Finite Group Databasebeta

← #10824909

Subgroups — #10824909

Order 768.

744 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#441C211yesmaximal, characteristic, complemented
2C4×#942C212yesmaximal
3#78489C211yesmaximal, characteristic
4#72074C211yesmaximal, characteristic
5#72070C211yesmaximal, characteristic, complemented
6C4×#836C212yesmaximal, complemented
7#70593C211yesmaximal, characteristic
8#70573C211yesmaximal, characteristic, complemented
9#70265C212yesmaximal, complemented
10#70264C212yesmaximal
11#69840C211yesmaximal, characteristic, complemented
12#23720—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2212yes—
15C2×#942C2212yes—
16C4×#235C2211yescharacteristic
17C3×#441—22no—
18C3×#441—22no—
19C2×C3×#106C2211yescharacteristic
20C2×C3×#103C2212yes—
21C2×C3×#97C2211yescharacteristic
22C2×#836C2212yes—
23C2×#836C2212yes—
24#5225C2212yes—
25#5110C2211yescharacteristic
26#5110C2211yescharacteristic
27C2×#819C2212yes—
28#5102C2211yescharacteristic
29#5102C2211yescharacteristic
30#5030—22no—
31C4×#216—22no—
32#5026—22no—
33C4×#215—22no—
34#4966C2212yes—
35#4964C2212yes—
36C4×#208C2211yescharacteristic
37C4×#208C2212yes—
38#4709—22no—
39#4705—22no—
40#4552C2211yescharacteristic
41#4552C2211yescharacteristic
42#4530C2212yes—
43#4530C2212yes—
44C4×#137—32no—
45C2×#441D311yescharacteristic, complemented
46#2923—31no—
47#2090—31no—
48#2079—31no—
49#2064—32no—
50#2052—32no—
51#2047—31no—
52#2020—31no—
53#1778—31no—
54C4×#101—32no—
55C22×C3×Q8C2312yes—
56C2×#235C2311yescharacteristic
57C2×C3×Q16—24no—
58C2×C3×Q16—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42—22no—
68C2×C3×C42C2311yescharacteristic
69C2×#216—22no—
70#942—24no—
71#942—24no—
72C2×#215—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×#103—24no—
85C3×#97D412yes—
86C3×#97D412yes—
87#836—24no—
88#836—24no—
89#835—22no—
90#835—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×#137—32no—
105C22×Q16C2×D311yescharacteristic
106C4×#41—31no—
107C2×C4×Q8C2×D312yes—
108#441—22no—
109#441—22no—
110#441—62no—
111#440—62no—
112C4×Q16—62no—
113C4×SD16—62no—
114#433—62no—
115#432—62no—
116C2×#108—32no—
117#428—32no—
118C2×#106C2×D311yescharacteristic
119#424—31no—
120#424—31no—
121#419—32no—
122#417—32no—
123C2×#103C2×D312yes—
124C2×#101—32no—
125C2×#101—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—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8#4114yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4#4112yes—
144C22×C3×C4C2411yescharacteristic
145C2×#67—21no—
146#235—22no—
147#235—22no—
148C2×D12C2×D411yescharacteristic
149C2×C4×D3—21no—
150C2×C4×D3—22no—
151C2×Dic6C2×D411yescharacteristic
152C3×Q16—28no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8C2×D412yes—
156C2×C3×C8C2×D412yes—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—42no—
161C3×C42—42no—
162C3×C42—42no—
163C3×C42C2×D412yes—
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166#217—21no—
167#216—28no—
168#215—28no—
169(C2×D3)⋊C4—22no—
170C4⋊Dic3C2×D412yes—
171Dic3⋊C4—22no—
172C4×Dic3—22no—
173C4×Dic3—21no—
174#208—22no—
175#208—22no—
176#208—22no—
177#208—22no—
178#208—22no—
179#208#4114yes—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×#41—31no—
185C22×Q8C22×D312yes—
186#137—64no—
187#137—64no—
188C2×Q16—24no—
189C2×Q16—22no—
190C2×Q16—62no—
191C2×SD16—62no—
192C2×M16—32no—
193C2×M16—32no—
194C2×M16—31no—
195C2×M16—31no—
196C22×C8C22×D311yescharacteristic
197C22×C8C22×D311yescharacteristic
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—32no—
203C4×D4—32no—
204C4×D4—62no—
205#117—31no—
206#117—61no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C22×D312yes—
209C2×C42—22no—
210C2×C42—61no—
211C2×C42C22×D311yescharacteristic
212#108—64no—
213#107—62no—
214#106—22no—
215#106—62no—
216C4⋊C8—62no—
217C4⋊C8—62no—
218#103—24no—
219#103—62no—
220#103—62no—
221#103—62no—
222#102—62no—
223#101—64no—
224#101—64no—
225C22⋊C8—62no—
226C22⋊C8—62no—
227#97—62no—
228#97—61no—
229#97#6712yes—
230#97#6712yes—
231C4×C8—62no—
232C4×C8—61no—
233C23×C3C2×#4111yescharacteristic
234C22×D3—21no—
235C3×Q8—24no—
236C3×Q8—24no—
237C3×Q8—42no—
238C3×Q8—42no—
239C2×C3×C4—24no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—82no—
245C2×C3×C4#12112yes—
246C2×C3×C4#12112yes—
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C4C22×D411yescharacteristic
249C2×C3×C4C2×#4112yes—
250C2×C3×C4C2×#4112yes—
251#67—42no—
252C2×Dic3—21no—
253C2×Dic3—22no—
254D12—22no—
255C4×D3—41no—
256C4×D3—41no—
257C4×D3—82no—
258Dic6—22no—
259C3×C8—22no—
260C3×C8—22no—
261C3⋊C8—24no—
262C3⋊C8—41no—
263C3⋊C8—42no—
264C3⋊C8—41no—
265#41—62no—
266#41—62no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—31no—
273C2×Q8#23514yes—
274C2×D4—31no—
275C2×D4—61no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—61no—
279C22×C4—62no—
280C22×C4#23712yes—
281C22×C4C23×D311yescharacteristic
282Q16—28no—
283Q16—68no—
284SD16—68no—
285M16—34no—
286M16—62no—
287M16—62no—
288M16—62no—
289M16—62no—
290M16—62no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8C2×#6712yes—
298C2×C8C2×#6712yes—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—32no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—62no—
305C22⋊C4—61no—
306C22⋊C4—62no—
307C42—42no—
308C42—42no—
309C42—62no—
310C42—61no—
311C42D3×D412yes—
312C42C2×#6711yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—22no—
315C22×C3C2×#12111yescharacteristic
316C22×C3#52912yes—
317C2×D3—41no—
318C2×D3—42no—
319C3×C4—22no—
320C3×C4—22no—
321C3×C4—44no—
322C3×C4—82no—
323C3×C4#52511yescharacteristic
324C3×C4#52512yes—
325C3×C4#53211yescharacteristic
326Dic3—41no—
327Dic3—42no—
328C23—61no—
329C23C2×#23711yescharacteristic
330Q8—24no—
331Q8—24no—
332Q8—42no—
333Q8—42no—
334Q8—62no—
335D4—62no—
336D4—122no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—62no—
343C2×C4—61no—
344C2×C4—82no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—242no—
348C2×C4#93812yes—
349C2×C4#93812yes—
350C2×C4C2×#23512yes—
351C2×C4C2×D3×D411yescharacteristic
352C2×C4D3×#4112yes—
353C2×C4C22×#6711yescharacteristic
354C8—22no—
355C8—22no—
356C8—64no—
357C8—121no—
358C8—122no—
359C8—121no—
360C2×C3—22no—
361C2×C3#239211yescharacteristic
362C2×C3#306912yes—
363D3—81no—
364C22—22no—
365C22—122no—
366C22—121no—
367C22#553812yes—
368C22C2×#93811yescharacteristic
369C4—22no—
370C4—22no—
371C4—44no—
372C4—82no—
373C4—122no—
374C4—121no—
375C4#554712yes—
376C4#579111yescharacteristic
377C4#583211yescharacteristic
378C3#2372011yescharacteristic, complemented, Sylow
379C2—22no—
380C2—241no—
381C2#8240111yescharacteristic
382C2#8399812yes—
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#441C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenterC2×#441D3
hypocenterC3#23720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
1. C2×C3×#441C2semidirect
5. #72070C2semidirect
6. C4×#836C2semidirect
8. #70573C2semidirect
9. #70265C2semidirect
11. #69840C2semidirect
45. C2×#441D3semidirect
378. C3#23720semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4412
C4×#9422
#784892
#720742
#720702
C4×#8362
#705932
#705732
#702652
#702642
#698402
#237203