Finite Group Databasebeta

← #10896435

Subgroups — #10896435

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73390C211yesmaximal, characteristic, complemented
2#73387C211yesmaximal, characteristic
3C3×#1832C211yesmaximal, characteristic, complemented
4#70458C212yesmaximal
5#70454C212yesmaximal, complemented
6#68692C211yesmaximal, characteristic, complemented
7#68690C211yesmaximal, characteristic, complemented
8#68684C212yesmaximal, complemented
9#68682C211yesmaximal, characteristic
10#68679C211yesmaximal, characteristic
11#68671C212yesmaximal
12#29269—31nomaximal, Sylow
13C2×#875C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15C2×#870C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C2×C3×C4⋊C8C2212yes—
18C3×#376C2211yescharacteristic
19C2×#821C2212yes—
20C2×#821C2212yes—
21C2×#817C2212yes—
22C2×#817C2212yes—
23C2×#816C2212yes—
24C2×#816C2212yes—
25#4939C2211yescharacteristic
26#4935C2211yescharacteristic
27#4935C2211yescharacteristic
28C4×C4⋊Dic3C2212yes—
29#4929C2211yescharacteristic
30#4928C2212yes—
31#4670—22no—
32#4668—22no—
33#4655—22no—
34#4647—22no—
35C3×#336C2212yes—
36C3×#336C2211yescharacteristic
37C3×#336C2211yescharacteristic
38#4551C2212yes—
39#4551C2212yes—
40#2330—31no—
41#2326—31no—
42#2096—31no—
43#2094—31no—
44#2026—31no—
45#2025—31no—
46#1983—32no—
47#1948—32no—
48#1865—32no—
49#1864—32no—
50#1832D311yescharacteristic, complemented
51C22×D12C2311yescharacteristic
52C22×C4×D3—21no—
53C22×Dic6C2311yescharacteristic
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C2×C3×C42C2311yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3—22no—
64C2×C4⋊Dic3C2312yes—
65C2×C4⋊Dic3C2311yescharacteristic
66C2×C4⋊Dic3C2311yescharacteristic
67C2×C4⋊Dic3C2312yes—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—22no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—22no—
73#875—22no—
74#875Q814yes—
75C4×D12—42no—
76#870—24no—
77C4×Dic6—42no—
78C3×C4⋊C8—24no—
79C3×#95D412yes—
80C3×#95D412yes—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84#821—24no—
85#821—42no—
86#817—42no—
87#817—42no—
88#816—24no—
89#816—42no—
90C2×#125—31no—
91C2×#124—31no—
92C2×C4×Q8—31no—
93C2×C4×D4—31no—
94C2×#106—32no—
95C2×#106—32no—
96C2×C4⋊C8C2×D312yes—
97C2×#103—32no—
98C2×#103—32no—
99C2×#102—32no—
100C2×#102—32no—
101#391—31no—
102#390—31no—
103#389—32no—
104#389—62no—
105#388—62no—
106#387—62no—
107#385—62no—
108#384—31no—
109#384—31no—
110C4×C4⋊C4—32no—
111#376C2×D311yescharacteristic
112#340—32no—
113#340—32no—
114#336C2×D311yescharacteristic
115#336C2×D311yescharacteristic
116#336C2×D312yes—
117C23×D3—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2411yescharacteristic
123C22×Dic3—22no—
124C22×Dic3—21no—
125C22×Dic3—22no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—21no—
131C2×D12C2×Q812yes—
132C2×C4×D3—42no—
133C2×C4×D3—82no—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—21no—
137C2×Dic6C2×Q812yes—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C3×C42—22no—
145C3×C42—22no—
146C3×C42C2×Q812yes—
147C3×C42#4112yes—
148(C2×D3)⋊C4—28no—
149(C2×D3)⋊C4—42no—
150(C2×D3)⋊C4—82no—
151C4⋊Dic3—22no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—82no—
157C4⋊Dic3—82no—
158C4⋊Dic3#4114yes—
159Dic3⋊C4—28no—
160Dic3⋊C4—42no—
161Dic3⋊C4—82no—
162C4×Dic3—42no—
163C4×Dic3—82no—
164C22×Q8—31no—
165C22×D4—31no—
166C23×C4—61no—
167C22×C8C22×D312yes—
168C22×C8C22×D312yes—
169#125—64no—
170#124—34no—
171#124—62no—
172C4×Q8—122no—
173C4×D4—122no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C42—61no—
187C2×C42—62no—
188C2×C42C22×D311yescharacteristic
189C2×C42C22×D311yescharacteristic
190C2×C42C22×D311yescharacteristic
191#106—122no—
192#106—122no—
193C4⋊C8—24no—
194#103—64no—
195#103—122no—
196#102—64no—
197#102—122no—
198#95—62no—
199#95—62no—
200#95—62no—
201#95D1212yes—
202#95D1212yes—
203C23×C3C22×D411yescharacteristic
204C22×D3—22no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—42no—
208C22×D3—42no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4C22⋊Q814yes—
219C2×C3×C4C22×Q811yescharacteristic
220C2×C3×C4C2×#4111yescharacteristic
221C2×C3×C4C2×#4112yes—
222C2×C3×C4H(5,𝔽2)12yes—
223C2×Dic3—22no—
224C2×Dic3—24no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236D12—24no—
237D12—42no—
238C4×D3—84no—
239Dic6—24no—
240Dic6—42no—
241C3×C8—42no—
242C3×C8—42no—
243C24—61no—
244C2×Q8—32no—
245C2×Q8—61no—
246C2×Q8—62no—
247C2×Q8—62no—
248C2×D4—32no—
249C2×D4—61no—
250C2×D4—62no—
251C2×D4—62no—
252C22×C4—61no—
253C22×C4—62no—
254C22×C4—62no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—122no—
258C22×C4—242no—
259C22×C4C2×D1211yescharacteristic
260C22×C4C2×D1212yes—
261C22×C4C2×D1211yescharacteristic
262C22×C4C2×D1212yes—
263C22×C4C23×D311yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C4⋊C4—34no—
271C4⋊C4—68no—
272C4⋊C4—62no—
273C4⋊C4—62no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—122no—
278C4⋊C4—242no—
279C4⋊C4—242no—
280C4⋊C4—242no—
281C22⋊C4—68no—
282C22⋊C4—122no—
283C22⋊C4—242no—
284C42—22no—
285C42—22no—
286C42—122no—
287C42—242no—
288C42#23712yes—
289C42D3×Q812yes—
290C22×C3C2×C22⋊Q811yescharacteristic
291C22×C3#52511yescharacteristic
292C22×C3#53511yescharacteristic
293C22×C3#57712yes—
294C22×C3#57712yes—
295C2×D3—44no—
296C2×D3—42no—
297C2×D3—82no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—24no—
301C3×C4—42no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4#55412yes—
305C3×C4#55912yes—
306Dic3—42no—
307Dic3—44no—
308Dic3—82no—
309Dic3—81no—
310Dic3—81no—
311C23—62no—
312C23—122no—
313C23—122no—
314C23—121no—
315C23—121no—
316C23C22×D1211yescharacteristic
317Q8—64no—
318Q8—122no—
319D4—64no—
320D4—122no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—62no—
331C2×C4—64no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—122no—
343C2×C4—244no—
344C2×C4#89714yes—
345C2×C4C2×#23711yescharacteristic
346C2×C4C2×D3×Q811yescharacteristic
347C2×C4D3×#4112yes—
348C2×C4#100912yes—
349C8—42no—
350C8—42no—
351C2×C3#258011yescharacteristic
352C2×C3#308412yes—
353C2×C3#312512yes—
354C2×C3#322111yescharacteristic
355C2×C3#322211yescharacteristic
356D3—82no—
357C22—122no—
358C22—124no—
359C22—242no—
360C22—241no—
361C22—241no—
362C22C2×#89711yescharacteristic
363C22#552411yescharacteristic
364C22#552511yescharacteristic
365C22#575112yes—
366C22#575112yes—
367C4—24no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—124no—
372C4—122no—
373C4—241no—
374C4—241no—
375C4—242no—
376C4#568112yes—
377C4#569012yes—
378C3#2926911yescharacteristic, complemented, Sylow
379C2—242no—
380C2#7419011yescharacteristic
381C2#7909212yes—
382C2#7938912yes—
383C2#7939111yescharacteristic
384C2#7939711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1832C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1832D3
hypocenterC3#29269
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #1832

Complements

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

NH ≅ G/Ntype
1. #73390C2semidirect
3. C3×#1832C2semidirect
5. #70454C2semidirect
6. #68692C2semidirect
7. #68690C2semidirect
8. #68684C2semidirect
50. #1832D3semidirect
378. C3#29269semidirect

Maximal subgroups

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

subgroupindex
#733902
#733872
C3×#18322
#704582
#704542
#686922
#686902
#686842
#686822
#686792
#686712
#292693