Finite Group Databasebeta

← #10824898

Subgroups — #10824898

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×C3×#442C211yesmaximal, characteristic, complemented
2C4×#932C212yesmaximal, complemented
3#78489C211yesmaximal, characteristic, complemented
4#72046C211yesmaximal, characteristic
5#72042C211yesmaximal, characteristic, complemented
6C4×#833C212yesmaximal, complemented
7#70593C211yesmaximal, characteristic, complemented
8#70573C211yesmaximal, characteristic, complemented
9#70197C212yesmaximal, complemented
10#70196C212yesmaximal, complemented
11#69840C211yesmaximal, characteristic, complemented
12#23719—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×#932C2212yes—
16C4×#235C2211yescharacteristic
17C3×#442—22no—
18C3×#442—22no—
19C2×C3×#106C2211yescharacteristic
20C2×C3×#102C2212yes—
21C2×C3×#97C2211yescharacteristic
22C2×#833C2212yes—
23C2×#833C2212yes—
24#5215C2212yes—
25#5110C2211yescharacteristic
26#5110C2211yescharacteristic
27C2×#819C2212yes—
28#5102C2211yescharacteristic
29#5102C2211yescharacteristic
30#5018—22no—
31C4×#214—22no—
32#5014—22no—
33C4×#213—22no—
34#4966C2212yes—
35#4964C2212yes—
36C4×#208C2212yes—
37C4×#208C2211yescharacteristic
38#4709—22no—
39#4705—22no—
40#4552C2211yescharacteristic
41#4552C2211yescharacteristic
42#4530C2212yes—
43#4530C2212yes—
44C4×AGL(1,ℤ/8)—32no—
45C2×#442D311yescharacteristic, complemented
46#2923—31no—
47#2090—31no—
48#2079—31no—
49#2063—32no—
50#2051—32no—
51#2045—31no—
52#2018—31no—
53#1778—31no—
54C4×#100—32no—
55C22×C3×D4C2312yes—
56C23×C3×C4—22no—
57C2×#235C2311yescharacteristic
58C2×C3×D8—24no—
59C2×C3×D8—22no—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C42C2311yescharacteristic
69C2×#214—22no—
70#932—24no—
71#932—24no—
72C2×#213—22no—
73C4×#67—22no—
74#926C2311yescharacteristic
75C2×#208C2312yes—
76C2×#208C2311yescharacteristic
77C2×#208C2311yescharacteristic
78C2×#208C2312yes—
79C4×D12D412yes—
80#872—21no—
81C42×D3—21no—
82C4×Dic6D412yes—
83C3×#106—22no—
84C3×#102—24no—
85C3×#97D412yes—
86C3×#97D412yes—
87#833—24no—
88#833—24no—
89#832—22no—
90#832—22no—
91(C2×D3)⋊C8—22no—
92(C2×D3)⋊C8—22no—
93#819—24no—
94Dic3⋊C8—22no—
95Dic3⋊C8—22no—
96#810—22no—
97#809—22no—
98#808—22no—
99#807—22no—
100#803—21no—
101#803—22no—
102C4×C3⋊C8—21no—
103C4×C3⋊C8—22no—
104C2×AGL(1,ℤ/8)—32no—
105C22×D8C2×D311yescharacteristic
106C4×#41—31no—
107C2×C4×D4C2×D312yes—
108#442—22no—
109#442—22no—
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#417—32no—
123C2×#102C2×D312yes—
124C2×#100—32no—
125C2×#100—32no—
126#408—31no—
127#408—31no—
128C4×M16—31no—
129C4×M16—32no—
130C2×#97C2×D311yescharacteristic
131#343—32no—
132#343—32no—
133#341—31no—
134#341—31no—
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×C4—22no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4—84no—
146C22×C3×C4#4112yes—
147C22×C3×C4C2411yescharacteristic
148C2×#67—21no—
149#235—22no—
150#235—22no—
151C2×D12C2×D411yescharacteristic
152C2×C4×D3—21no—
153C2×C4×D3—22no—
154C2×Dic6C2×D411yescharacteristic
155C3×D8—28no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8C2×D412yes—
159C2×C3×C8C2×D412yes—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C22⋊C4—42no—
163C3×C22⋊C4—42no—
164C3×C42C2×D412yes—
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167#217—21no—
168#214—28no—
169#213—28no—
170(C2×D3)⋊C4—22no—
171C4⋊Dic3C2×D412yes—
172Dic3⋊C4—22no—
173C4×Dic3—22no—
174C4×Dic3—21no—
175#208—22no—
176#208—22no—
177#208—22no—
178#208—22no—
179#208—22no—
180#208#4114yes—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×#41—31no—
186C22×D4C22×D312yes—
187C23×C4—22no—
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—32no—
196C2×M16—31no—
197C2×M16—31no—
198C22×C8C22×D311yescharacteristic
199C22×C8C22×D311yescharacteristic
200C4×Q8—32no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—32no—
206C4×D4—62no—
207#117—31no—
208#117—61no—
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—22no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213#108—64no—
214#107—62no—
215#106—22no—
216#106—62no—
217C4⋊C8—62no—
218C4⋊C8—62no—
219#103—62no—
220#102—24no—
221#102—62no—
222#102—62no—
223#102—62no—
224#100—64no—
225#100—64no—
226C22⋊C8—62no—
227C22⋊C8—62no—
228#97—62no—
229#97—61no—
230#97#6712yes—
231#97#6712yes—
232C4×C8—62no—
233C4×C8—61no—
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—42no—
246C2×C3×C4—42no—
247C2×C3×C4—84no—
248C2×C3×C4—84no—
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—
256#67—42no—
257C2×Dic3—21no—
258C2×Dic3—22no—
259D12—22no—
260C4×D3—41no—
261C4×D3—41no—
262C4×D3—82no—
263Dic6—22no—
264C3×C8—22no—
265C3×C8—22no—
266C3⋊C8—24no—
267C3⋊C8—41no—
268C3⋊C8—42no—
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#23514yes—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—84no—
288C22×C4#23712yes—
289C22×C4C23×D311yescharacteristic
290SD16—68no—
291D8—28no—
292D8—68no—
293M16—34no—
294M16—62no—
295M16—62no—
296M16—62no—
297M16—62no—
298M16—62no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—62no—
302C2×C8—62no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8C2×#6712yes—
306C2×C8C2×#6712yes—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—32no—
310C4⋊C4—62no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—61no—
314C22⋊C4—62no—
315C42—62no—
316C42—61no—
317C42D3×D412yes—
318C42C2×#6711yescharacteristic
319C42C2×#6711yescharacteristic
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—22no—
331C3×C4—22no—
332C3×C4—82no—
333C3×C4#52511yescharacteristic
334C3×C4#52512yes—
335C3×C4#53211yescharacteristic
336Dic3—41no—
337Dic3—42no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—61no—
342C23—82no—
343C23—82no—
344C23C2×#23711yescharacteristic
345Q8—62no—
346D4—24no—
347D4—24no—
348D4—42no—
349D4—42no—
350D4—62no—
351D4—122no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—62no—
355C2×C4—61no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—242no—
362C2×C4#93812yes—
363C2×C4#93812yes—
364C2×C4C2×#23512yes—
365C2×C4C2×D3×D411yescharacteristic
366C2×C4D3×#4112yes—
367C2×C4C22×#6711yescharacteristic
368C8—22no—
369C8—22no—
370C8—64no—
371C8—121no—
372C8—122no—
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—121no—
386C22—122no—
387C22#553812yes—
388C22C2×#93811yescharacteristic
389C4—22no—
390C4—22no—
391C4—82no—
392C4—121no—
393C4—122no—
394C4#554712yes—
395C4#579111yescharacteristic
396C4#583211yescharacteristic
397C3#2371911yescharacteristic, complemented, Sylow
398C2—22no—
399C2—84no—
400C2—241no—
401C2#8240111yescharacteristic
402C2#8396312yes—
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC22×C3C2×#121
prime radical#72046C2
hypercenterC2×#442D3
hypocenterC3#23719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×C3×#442C2semidirect
2. C4×#932C2semidirect
3. #78489C2semidirect
5. #72042C2semidirect
6. C4×#833C2semidirect
7. #70593C2semidirect
8. #70573C2semidirect
9. #70197C2semidirect
10. #70196C2semidirect
11. #69840C2semidirect
45. C2×#442D3semidirect
397. C3#23719semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4422
C4×#9322
#784892
#720462
#720422
C4×#8332
#705932
#705732
#701972
#701962
#698402
#237193