Finite Group Databasebeta

← #10706232

Subgroups — #10706232

Order 768.

764 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82403C211yesmaximal, characteristic
2C3×#1800C211yesmaximal, characteristic, complemented
3#71600C211yesmaximal, characteristic, complemented
4#67493C214yesmaximal, complemented
5#11880—31nomaximal, Sylow
6C23×D12C411yescharacteristic
7C2×C3×C4×D4C2211yescharacteristic
8C2×#937—21no—
9C2×#930—21no—
10C3×#423—21no—
11C2×C3×#100—21no—
12#5212C411yescharacteristic
13C2×#830—21no—
14#4991C2212yes—
15#4960C412yes—
16#4955—21no—
17#4939—21no—
18C3×#343C2212yescomplemented
19#4528C2212yescomplemented
20#4525—21no—
21#2408—31no—
22#1800D311yescharacteristic, complemented
23#1459—31no—
24#1345—34no—
25C24×D3—21no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4C2×C411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C22×#67—22no—
30C22×D12—21no—
31C22×D12—44no—
32C22×D12—41no—
33C22×D12—41no—
34C22×D12C2×C412yes—
35C2×C3×M16—22no—
36C3×C4×D4—41no—
37C3×C4×D4—41no—
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×#217—21no—
43#937—42no—
44#937—42no—
45#930—44no—
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—22no—
49C2×C4⋊Dic3—21no—
50C22×C3⋊C8—41no—
51C3×C4⋊C8—41no—
52C3×C4⋊C8—41no—
53C3×#100—42no—
54C3⋊#95C2×C412yes—
55#830—22no—
56#830—41no—
57C23×D4—31no—
58C2×#121—61no—
59C2×#120—61no—
60C2×C4×D4C2×D311yescharacteristic
61#423—21no—
62C2×#100—21no—
63C2×C22⋊C8—61no—
64#394—32no—
65#392—32no—
66#390—61no—
67#380—31no—
68#379—61no—
69#343C2×D312yescomplemented
70#324—32no—
71#323—61no—
72C24×C3C22⋊C411yescharacteristic
73C24×C3C22⋊C411yescharacteristic
74C23×D3—22no—
75C23×D3—44no—
76C23×D3—42no—
77C23×D3—42no—
78C23×D3—42no—
79C23×D3—82no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4D812yes—
83C2×C3×D4SD1612yes—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—42no—
89C22×C3×C4—42no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C2×#67—44no—
96C2×#67—44no—
97C2×#67—82no—
98C22×Dic3—22no—
99C2×D12—24no—
100C2×D12—42no—
101C2×D12—42no—
102C2×D12—48no—
103C2×D12—44no—
104C2×D12—41no—
105C2×D12—41no—
106C2×D12—44no—
107C2×D12—44no—
108C3×M16—42no—
109C3×M16—42no—
110C2×C3×C8—42no—
111C3×C4⋊C4—41no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C42—21no—
115C3×C42—21no—
116#217—42no—
117(C2×D3)⋊C4—44no—
118(C2×D3)⋊C4—42no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—82no—
121C4⋊Dic3—42no—
122C2×C3⋊C8—42no—
123C2×C3⋊C8—82no—
124C25—61no—
125C22×D4—32no—
126C22×D4—62no—
127C22×D4—61no—
128C22×D4—121no—
129C22×D4—121no—
130C22×D4—124no—
131C22×D4#6711yescharacteristic
132C23×C4C4×D311yescharacteristic
133C23×C4D1211yescharacteristic
134C2×M16—22no—
135C22×C8—121no—
136#121—124no—
137#120—122no—
138#120—122no—
139C4×D4—41no—
140C4×D4—41no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4C22×D311yescharacteristic
143C2×C22⋊C4—61no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4—62no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4C4×D311yescharacteristic
148C2×C22⋊C4D1211yescharacteristic
149C2×C42#6711yescharacteristic
150C4⋊C8—41no—
151C4⋊C8—41no—
152#100—42no—
153C22⋊C8—62no—
154C22⋊C8—121no—
155#95—32no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3#10212yes—
165C23×C3#10212yes—
166C23×C3C2×C22⋊C411yescharacteristic
167C22×D3—24no—
168C22×D3—42no—
169C22×D3—42no—
170C22×D3—42no—
171C22×D3—48no—
172C22×D3—42no—
173C22×D3—42no—
174C22×D3—42no—
175C22×D3—88no—
176C22×D3—88no—
177C22×D3—84no—
178C22×D3—84no—
179C3×D4—42no—
180C3×D4—42no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—22no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4#9911yescharacteristic
195C2×C3×C4#9911yescharacteristic
196C2×C3×C4C2×D811yescharacteristic
197C2×C3×C4C2×SD1611yescharacteristic
198#67—88no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202D12—44no—
203D12—44no—
204D12—44no—
205D12—44no—
206C3×C8—82no—
207C3⋊C8—82no—
208C24—62no—
209C24—122no—
210C24—122no—
211C24—124no—
212C24—122no—
213C24—242no—
214C24(C2×D3)⋊C411yescharacteristic
215C24(C2×D3)⋊C411yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—64no—
219C2×D4—122no—
220C2×D4—122no—
221C2×D4—128no—
222C2×D4—124no—
223C2×D4—124no—
224C2×D4—124no—
225C2×D4—124no—
226C2×D4—124no—
227C2×D4—121no—
228C2×D4—121no—
229C2×D4—242no—
230C2×D4#21312yes—
231C2×D4#21512yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—62no—
239C22×C4(C2×D3)⋊C411yescharacteristic
240C22×C4(C2×D3)⋊C411yescharacteristic
241C22×C4C2×C4×D311yescharacteristic
242C22×C4C2×D1211yescharacteristic
243C22×C4C2×#6711yescharacteristic
244M16—42no—
245M16—42no—
246C2×C8—42no—
247C2×C8—122no—
248C2×C8—242no—
249C4⋊C4—41no—
250C4⋊C4—122no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—124no—
254C22⋊C4—122no—
255C22⋊C4—122no—
256C22⋊C4—122no—
257C22⋊C4—242no—
258C42—21no—
259C42—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—81no—
272C22×C3C2×#9911yescharacteristic
273C22×C3C2×#10211yescharacteristic
274C22×C3#42111yescharacteristic
275C2×D3—44no—
276C2×D3—44no—
277C2×D3—44no—
278C2×D3—82no—
279C2×D3—82no—
280C2×D3—816no—
281C2×D3—82no—
282C2×D3—82no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4AΣL(1,𝔽16)12yes—
290Dic3—82no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—64no—
300C23—122no—
301C23—122no—
302C23—122no—
303C23—128no—
304C23—122no—
305C23—122no—
306C23—122no—
307C23—244no—
308C23—248no—
309C23—248no—
310C23—244no—
311C23#80912yes—
312C23#80912yes—
313C23C2×(C2×D3)⋊C411yescharacteristic
314D4—42no—
315D4—42no—
316D4—124no—
317D4—124no—
318D4—124no—
319D4—124no—
320D4—248no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4#80611yescharacteristic
338C2×C4#80611yescharacteristic
339C2×C4C2×#21311yescharacteristic
340C2×C4C2×#21511yescharacteristic
341C8—82no—
342C8—242no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—42no—
346C2×C3—42no—
347C2×C3#151911yescharacteristic
348C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
349C2×C3#213411yescharacteristic
350D3—84no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22—124no—
364C22—124no—
365C22—124no—
366C22—242no—
367C22—242no—
368C22—2416no—
369C22—242no—
370C22—242no—
371C22C2×#80611yescharacteristic
372C22C2×#80911yescharacteristic
373C22#500511yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—242no—
381C4#447512yes—
382C3#1188011yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—244no—
388C2#6950911yescharacteristic
389C2C2×#447511yescharacteristic
390C2#7014911yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1800C2
socleC22×C3C2×#99
prime radical#82403C2
cosocleC2×C3×C4⋊C4C23
hypercenter#1800D3
hypocenterC3#11880
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. C3×#1800C2semidirect
3. #71600C2semidirect
4. #67493C2semidirect
18. C3×#343C22semidirect
19. #4528C22semidirect
22. #1800D3semidirect
69. #343C2×D3semidirect
382. C3#11880semidirect

Maximal subgroups

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

subgroupindex
#824032
C3×#18002
#716002
#674932
#118803