Finite Group Databasebeta

← #10824900

Subgroups — #10824900

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
1C4×#908C212yesmaximal, complemented
2C2×#4711C211yesmaximal, characteristic, complemented
3#78489C211yesmaximal, characteristic, complemented
4C3×#1778C211yesmaximal, characteristic, complemented
5#70715C212yesmaximal, complemented
6#70714C212yesmaximal, complemented
7C4×#823C212yesmaximal, complemented
8#68942C211yesmaximal, characteristic
9#68932C211yesmaximal, characteristic, complemented
10#68924C211yesmaximal, characteristic, complemented
11#68922C211yesmaximal, characteristic, complemented
12#23719—31nomaximal, Sylow
13C2×#908C2212yes—
14C22×D24C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C2×C4×D12C2212yes—
17C2×C3×#108C2212yes—
18C3×C4×M16C2212yes—
19C2×C3×#97C2211yescharacteristic
20#5133C2212yes—
21#5131C2212yes—
22C2×#823C2212yes—
23C2×#823C2212yes—
24#5121C2212yes—
25C2×#821C2212yes—
26#5110C2211yescharacteristic
27#5110C2211yescharacteristic
28C2×#817C2211yescharacteristic
29#5102C2211yescharacteristic
30#5102C2211yescharacteristic
31C4×#208C2211yescharacteristic
32#4711—22no—
33#4711—22no—
34#4711—22no—
35#4710—22no—
36#4709—22no—
37#4705—22no—
38C4×D24—22no—
39C4×#204—22no—
40C3×#341C2211yescharacteristic
41C3×#341C2211yescharacteristic
42#4554C2212yes—
43#4554C2212yes—
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#2045—31no—
52#2018—31no—
53#1778D311yescharacteristic, complemented
54C4×#100—32no—
55C2×#235C2311yescharacteristic
56C22×D12C2312yes—
57C22×C4×D3—22no—
58C2×C3×M16C2312yes—
59C2×C3×M16C2312yes—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C4×#67—22no—
64C2×(C2×D3)⋊C4—22no—
65#926C2311yescharacteristic
66C2×C4⋊Dic3C2312yes—
67C2×#208C2311yescharacteristic
68C2×#208C2311yescharacteristic
69#908—24no—
70#908—24no—
71C2×D24—22no—
72C2×D24—24no—
73C2×D24—22no—
74C2×#204—22no—
75C4×D12—22no—
76C4×D12—22no—
77C4×D12—22no—
78C4×D12—22no—
79C4×D12D412yes—
80#872—21no—
81C42×D3—21no—
82C4×Dic6D412yes—
83C3×#108—24no—
84C3×#97—22no—
85C3×#97D412yes—
86C3×#97D412yes—
87C3×C4×C8—22no—
88#823—24no—
89#823—24no—
90#821—22no—
91#821—24no—
92#821—22no—
93#821—22no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—22no—
96#818—22no—
97#817—22no—
98#817—22no—
99#816—22no—
100Dic3⋊C8—22no—
101Dic3⋊C8—22no—
102#803—21no—
103C4×C3⋊C8—21no—
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×#108C2×D312yes—
117#428—32no—
118C2×#106—31no—
119#424—31no—
120#424—31no—
121#418—32no—
122#417—32no—
123C2×#102—32no—
124C2×#100—32no—
125C2×#100—32no—
126#408—31no—
127#408—31no—
128C4×M16—31no—
129C4×M16C2×D312yes—
130C2×#97C2×D311yescharacteristic
131#343—32no—
132#343—32no—
133#341C2×D311yescharacteristic
134#341C2×D311yescharacteristic
135C23×D3—22no—
136C22×C3×C4#4112yes—
137C22×C3×C4C2411yescharacteristic
138C2×#67—21no—
139C22×Dic3—22no—
140#235—22no—
141#235—22no—
142C2×D12—22no—
143C2×D12—22no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—22no—
147C2×D12C2×D411yescharacteristic
148C2×D12#4114yes—
149C2×C4×D3—21no—
150C2×C4×D3—22no—
151C2×C4×D3—42no—
152C2×C4×D3—42no—
153C2×C4×D3—84no—
154C2×Dic6C2×D411yescharacteristic
155C3×M16—22no—
156C3×M16—22no—
157C3×M16—22no—
158C3×M16#4114yes—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8C2×D412yes—
164C2×C3×C8C2×D412yes—
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D412yes—
168#217—21no—
169(C2×D3)⋊C4—22no—
170(C2×D3)⋊C4—42no—
171(C2×D3)⋊C4—42no—
172C4⋊Dic3—22no—
173C4⋊Dic3—22no—
174C4⋊Dic3C2×D412yes—
175Dic3⋊C4—22no—
176C4×Dic3—22no—
177C4×Dic3—21no—
178#208—22no—
179#208—22no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182D24—28no—
183D24—28no—
184#204—28no—
185C2×#41—31no—
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—31no—
195C2×M16—31no—
196C2×M16C22×D312yes—
197C2×M16C22×D312yes—
198C22×C8C22×D311yescharacteristic
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⋊C4—32no—
210C2×C22⋊C4—62no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213#108—24no—
214#107—62no—
215#106—62no—
216#106—62no—
217C4⋊C8—62no—
218C4⋊C8—62no—
219#103—62no—
220#102—62no—
221#102—62no—
222#102—62no—
223#102—64no—
224#100—64no—
225#100—64no—
226C22⋊C8—62no—
227C22⋊C8—62no—
228#97—22no—
229#97—61no—
230#97D1212yes—
231#97D1212yes—
232C4×C8—22no—
233C4×C8—61no—
234C23×C3C2×#4111yescharacteristic
235C22×D3—21no—
236C22×D3—24no—
237C22×D3—42no—
238C22×D3—42no—
239C22×D3—82no—
240C22×D3—82no—
241C2×C3×C4—42no—
242C2×C3×C4#12112yes—
243C2×C3×C4#12112yes—
244C2×C3×C4C22×D411yescharacteristic
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C2×#4112yes—
247C2×C3×C4C2×#4112yes—
248#67—42no—
249C2×Dic3—21no—
250C2×Dic3—22no—
251C2×Dic3—42no—
252C2×Dic3—82no—
253D12—24no—
254D12—24no—
255D12—22no—
256D12—42no—
257D12—42no—
258C4×D3—41no—
259C4×D3—41no—
260C4×D3—82no—
261C4×D3—84no—
262C4×D3—84no—
263Dic6—22no—
264C3×C8—24no—
265C3×C8—22no—
266C3×C8—22no—
267C3×C8—42no—
268C3⋊C8—41no—
269C3⋊C8—41no—
270C24—62no—
271#41—62no—
272#41—62no—
273C2×Q8—31no—
274C2×D4—31no—
275C2×D4—34no—
276C2×D4—61no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—62no—
282C22×C4—61no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4—244no—
288C22×C4#23912yes—
289C22×C4C23×D311yescharacteristic
290SD16—68no—
291D8—68no—
292D8—68no—
293M16—22no—
294M16—22no—
295M16—22no—
296M16—62no—
297M16—62no—
298M16#23514yes—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8C2×D1212yes—
306C2×C8C2×D1212yes—
307C4⋊C4—32no—
308C4⋊C4—62no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C22⋊C4—61no—
312C22⋊C4—62no—
313C22⋊C4—122no—
314C22⋊C4—122no—
315C42—62no—
316C42—61no—
317C42C2×D1211yescharacteristic
318C42C2×D1211yescharacteristic
319C42D3×D412yes—
320C22×C3—22no—
321C22×C3C2×#12111yescharacteristic
322C22×C3#52912yes—
323C2×D3—41no—
324C2×D3—44no—
325C2×D3—42no—
326C2×D3—84no—
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—84no—
330C3×C4—22no—
331C3×C4—22no—
332C3×C4#52511yescharacteristic
333C3×C4#52512yes—
334C3×C4#53211yescharacteristic
335Dic3—41no—
336Dic3—42no—
337Dic3—82no—
338C23—61no—
339C23—64no—
340C23—122no—
341C23—122no—
342C23—242no—
343C23—242no—
344C23C2×#23911yescharacteristic
345Q8—62no—
346D4—64no—
347D4—62no—
348D4—64no—
349D4—122no—
350D4—122no—
351D4—122no—
352C2×C4—42no—
353C2×C4—62no—
354C2×C4—61no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—122no—
358C2×C4—242no—
359C2×C4—244no—
360C2×C4—244no—
361C2×C4—242no—
362C2×C4#89512yes—
363C2×C4#89512yes—
364C2×C4C22×D1211yescharacteristic
365C2×C4C2×#23512yes—
366C2×C4C2×D3×D411yescharacteristic
367C2×C4D3×#4112yes—
368C8—22no—
369C8—22no—
370C8—24no—
371C8—42no—
372C8—121no—
373C8—121no—
374C2×C3—22no—
375C2×C3#239211yescharacteristic
376C2×C3#306812yes—
377D3—81no—
378D3—84no—
379C22—22no—
380C22—121no—
381C22—124no—
382C22—122no—
383C22—244no—
384C22—244no—
385C22—242no—
386C22—242no—
387C22C2×#89511yescharacteristic
388C22#558012yes—
389C4—22no—
390C4—22no—
391C4—121no—
392C4—122no—
393C4—242no—
394C4#547711yescharacteristic
395C4#552411yescharacteristic
396C4#554712yes—
397C3#2371911yescharacteristic, complemented, Sylow
398C2—22no—
399C2—241no—
400C2—244no—
401C2#7377611yescharacteristic
402C2#7926112yes—
subgroupG/H
centerC22C2×#895
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1778C2
socleC22×C3C2×#121
prime radical#68942C2
hypercenter#1778D3
hypocenterC3#23719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1778

Complements

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

NH ≅ G/Ntype
1. C4×#908C2semidirect
2. C2×#4711C2semidirect
3. #78489C2semidirect
4. C3×#1778C2semidirect
5. #70715C2semidirect
6. #70714C2semidirect
7. C4×#823C2semidirect
9. #68932C2semidirect
10. #68924C2semidirect
11. #68922C2semidirect
53. #1778D3semidirect
397. C3#23719semidirect

Maximal subgroups

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

subgroupindex
C4×#9082
C2×#47112
#784892
C3×#17782
#707152
#707142
C4×#8232
#689422
#689322
#689242
#689222
#237193