Finite Group Databasebeta

← #10877356

Subgroups — #10877356

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#78655C211yesmaximal, characteristic
3C3×#1858C211yesmaximal, characteristic, complemented
4#72029C211yesmaximal, characteristic
5#70729C212yesmaximal, complemented
6#70725C212yesmaximal, complemented
7#70136C211yesmaximal, characteristic
8#70134C211yesmaximal, characteristic, complemented
9#70001C211yesmaximal, characteristic, complemented
10#68939C212yesmaximal, complemented
11#68936C212yesmaximal, complemented
12#27691—31nomaximal, Sylow
13C3×#568C2211yescharacteristic
14#5828C2211yescharacteristic
15#5819C2211yescharacteristic
16C2×#948C2211yescharacteristic
17C22×D3×C8C2211yescharacteristic
18C3×#430C2212yes—
19C3×#413C2211yescharacteristic
20C3×#408C2211yescharacteristic
21#5237C2211yescharacteristic
22#5126C2211yescharacteristic
23#5126C2212yes—
24#5126C2212yes—
25#5124—22no—
26#5124C2212yes—
27#5124C2212yes—
28C2×#822—22no—
29C2×#822C2212yes—
30C2×#822C2212yes—
31#5122C2211yescharacteristic
32#5102C2211yescharacteristic
33#4728—22no—
34#67⋊C8—22no—
35C3×#341C2212yes—
36C3⋊#338C2212yes—
37#4559C2212yes—
38#4556C2212yes—
39#4552C2212yes—
40#2929—31no—
41#2916—31no—
42#2103—32no—
43#2091—32no—
44#1953—32no—
45#1891—31no—
46#1891—31no—
47#1858—31no—
48#1858D311yescharacteristic, complemented
49#1813—31no—
50#1810—32no—
51C2×C3×#41C2311yescharacteristic
52C22×C4×D3C2311yescharacteristic
53C3×#131—42no—
54C2×C3×M16—22no—
55C2×C3×M16C2311yescharacteristic
56C2×C3×M16C2312yes—
57C2×C3×M16C2312yes—
58C22×C3×C8C2311yescharacteristic
59#948—24no—
60#948—42no—
61#946—21no—
62#938—21no—
63#937Q812yes—
64#935Q812yes—
65C4×#67—22no—
66#926C2311yescharacteristic
67C2×#208—22no—
68C2×#208D412yes—
69C2×#208C2312yes—
70C2×#208C2312yes—
71C2×#208C2311yescharacteristic
72C22×C3⋊C8D412yes—
73C22×C3⋊C8C2311yescharacteristic
74C2×D3×C8—21no—
75C2×D3×C8—21no—
76C2×D3×C8—24no—
77C3×#108—42no—
78C3×#101—21no—
79C3×#100—21no—
80C3×C22⋊C8—22no—
81#836—21no—
82#833—21no—
83#822—22no—
84#822—24no—
85#822—22no—
86#822—24no—
87#822—42no—
88(C2×D3)⋊C8—22no—
89(C2×D3)⋊C8—42no—
90(C2×D3)⋊C8—42no—
91#815—21no—
92Dic3⋊C8—22no—
93C8×Dic3—21no—
94#568—31no—
95#568C2×D311yescharacteristic
96C2×#131—31no—
97C23×C8—31no—
98#525—31no—
99#435—62no—
100C8×D4—62no—
101#430—32no—
102#430—32no—
103#430—62no—
104#430C2×D312yes—
105C2×#108—32no—
106C2×#108—32no—
107C2×#108—62no—
108#424—31no—
109#413—31no—
110#413C2×D311yescharacteristic
111#408C2×D311yescharacteristic
112#407—31no—
113#407—32no—
114#407—32no—
115#405—31no—
116#344—32no—
117#344—32no—
118#341—32no—
119#341C2×D312yes—
120#338—32no—
121C23×D3C2×Q811yescharacteristic
122C3×#41—22no—
123C3×#41—22no—
124C2×C3×Q8#4111yescharacteristic
125C2×C3×D4C2×Q812yes—
126C2×C3×D4#4111yescharacteristic
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2411yescharacteristic
129C2×#67—22no—
130C22×Dic3C2×Q811yescharacteristic
131C2×C4×D3—21no—
132C2×C4×D3—22no—
133C2×C4×D3—21no—
134C2×C4×D3—22no—
135C3×M16—22no—
136C3×M16—22no—
137C3×M16—22no—
138C3×M16—22no—
139C3×M16—22no—
140C3×M16—42no—
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—42no—
147#217—22no—
148#217C2×Q812yes—
149(C2×D3)⋊C4—22no—
150C4⋊Dic3—21no—
151Dic3⋊C4—22no—
152C4×Dic3—21no—
153#208—22no—
154#208—24no—
155#208—22no—
156#208—22no—
157#208—22no—
158#208—22no—
159#208—42no—
160C2×C3⋊C8—24no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—42no—
164C2×C3⋊C8C2×D412yes—
165C2×C3⋊C8C2×D411yescharacteristic
166C2×C3⋊C8C2×D411yescharacteristic
167C2×C3⋊C8#4114yes—
168D3×C8—42no—
169D3×C8—42no—
170C2×#41C22×D311yescharacteristic
171C23×C4—31no—
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—62no—
181C2×M16C22×D312yes—
182C2×M16C22×D312yes—
183C2×M16C22×D311yescharacteristic
184C22×C8—31no—
185C22×C8—32no—
186C22×C8—64no—
187C22×C8—61no—
188C22×C8—61no—
189C22×C8C22×D311yescharacteristic
190#123—32no—
191C22⋊Q8—61no—
192#121—61no—
193#120—32no—
194C4×D4—62no—
195#117—31no—
196#108—42no—
197#108—64no—
198#108—62no—
199#108—62no—
200#108—64no—
201#108—122no—
202C4⋊C8—62no—
203#101—21no—
204#101—61no—
205#100—21no—
206#100—61no—
207C22⋊C8—22no—
208C22⋊C8—62no—
209C22⋊C8—122no—
210C22⋊C8—122no—
211#97—61no—
212C4×C8—61no—
213C23×C3C22⋊Q812yes—
214C23×C3C22×Q811yescharacteristic
215C22×D3—21no—
216C22×D3—22no—
217C22×D3—42no—
218C3×Q8—22no—
219C3×D4—24no—
220C3×D4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4C22⋊Q812yes—
224C2×C3×C4C22×D411yescharacteristic
225C2×C3×C4C2×#4111yescharacteristic
226C2×C3×C4C2×#4112yes—
227C2×C3×C4H(5,𝔽2)12yes—
228#67—44no—
229C2×Dic3—22no—
230C2×Dic3—21no—
231C2×Dic3—22no—
232C4×D3—41no—
233C4×D3—41no—
234C4×D3—42no—
235C3×C8—41no—
236C3×C8—42no—
237C3×C8—41no—
238C3×C8—42no—
239C3⋊C8—22no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C3⋊C8#12312yes—
243C3⋊C8#12312yes—
244C24—31no—
245#41—22no—
246#41—22no—
247C2×Q8#23511yescharacteristic
248C2×D4—62no—
249C2×D4#23511yescharacteristic
250C2×D4D3×Q812yes—
251C22×C4—31no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—62no—
256C22×C4D3×D412yes—
257C22×C4C23×D311yescharacteristic
258M16—22no—
259M16—22no—
260M16—22no—
261M16—22no—
262M16—22no—
263M16—42no—
264M16—64no—
265M16—62no—
266M16—62no—
267M16—62no—
268M16—62no—
269M16—62no—
270M16—122no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—31no—
277C2×C8—31no—
278C2×C8—34no—
279C2×C8—32no—
280C2×C8—42no—
281C2×C8—64no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—122no—
285C2×C8—122no—
286C2×C8—122no—
287C4⋊C4—62no—
288C4⋊C4—61no—
289C22⋊C4—32no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C42—61no—
293C22×C3—22no—
294C22×C3—41no—
295C22×C3—42no—
296C22×C3C2×C22⋊Q811yescharacteristic
297C22×C3#55412yes—
298C2×D3—22no—
299C2×D3—24no—
300C2×D3—41no—
301C2×D3—42no—
302C2×D3—41no—
303C3×C4—22no—
304C3×C4C2×#12311yescharacteristic
305C3×C4#53511yescharacteristic
306C3×C4#55312yes—
307Dic3—42no—
308Dic3—41no—
309C23—62no—
310C23—61no—
311C23—122no—
312C23#89612yes—
313C23C2×D3×Q811yescharacteristic
314Q8—22no—
315D4—24no—
316D4—22no—
317D4—124no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—61no—
323C2×C4—121no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4#89612yes—
327C2×C4C2×#23511yescharacteristic
328C2×C4C2×D3×D411yescharacteristic
329C2×C4#100412yes—
330C2×C4D3×#4112yes—
331C8—32no—
332C8—32no—
333C8—41no—
334C8—42no—
335C8—42no—
336C8—41no—
337C8—62no—
338C8—122no—
339C8—122no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—42no—
343C2×C3#248711yescharacteristic
344D3—42no—
345C22—22no—
346C22—41no—
347C22—42no—
348C22—62no—
349C22—64no—
350C22—121no—
351C22—122no—
352C22—121no—
353C22C2×#89611yescharacteristic
354C22#563612yes—
355C4—22no—
356C4—122no—
357C4—121no—
358C4#554012yes—
359C4#559211yescharacteristic
360C4D3×#12311yescharacteristic
361C3#2769111yescharacteristic, complemented, Sylow
362C2—21no—
363C2—22no—
364C2—42no—
365C2—122no—
366C2#7445711yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1858C2
socleC2×C3#2487
prime radical#5828C22
hypercenter#1858D3
hypocenterC3#27691
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1858

Complements

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

NH ≅ G/Ntype
1. #83810C2semidirect
3. C3×#1858C2semidirect
5. #70729C2semidirect
6. #70725C2semidirect
8. #70134C2semidirect
9. #70001C2semidirect
10. #68939C2semidirect
11. #68936C2semidirect
48. #1858D3semidirect
361. C3#27691semidirect

Maximal subgroups

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

subgroupindex
#838102
#786552
C3×#18582
#720292
#707292
#707252
#701362
#701342
#700012
#689392
#689362
#276913