4—32no—
202C4×D4—62no—
203C4×D4—62no—
204C4×D4—62no—
205C4×D4—62no—
206C4×D4—62no—
207#117—61no—
208#117C22×D311yescharacteristic
209C2×C4⋊C4—32no—
210C2×C22⋊C4—62no—
211C2×C42—31no—
212C2×C42—61no—
213#108—64no—
214#107—62no—
215#106—62no—
216#106—62no—
217C4⋊C8—22no—
218C4⋊C8—62no—
219#103—62no—
220#102—22no—
221#102—22no—
222#102—62no—
223#102—64no—
224#100—24no—
225#100—64no—
226C22⋊C8—62no—
227C22⋊C8—62no—
228#97—32no—
229#97—32no—
230#97—61no—
231#97—62no—
232C4×C8—61no—
233C4×C8—62no—
234C23×C3—24no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—82no—
238C23×C3—82no—
239C23×C3C2×#4111yescharacteristic
240C22×D3—21no—
241C3×D4—24no—
242C3×D4—24no—
243C3×D4—42no—
244C3×D4—42no—
245C2×C3×C4—22no—
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×C3×C4C2×#4112yes—
250#67—42no—
251C2×Dic3—21no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—84no—
255C2×Dic3—84no—
256C2×Dic3—82no—
257C2×Dic3#12112yes—
258C2×Dic3#12112yes—
259D12—22no—
260C4×D3—41no—
261C4×D3—41no—
262C4×D3—82no—
263Dic6—22no—
264C3×C8—22no—
265C3×C8—41no—
266C3×C8—42no—
267C3⋊C8—24no—
268C3⋊C8—22no—
269C3⋊C8—41no—
270C24—22no—
271#41—62no—
272#41—62no—
273C2×Q8—31no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—31no—
280C2×D4—61no—
281C2×D4#23714yes—
282C22×C4—32no—
283C22×C4—62no—
284C22×C4—61no—
285C22×C4—62no—
286C22×C4—122no—
287C22×C4—122no—
288C22×C4—244no—
289C22×C4C23×D311yescharacteristic
290SD16—68no—
291D8—28no—
292D8—68no—
293M16—22no—
294M16—22no—
295M16—22no—
296M16—34no—
297M16—62no—
298M16—62no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—32no—
303C2×C8—62no—
304C2×C8—62no—
305C2×C8—61no—
306C2×C8D3×D412yes—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—62no—
310C4⋊C4D3×D412yes—
311C22⋊C4—21no—
312C22⋊C4—62no—
313C22⋊C4—122no—
314C22⋊C4—122no—
315C42—21no—
316C42—31no—
317C42—32no—
318C42—31no—
319C42—62no—
320C22×C3—22no—
321C22×C3—44no—
322C22×C3—84no—
323C22×C3—84no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3C2×#12111yescharacteristic
327C22×C3#52912yes—
328C2×D3—41no—
329C2×D3—42no—
330C3×C4—42no—
331C3×C4#52511yescharacteristic
332C3×C4#52512yes—
333C3×C4#53211yescharacteristic
334Dic3—22no—
335Dic3—22no—
336Dic3—41no—
337Dic3—82no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—61no—
342C23—82no—
343C23—82no—
344C23D3×#4111yescharacteristic
345Q8—62no—
346D4—24no—
347D4—24no—
348D4—42no—
349D4—42no—
350D4—62no—
351D4—122no—
352C2×C4—22no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—61no—
356C2×C4—122no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—242no—
361C2×C4—244no—
362C2×C4—244no—
363C2×C4—242no—
364C2×C4C2×D3×D411yescharacteristic
365C2×C4C2×D3×D411yescharacteristic
366C2×C4C2×#23712yes—
367C2×C4D3×#4112yes—
368C8—22no—
369C8—41no—
370C8—42no—
371C8—62no—
372C8—64no—
373C8—121no—
374C2×C3—22no—
375C2×C3—84no—
376C2×C3#239211yescharacteristic
377C2×C3#306812yes—
378D3—81no—
379C22—22no—
380C22—44no—
381C22—84no—
382C22—84no—
383C22—82no—
384C22—82no—
385C22—122no—
386C22—121no—
387C22D3×#12111yescharacteristic
388C22#564912yes—
389C4—42no—
390C4—62no—
391C4—62no—
392C4—121no—
393C4—242no—
394C4#558611yescharacteristic
395C4#560612yes—
396C4#566811yescharacteristic
397C3#2371911yescharacteristic, complemented, Sylow
398C2—22no—
399C2—84no—
400C2—241no—
401C2#7689111yescharacteristic
402C2#8093812yes—
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2045C2
socleC22×C3C2×#121
prime radical#72046C2
hypercenter#2045D3
hypocenterC3#23719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2045

Complements

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

NH ≅ G/Ntype
1. C2×#5152C2semidirect
2. #79071C2semidirect
3. #78914C2semidirect
4. C3×#2045C2semidirect
6. #70533C2semidirect
7. #70317C2semidirect
8. #70092C2semidirect
9. Dic3×#100C2semidirect
10. #68922C2semidirect
11. #68917C2semidirect
51. #2045D3semidirect
397. C3#23719semidirect

Maximal subgroups

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

subgroupindex
C2×#51522
#790712
#789142
C3×#20452
#720462
#705332
#703172
#700922
Dic3×#1002
#689222
#689172
#237193