Finite Group Databasebeta

← #10705707

Subgroups — #10705707

Order 768.

488 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 73 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#78358C211yesmaximal, characteristic
2D3×#343C211yesmaximal, characteristic, complemented
3#68144C211yesmaximal, characteristic, complemented
4C3×#1353C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67493C211yesmaximal, characteristic, complemented
7#67456C211yesmaximal, characteristic, complemented
8#11766—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#895C411yescharacteristic
11C2×D3×C4⋊C4C2211yescharacteristic
12C3×#400C2211yescharacteristic
13#4991C2211yescharacteristic
14#4985C411yescharacteristic
15#4726—21no—
16#4672C411yescharacteristic
17#4670—21no—
18#4669—21no—
19#4666C411yescharacteristic
20#4665—21no—
21#4665—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4469—21no—
28#2591—31no—
29C2×#343—31no—
30#1463—31no—
31#1353—31no—
32#1353D311yescharacteristic, complemented
33#1345—31no—
34#1345—31no—
35C22×D12—21no—
36C22×D12C2×C411yescharacteristic
37C22×C4×D3C2×C411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C22×C4×D3D411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4C2×C411yescharacteristic
51C2×Dic3⋊C4D411yescharacteristic
52C2×C4×Dic3—21no—
53C2×#208—21no—
54D3×M16—24no—
55D3×M16—42no—
56C2×ΓL(1,𝔽25)—41no—
57C2×ΓL(1,𝔽25)—41no—
58C2×D3×C8—41no—
59#895—41no—
60#895—41no—
61D3×C4⋊C4—41no—
62D3×C4⋊C4—41no—
63C3×C22⋊C8—21no—
64C3×#95—21no—
65C3×#95—21no—
66C3×#95C2×C411yescharacteristic
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95C2×C411yescharacteristic
70#830—21no—
71(C2×D3)⋊C8—41no—
72C22×M16—61no—
73C2×#121—31no—
74C22×C4⋊C4—31no—
75#407—61no—
76#400—31no—
77#400C2×D311yescharacteristic
78#394—31no—
79#394—31no—
80#388—61no—
81#388—61no—
82#386—61no—
83#386—61no—
84C4×C22⋊C4—31no—
85#343—31no—
86#343—61no—
87#343C2×D311yescharacteristic
88#324—31no—
89#324—61no—
90#324C2×D311yescharacteristic
91C23×D3—21no—
92C23×D3C22⋊C411yescharacteristic
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3C22⋊C411yescharacteristic
100C22×Dic3C22⋊C411yescharacteristic
101C22×Dic3C22⋊C411yescharacteristic
102C2×D12—41no—
103C2×D12—42no—
104C2×D12—42no—
105C2×D12—41no—
106C2×D12—41no—
107C2×D12—81no—
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—22no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×C4×D3—81no—
118C2×C4×D3—81no—
119C3×M16—22no—
120C3×M16—41no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125(C2×D3)⋊C4—42no—
126(C2×D3)⋊C4—42no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—42no—
130(C2×D3)⋊C4—81no—
131(C2×D3)⋊C4—81no—
132C4⋊Dic3—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—41no—
135C4×Dic3—81no—
136#208—22no—
137#208—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140ΓL(1,𝔽25)—44no—
141ΓL(1,𝔽25)—82no—
142D3×C8—44no—
143C22×D4—31no—
144C22×D4—61no—
145C23×C4—31no—
146C23×C4—31no—
147C23×C4—31no—
148C2×M16—21no—
149C2×M16—64no—
150C2×M16—61no—
151C2×M16—121no—
152C2×M16—121no—
153C2×M16—122no—
154C22×C8—121no—
155#121—121no—
156#121—121no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—121no—
161C2×C4⋊C4—121no—
162C2×C4⋊C4C4×D311yescharacteristic
163C2×C4⋊C4C22×D311yescharacteristic
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C42—61no—
171C22⋊C8—21no—
172C22⋊C8—61no—
173C22⋊C8—121no—
174#95—21no—
175#95—21no—
176#95—31no—
177#95—61no—
178#95—61no—
179#95C4×D311yescharacteristic
180C23×C3C2×C22⋊C411yescharacteristic
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—22no—
184C22×D3—41no—
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3—41no—
191C22×D3—41no—
192C22×D3#9911yescharacteristic
193C22×D3#9911yescharacteristic
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
207C2×C3×C4#13711yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3#10412yes—
218D12—82no—
219D12—82no—
220D12—82no—
221C4×D3—22no—
222C4×D3—22no—
223C4×D3—22no—
224C4×D3—22no—
225C4×D3—42no—
226C4×D3—81no—
227C4×D3—81no—
228C4×D3—81no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24—31no—
234C24—61no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—122no—
239C2×D4—122no—
240C2×D4—241no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—62no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—241no—
257C22×C4—241no—
258C22×C4C2×C4×D311yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261M16—22no—
262M16—41no—
263M16—62no—
264M16—124no—
265M16—121no—
266M16—242no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—124no—
270C2×C8—121no—
271C2×C8—121no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—122no—
278C22⋊C4—122no—
279C22⋊C4—122no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—241no—
283C22⋊C4—241no—
284C42—241no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3C2×#9911yescharacteristic
288C22×C3#41711yescharacteristic
289C22×C3C2×#10411yescharacteristic
290C2×D3—21no—
291C2×D3—21no—
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—21no—
295C2×D3—42no—
296C2×D3—42no—
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C2×D3—81no—
303C2×D3—81no—
304C2×D3#35512yes—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—81no—
310C3×C4—81no—
311Dic3—22no—
312Dic3—41no—
313Dic3—41no—
314Dic3—81no—
315C23—31no—
316C23—31no—
317C23—62no—
318C23—61no—
319C23—61no—
320C23—121no—
321C23—121no—
322C23—121no—
323C23—121no—
324C23—121no—
325C23—121no—
326C23—121no—
327C23—121no—
328C23D3×C22⋊C411yescharacteristic
329D4—242no—
330D4—242no—
331D4—242no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—32no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4—241no—
362C2×C4#91411yescharacteristic
363C2×C4#91811yescharacteristic
364C8—42no—
365C8—81no—
366C8—122no—
367C8—241no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3#150411yescharacteristic
371C2×C3#212111yescharacteristic
372C2×C3C2×#35511yescharacteristic
373D3—22no—
374D3—41no—
375D3—161no—
376C22—21no—
377C22—21no—
378C22—32no—
379C22—61no—
380C22—61no—
381C22—61no—
382C22—61no—
383C22—61no—
384C22—122no—
385C22—122no—
386C22—241no—
387C22—241no—
388C22—241no—
389C22—241no—
390C22—241no—
391C22—241no—
392C22—241no—
393C22D3×#9911yescharacteristic
394C22#480311yescharacteristic
395C22D3×#10411yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—62no—
401C4—81no—
402C4—81no—
403C4—121no—
404C4—121no—
405C4—241no—
406C3#1176611yescharacteristic, complemented, Sylow
407C2—21no—
408C2—21no—
409C2—62no—
410C2—121no—
411C2—481no—
412C2#6833411yescharacteristic
413C2#6914311yescharacteristic
414C2D3×#35511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3C2×#99
prime radicalC2×#895C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1353D3
hypocenterC3#11766
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
2. D3×#343C2semidirect
3. #68144C2semidirect
4. C3×#1353C2semidirect
5. #67652C2semidirect
6. #67493C2semidirect
7. #67456C2semidirect
32. #1353D3semidirect
406. C3#11766semidirect

Maximal subgroups

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

subgroupindex
#783582
D3×#3432
#681442
C3×#13532
#676522
#674932
#674562
#117663