Finite Group Databasebeta

← #10824897

Subgroups — #10824897

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×#4869C211yesmaximal, characteristic, complemented
2#78866C212yesmaximal, complemented
3#78649C211yesmaximal, characteristic, complemented
4C3×#2079C211yesmaximal, characteristic, complemented
5#70573C211yesmaximal, characteristic, complemented
6#70307C212yesmaximal, complemented
7#70006C211yesmaximal, characteristic
8#68996C212yesmaximal, complemented
9#68972C212yesmaximal, complemented
10#68932C211yesmaximal, characteristic, complemented
11#68917C211yesmaximal, characteristic, complemented
12#23719—31nomaximal, Sylow
13Dic3×#41C2211yescharacteristic
14C2×#949C2212yes—
15C22×D24C2211yescharacteristic
16C2×#893C2212yes—
17C2×C3×#106C2211yescharacteristic
18C3×#417C2212yes—
19C3×#408C2211yescharacteristic
20#5234C2211yescharacteristic
21#5131C2212yes—
22C2×#823C2212yes—
23C2×#823C2212yes—
24C2×#822C2212yes—
25#5119C2211yescharacteristic
26Dic3×M16C2212yes—
27Dic3×M16C2211yescharacteristic
28#5102C2211yescharacteristic
29C2×#815C2211yescharacteristic
30C2×#809C2212yes—
31#4964C2212yes—
32#4869—22no—
33#4869—22no—
34#4817—22no—
35#4788—22no—
36#4785—22no—
37#4756—22no—
38#4720—22no—
39#4719—22no—
40C3×#343C2212yes—
41C3×#341C2211yescharacteristic
42#4552C2211yescharacteristic
43#4530C2212yes—
44C4×AGL(1,ℤ/8)—32no—
45C2×#442—31no—
46#2923—31no—
47#2090—31no—
48#2079D311yescharacteristic, complemented
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×M16C2312yes—
59C2×C3×M16C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62#949—24no—
63#949—24no—
64Q8×Dic3D412yes—
65C2×#215—22no—
66D4×Dic3—22no—
67D4×Dic3D412yes—
68C2×#213—22no—
69C2×(C2×D3)⋊C4—22no—
70#926—21no—
71#926C2311yescharacteristic
72C2×C4×Dic3—21no—
73C2×C4×Dic3C2311yescharacteristic
74C2×#208C2311yescharacteristic
75C2×#208C2312yes—
76C22×C3⋊C8C2311yescharacteristic
77C2×D24—24no—
78C2×D24—22no—
79#893—22no—
80#893—22no—
81#893—22no—
82#893—22no—
83C3×#106—22no—
84C3×#103—22no—
85C3×#102—22no—
86C3×C22⋊C8—22no—
87#830—22no—
88#823—24no—
89#823—24no—
90#822—24no—
91#821—22no—
92#821—22no—
93#815—21no—
94#815—22no—
95#815D412yes—
96#815D412yes—
97Dic3⋊C8—22no—
98Dic3⋊C8—22no—
99C8×Dic3—21no—
100C8×Dic3—22no—
101#809—24no—
102#808—22no—
103#807—22no—
104C2×AGL(1,ℤ/8)—32no—
105C22×D8—31no—
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×#106C2×D311yescharacteristic
119#424—31no—
120#424—31no—
121#418—32no—
122#417C2×D312yes—
123C2×#102—32no—
124C2×#100—32no—
125C2×#100—32no—
126#408—31no—
127#408C2×D311yescharacteristic
128C4×M16—31no—
129C4×M16—32no—
130C2×#97—31no—
131#343—32no—
132#343C2×D312yes—
133#341—31no—
134#341C2×D311yescharacteristic
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×C4C2411yescharacteristic
144C22×Dic3—22no—
145C22×Dic3#4112yes—
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—
157C3×M16—22no—
158C2×C3×C8—22no—
159C2×C3×C8—21no—
160C2×C3×C8—22no—
161C2×C3×C8C2×D412yes—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164#217—21no—
165#217—22no—
166#215—28no—
167#213—28no—
168(C2×D3)⋊C4—42no—
169(C2×D3)⋊C4—42no—
170C4⋊Dic3—22no—
171C4⋊Dic3C2×D412yes—
172C4×Dic3—21no—
173C4×Dic3—22no—
174C4×Dic3C2×D412yes—
175C4×Dic3C2×D411yescharacteristic
176C4×Dic3C2×D411yescharacteristic
177#208—22no—
178#208—22no—
179#208#4114yes—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8C2×D412yes—
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—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—31no—
208#117—61no—
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—62no—
211C2×C42—31no—
212C2×C42—61no—
213#108—64no—
214#107—62no—
215#106—22no—
216#106—62no—
217C4⋊C8—62no—
218C4⋊C8—62no—
219#103—22no—
220#102—22no—
221#102—64no—
222#102—62no—
223#102—62no—
224#100—64no—
225#100—64no—
226C22⋊C8—22no—
227C22⋊C8—62no—
228#97—32no—
229#97—32no—
230#97—61no—
231#97—62no—
232C4×C8—61no—
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—21no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—82no—
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C22×D411yescharacteristic
251C2×C3×C4C2×#4112yes—
252C2×C3×C4C2×#4112yes—
253C2×Dic3—22no—
254C2×Dic3—42no—
255C2×Dic3—82no—
256C2×Dic3#12112yes—
257C2×Dic3#12112yes—
258D12—24no—
259D12—24no—
260D12—42no—
261D12—42no—
262C4×D3—84no—
263C4×D3—84no—
264C3×C8—22no—
265C3×C8—42no—
266C3×C8—41no—
267C3⋊C8—24no—
268C3⋊C8—22no—
269C3⋊C8—41no—
270C24—62no—
271#41—22no—
272#41—22no—
273C2×Q8D3×D411yescharacteristic
274C2×D4—21no—
275C2×D4—34no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4D3×D411yescharacteristic
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—32no—
285C22×C4—62no—
286C22×C4—122no—
287C22×C4—122no—
288C22×C4—244no—
289C22×C4C23×D311yescharacteristic
290SD16—68no—
291D8—68no—
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—22no—
308C4⋊C4—22no—
309C4⋊C4—32no—
310C4⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—61no—
313C22⋊C4—122no—
314C22⋊C4—122no—
315C42—31no—
316C42—32no—
317C42—31no—
318C42—62no—
319C42—61no—
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—41no—
331C3×C4—82no—
332C3×C4#52512yes—
333C3×C4#52511yescharacteristic
334C3×C4#53211yescharacteristic
335Dic3—22no—
336Dic3—22no—
337Dic3—42no—
338C23—21no—
339C23—64no—
340C23—122no—
341C23—122no—
342C23—242no—
343C23—242no—
344C23D3×#4111yescharacteristic
345Q8—22no—
346D4—22no—
347D4—42no—
348D4—64no—
349D4—64no—
350D4—122no—
351D4—122no—
352C2×C4—21no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—62no—
359C2×C4—82no—
360C2×C4—122no—
361C2×C4—244no—
362C2×C4—244no—
363C2×C4—242no—
364C2×C4C2×D3×D411yescharacteristic
365C2×C4C2×D3×D411yescharacteristic
366C2×C4D3×#4112yes—
367C2×C4D3×#4112yes—
368C8—22no—
369C8—42no—
370C8—41no—
371C8—64no—
372C8—62no—
373C8—121no—
374C2×C3—22no—
375C2×C3—81no—
376C2×C3#239211yescharacteristic
377C2×C3#306812yes—
378D3—84no—
379C22—22no—
380C22—42no—
381C22—41no—
382C22—124no—
383C22—244no—
384C22—244no—
385C22—242no—
386C22—242no—
387C22D3×#12111yescharacteristic
388C22#570612yes—
389C4—41no—
390C4—62no—
391C4—62no—
392C4—82no—
393C4—122no—
394C4#565611yescharacteristic
395C4#565612yes—
396C4#566811yescharacteristic
397C3#2371911yescharacteristic, complemented, Sylow
398C2—22no—
399C2—81no—
400C2—244no—
401C2#7745711yescharacteristic
402C2#8093012yes—
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2079C2
socleC22×C3C2×#121
prime radical#70006C2
hypercenter#2079D3
hypocenterC3#23719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2079

Complements

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

NH ≅ G/Ntype
1. C2×#4869C2semidirect
2. #78866C2semidirect
3. #78649C2semidirect
4. C3×#2079C2semidirect
5. #70573C2semidirect
6. #70307C2semidirect
8. #68996C2semidirect
9. #68972C2semidirect
10. #68932C2semidirect
11. #68917C2semidirect
48. #2079D3semidirect
397. C3#23719semidirect

Maximal subgroups

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

subgroupindex
C2×#48692
#788662
#786492
C3×#20792
#705732
#703072
#700062
#689962
#689722
#689322
#689172
#237193