Finite Group Databasebeta

← #10877354

Subgroups — #10877354

Order 768.

640 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83810C211yesmaximal, characteristic, complemented
2#82916C211yesmaximal, characteristic
3C3×#1891C211yesmaximal, characteristic
4#72027C211yesmaximal, characteristic, complemented
5#71714C211yesmaximal, characteristic
6#70135C211yesmaximal, characteristic, complemented
7#70134C211yesmaximal, characteristic, complemented
8#69987C212yesmaximal, complemented
9#69982C212yesmaximal, complemented
10#69016C212yesmaximal, complemented
11#69015C212yesmaximal, complemented
12#27691—31nomaximal, Sylow
13C3×#568C2211yescharacteristic
14C3×#525C2211yescharacteristic
15#5819C2211yescharacteristic
16C2×#948C2211yescharacteristic
17C23×C3⋊C8C2211yescharacteristic
18C3×#413C2211yescharacteristic
19C3×#407C2212yes—
20#5237C2211yescharacteristic
21#5234C2211yescharacteristic
22#5207C2211yescharacteristic
23#5207C2212yes—
24#5125C2212yes—
25#5124—22no—
26#5124C2212yes—
27#5124C2212yes—
28C2×#822—22no—
29C2×#822C2212yes—
30C2×#822C2212yes—
31#5011—22no—
32D4×C3⋊C8—22no—
33#4999C2211yescharacteristic
34#4997C2211yescharacteristic
35C3×#344C2212yes—
36C3⋊#338C2212yes—
37#4558C2212yes—
38#4558C2212yes—
39#4533C2212yes—
40#2929—31no—
41#2916—31no—
42#2103—32no—
43#2091—32no—
44#1953—32no—
45#1891—31no—
46#1891D311yescharacteristic
47#1858—31no—
48#1858—31no—
49#1813—31no—
50#1810—32no—
51C2×C3×#41C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C3×#131—42no—
54C2×C3×M16—22no—
55C2×C3×M16C2312yes—
56C2×C3×M16C2312yes—
57C3×#123Q812yes—
58C3×C22⋊Q8—21no—
59C3×#121—21no—
60C3×#120Q812yes—
61C3×C4×D4—22no—
62C3×#117C2311yescharacteristic
63#948—24no—
64#948—42no—
65C2×#208—22no—
66C2×#208D412yes—
67C2×#208C2312yes—
68C2×#208C2312yes—
69C2×#208C2311yescharacteristic
70C2×#208C2311yescharacteristic
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C22×C3⋊C8—24no—
74C22×C3⋊C8D412yes—
75C22×C3⋊C8C2311yescharacteristic
76C22×C3⋊C8C2311yescharacteristic
77C3×#101—21no—
78C3×#100—21no—
79C3×C22⋊C8—42no—
80#836—21no—
81#833—21no—
82#830—22no—
83#830—22no—
84#830—42no—
85#822—22no—
86#822—24no—
87#822—22no—
88#822—24no—
89#822—42no—
90#819—42no—
91C4⋊(C3⋊C8)—22no—
92#803—21no—
93C4×C3⋊C8—21no—
94#568—31no—
95#568C2×D311yescharacteristic
96C2×#131—31no—
97C23×C8—31no—
98#525C2×D311yescharacteristic
99#435—62no—
100C8×D4—62no—
101#430—32no—
102#430—32no—
103#430—32no—
104#430—62no—
105C2×#108—32no—
106C2×#108—32no—
107C2×#108—62no—
108#424—31no—
109#413—31no—
110#413C2×D311yescharacteristic
111#408—31no—
112#407—31no—
113#407—32no—
114#407C2×D312yes—
115#405—31no—
116#344—32no—
117#344C2×D312yes—
118#341—32no—
119#341—32no—
120#338—32no—
121C24×C3C2×Q811yescharacteristic
122C3×#41—22no—
123C3×#41—22no—
124C2×C3×Q8#4111yescharacteristic
125C2×C3×D4—22no—
126C2×C3×D4C2×Q812yes—
127C2×C3×D4#4111yescharacteristic
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×Q811yescharacteristic
134C22×C3×C4C2411yescharacteristic
135C3×M16—22no—
136C3×M16—22no—
137C3×M16—22no—
138C3×M16—22no—
139C3×M16—42no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—42no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—21no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4C2×Q812yes—
148C3×C42—21no—
149#208—22no—
150#208—24no—
151#208—22no—
152#208—22no—
153#208—22no—
154#208—22no—
155#208—22no—
156#208—42no—
157C2×C3⋊C8—24no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—42no—
164C2×C3⋊C8—42no—
165C2×C3⋊C8—42no—
166C2×C3⋊C8C2×D412yes—
167C2×C3⋊C8C2×D411yescharacteristic
168C2×C3⋊C8C2×D411yescharacteristic
169C2×C3⋊C8#4114yes—
170C2×#41C22×D311yescharacteristic
171C23×C4C22×D311yescharacteristic
172#131—42no—
173#131—64no—
174#131—122no—
175C2×M16—22no—
176C2×M16—32no—
177C2×M16—31no—
178C2×M16—32no—
179C2×M16—32no—
180C2×M16—31no—
181C2×M16—62no—
182C2×M16C22×D312yes—
183C2×M16C22×D312yes—
184C22×C8—31no—
185C22×C8—32no—
186C22×C8—31no—
187C22×C8—64no—
188C22×C8—61no—
189C22×C8—61no—
190#123Dic612yes—
191C22⋊Q8—21no—
192#121—21no—
193#120Dic612yes—
194C4×D4—22no—
195#117C22×D311yescharacteristic
196#108—64no—
197#108—62no—
198#108—62no—
199#108—64no—
200#108—122no—
201#108—122no—
202C4⋊C8—62no—
203#101—21no—
204#101—61no—
205#100—21no—
206#100—61no—
207C22⋊C8—42no—
208C22⋊C8—62no—
209C22⋊C8—62no—
210C22⋊C8—122no—
211#97—61no—
212C4×C8—61no—
213C23×C3—21no—
214C23×C3—22no—
215C23×C3—42no—
216C23×C3C22⋊Q812yes—
217C23×C3C22×Q811yescharacteristic
218C3×Q8—22no—
219C3×D4—24no—
220C3×D4—22no—
221C3×D4—44no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4C22⋊Q812yes—
231C2×C3×C4C22×D411yescharacteristic
232C2×C3×C4C2×#4111yescharacteristic
233C2×C3×C4C2×#4112yes—
234C2×C3×C4H(5,𝔽2)12yes—
235C3×C8—42no—
236C3×C8—42no—
237C3⋊C8—22no—
238C3⋊C8—41no—
239C3⋊C8—42no—
240C3⋊C8—41no—
241C3⋊C8—42no—
242C3⋊C8#12312yes—
243C3⋊C8#12312yes—
244C24C2×Dic611yescharacteristic
245#41—22no—
246#41—22no—
247C2×Q8#23711yescharacteristic
248C2×D4—22no—
249C2×D4C2×Dic612yes—
250C2×D4#23711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4C2×Dic611yescharacteristic
256C22×C4D3×D412yes—
257C22×C4C23×D311yescharacteristic
258M16—22no—
259M16—22no—
260M16—22no—
261M16—22no—
262M16—42no—
263M16—64no—
264M16—62no—
265M16—62no—
266M16—62no—
267M16—62no—
268M16—62no—
269M16—62no—
270M16—122no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—31no—
274C2×C8—31no—
275C2×C8—34no—
276C2×C8—32no—
277C2×C8—42no—
278C2×C8—64no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—62no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—122no—
285C2×C8—122no—
286C2×C8—122no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4C2×Dic612yes—
292C42—21no—
293C22×C3—22no—
294C22×C3—24no—
295C22×C3—22no—
296C22×C3—41no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—41no—
301C22×C3C2×C22⋊Q811yescharacteristic
302C22×C3#55412yes—
303C3×C4—22no—
304C3×C4—42no—
305C3×C4—41no—
306C3×C4C2×#12311yescharacteristic
307C3×C4#53511yescharacteristic
308C3×C4#55312yes—
309C23—21no—
310C23—22no—
311C23—42no—
312C23#87812yes—
313C23C22×Dic611yescharacteristic
314Q8—22no—
315D4—24no—
316D4—22no—
317D4—44no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4#87812yes—
327C2×C4C2×D3×D411yescharacteristic
328C2×C4C2×#23711yescharacteristic
329C2×C4#100412yes—
330C2×C4D3×#4112yes—
331C8—32no—
332C8—32no—
333C8—42no—
334C8—42no—
335C8—62no—
336C8—121no—
337C8—122no—
338C8—121no—
339C8—122no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—42no—
343C2×C3—42no—
344C2×C3#248711yescharacteristic
345C22—22no—
346C22—24no—
347C22—22no—
348C22—41no—
349C22—42no—
350C22—41no—
351C22—42no—
352C22—41no—
353C22C2×#87811yescharacteristic
354C22#553912yes—
355C4—22no—
356C4—42no—
357C4—41no—
358C4#559211yescharacteristic
359C4#560112yes—
360C4D3×#12311yescharacteristic
361C3#2769111yescharacteristic, complemented, Sylow
362C2—21no—
363C2—22no—
364C2—42no—
365C2—42no—
366C2#7473811yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1891C2
socleC2×C3#2487
prime radicalC3×#525C22
hypercenter#1891D3
hypocenterC3#27691
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1891

Complements

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

NH ≅ G/Ntype
1. #83810C2semidirect
4. #72027C2semidirect
6. #70135C2semidirect
7. #70134C2semidirect
8. #69987C2semidirect
9. #69982C2semidirect
10. #69016C2semidirect
11. #69015C2semidirect
361. C3#27691semidirect

Maximal subgroups

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

subgroupindex
#838102
#829162
C3×#18912
#720272
#717142
#701352
#701342
#699872
#699822
#690162
#690152
#276913