Finite Group Databasebeta

← #10705685

Subgroups — #10705685

Order 768.

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

normal subgroups only

This group has 622 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#77107C211yesmaximal, characteristic
2D3×#343C211yesmaximal, characteristic, complemented
3#68144C211yesmaximal, characteristic, complemented
4C3×#1345C211yesmaximal, characteristic, complemented
5#67647C211yesmaximal, characteristic, complemented
6#67494C211yesmaximal, characteristic, complemented
7#67457C211yesmaximal, characteristic, complemented
8#11762—31nomaximal, Sylow
9C22×#237C411yescharacteristic
10C2×D3×M16—21no—
11C2×D3×C4⋊C4C2211yescharacteristic
12C2×#883—21no—
13C3×#394C2211yescharacteristic
14#5222C411yescharacteristic
15#4979C2211yescharacteristic
16#4956—21no—
17#4950—21no—
18#4946—21no—
19#4726—21no—
20#4664C411yescharacteristic
21#4656C411yescharacteristic
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4469—21no—
28#2406—31no—
29C2×#343—31no—
30#1463—31no—
31#1354—31no—
32#1354—31no—
33#1345—31no—
34#1345D311yescharacteristic, complemented
35C22×C3×D4C2×C411yescharacteristic
36C22×#67—21no—
37C23×Dic3—21no—
38C2×#237—41no—
39C2×#237—41no—
40C2×#237—44no—
41C22×C4×D3C2×C411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×Dic6C2×C411yescharacteristic
45C2×C3×M16—21no—
46C2×C3×C4⋊C4C2311yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4C2×C411yescharacteristic
56C2×Dic3⋊C4D411yescharacteristic
57C2×#208—21no—
58D3×M16—24no—
59D3×M16—42no—
60C2×ΓL(1,𝔽25)—41no—
61C2×ΓL(1,𝔽25)—41no—
62C2×D3×C8—41no—
63D3×C4⋊C4—41no—
64D3×C4⋊C4—41no—
65#883—44no—
66C3×C22⋊C8—21no—
67C3×#95C2×C411yescharacteristic
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95C2×C411yescharacteristic
72#830—21no—
73(C2×D3)⋊C8—41no—
74C22×#41—31no—
75C22×M16—61no—
76C2×#123—61no—
77C22×C4⋊C4—31no—
78#407—61no—
79#395—31no—
80#395—31no—
81#394—31no—
82#394C2×D311yescharacteristic
83#386—61no—
84#385—61no—
85#385—61no—
86#381—31no—
87#343—31no—
88#343—61no—
89#343C2×D311yescharacteristic
90#324—31no—
91#324—61no—
92#324C2×D311yescharacteristic
93C24×C3—21no—
94C23×D3C22⋊C411yescharacteristic
95C2×C3×D4—41no—
96C2×C3×D4—42no—
97C22×C3×C4—21no—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C2×#67—24no—
102C2×#67—42no—
103C2×#67—41no—
104C2×#67—41no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—42no—
108C22×Dic3—42no—
109C22×Dic3—42no—
110C22×Dic3C22⋊C411yescharacteristic
111C22×Dic3C22⋊C411yescharacteristic
112C22×Dic3C22⋊C411yescharacteristic
113#237—44no—
114#237—44no—
115#237—44no—
116#237—44no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—22no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—42no—
127C2×C4×D3—42no—
128C2×Dic6—41no—
129C2×Dic6—42no—
130C3×M16—22no—
131C3×M16—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—42no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138#217—42no—
139#217—42no—
140(C2×D3)⋊C4—42no—
141C4⋊Dic3—41no—
142C4⋊Dic3—42no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—42no—
146#208—22no—
147#208—41no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150ΓL(1,𝔽25)—44no—
151ΓL(1,𝔽25)—82no—
152D3×C8—44no—
153C2×#41—121no—
154C2×#41—121no—
155C2×#41—124no—
156C22×Q8—31no—
157C22×D4—61no—
158C22×D4C4×D311yescharacteristic
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—31no—
162C23×C4—61no—
163C2×M16—21no—
164C2×M16—64no—
165C2×M16—61no—
166C2×M16—121no—
167C2×M16—121no—
168C2×M16—122no—
169C22×C8—121no—
170#123—124no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—121no—
177C2×C4⋊C4—121no—
178C2×C4⋊C4C22×D311yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C22⋊C8—21no—
185C22⋊C8—61no—
186C22⋊C8—121no—
187#95—31no—
188#95—61no—
189#95—61no—
190#95—61no—
191#95C4×D311yescharacteristic
192C23×C3—22no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3C2×C22⋊C411yescharacteristic
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—22no—
203C22×D3—41no—
204C22×D3#9911yescharacteristic
205C22×D3#9911yescharacteristic
206C3×D4—42no—
207C3×D4—42no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×C3×C4#13711yescharacteristic
219#67—44no—
220#67—44no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—22no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—41no—
227C2×Dic3—48no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3#10412yes—
235C4×D3—22no—
236C4×D3—22no—
237C4×D3—22no—
238C4×D3—22no—
239C4×D3—42no—
240C4×D3—42no—
241C4×D3—42no—
242C4×D3—81no—
243Dic6—42no—
244Dic6—42no—
245C3×C8—42no—
246C3×C8—81no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249C24—21no—
250C24—31no—
251#41—124no—
252#41—124no—
253#41—124no—
254#41—124no—
255C2×Q8—121no—
256C2×Q8—122no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—64no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—122no—
263C22×C4—21no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—62no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—122no—
279C22×C4—122no—
280C22×C4—122no—
281C22×C4—122no—
282C22×C4—122no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286M16—22no—
287M16—41no—
288M16—62no—
289M16—124no—
290M16—121no—
291M16—242no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—124no—
295C2×C8—121no—
296C2×C8—121no—
297C4⋊C4—41no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4—121no—
301C4⋊C4—122no—
302C4⋊C4—122no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—122no—
307C22⋊C4—122no—
308C22⋊C4—122no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3C2×#9911yescharacteristic
319C22×C3#41711yescharacteristic
320C22×C3C2×#10411yescharacteristic
321C2×D3—21no—
322C2×D3—21no—
323C2×D3—21no—
324C2×D3—21no—
325C2×D3—21no—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3AΣL(1,𝔽16)12yes—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—81no—
334Dic3—22no—
335Dic3—42no—
336Dic3—41no—
337Dic3—41no—
338Dic3—81no—
339C23—22no—
340C23—31no—
341C23—31no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—62no—
349C23—61no—
350C23—61no—
351C23—121no—
352C23D3×C22⋊C411yescharacteristic
353Q8—122no—
354Q8—122no—
355D4—42no—
356D4—42no—
357D4—124no—
358D4—124no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—32no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—62no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—128no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4#91111yescharacteristic
391C2×C4#91511yescharacteristic
392C8—42no—
393C8—81no—
394C8—122no—
395C8—241no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—82no—
399C2×C3#150411yescharacteristic
400C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
401C2×C3#213311yescharacteristic
402D3—22no—
403D3—41no—
404C22—21no—
405C22—21no—
406C22—32no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—61no—
411C22—61no—
412C22—61no—
413C22—61no—
414C22—61no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—122no—
420C22—122no—
421C22D3×#9911yescharacteristic
422C22#477111yescharacteristic
423C22D3×#10411yescharacteristic
424C4—21no—
425C4—21no—
426C4—41no—
427C4—41no—
428C4—62no—
429C4—81no—
430C4—121no—
431C4—122no—
432C4—121no—
433C4—241no—
434C3#1176211yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—62no—
438C2—82no—
439C2—121no—
440C2#6830211yescharacteristic
441C2D3×AΣL(1,𝔽16)11yescharacteristic
442C2#6916411yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1345C2
socleC22×C3C2×#99
prime radicalC22×#237C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1345D3
hypocenterC3#11762
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1345

Complements

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

NH ≅ G/Ntype
2. D3×#343C2semidirect
3. #68144C2semidirect
4. C3×#1345C2semidirect
5. #67647C2semidirect
6. #67494C2semidirect
7. #67457C2semidirect
34. #1345D3semidirect
434. C3#11762semidirect

Maximal subgroups

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

subgroupindex
#771072
D3×#3432
#681442
C3×#13452
#676472
#674942
#674572
#117623