Finite Group Databasebeta

← #10705652

Subgroups — #10705652

Order 768.

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

normal subgroups only

This group has 586 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#78105C211yesmaximal, characteristic
2D3×#342C211yesmaximal, characteristic
3#68120C211yesmaximal, characteristic
4C3×#1352C211yesmaximal, characteristic, complemented
5#67651C211yesmaximal, characteristic, complemented
6#67491C211yesmaximal, characteristic, complemented
7#67450C211yesmaximal, characteristic, complemented
8#11755—31nomaximal, Sylow
9C2×#892C411yescharacteristic, complemented
10C2×D3×C22⋊C4—21no—
11C2×D3×C22⋊C4C2211yescharacteristic
12C3×#399C2211yescharacteristic
13#4993C411yescharacteristic, complemented
14#4990—21no—
15#4989—21no—
16#4955—21no—
17#4953C2211yescharacteristic
18#4945—21no—
19#4726—21no—
20#4674C411yescharacteristic, complemented
21#4664C411yescharacteristic, complemented
22C3×#342C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4527—21no—
25#4526C2211yescharacteristic
26#4525C2211yescharacteristic
27#4468—21no—
28#2482—31no—
29C2×#342—31no—
30#1462—31no—
31#1352—31no—
32#1352D311yescharacteristic, complemented
33#1344—31no—
34#1344—31no—
35C24×D3D411yescharacteristic
36C22×C4×D3—41no—
37C22×C4×D3C2×C411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4C2311yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217D411yescharacteristic
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—41no—
52C2×(C2×D3)⋊C4C2×C411yescharacteristic
53C2×(C2×D3)⋊C4D411yescharacteristic
54C2×C4⋊Dic3C2×C411yescharacteristic, complemented
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3—21no—
57C2×ΓL(1,𝔽25)—41no—
58#892—41no—
59#892—41no—
60D3×C22⋊C4—24no—
61D3×C22⋊C4—41no—
62D3×C22⋊C4—41no—
63D3×C22⋊C4—42no—
64C3×C22⋊C8—21no—
65C3×#95C2×C411yescharacteristic, complemented
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95C2×C411yescharacteristic, complemented
70#830—21no—
71(C2×D3)⋊C8—41no—
72C2×#117—31no—
73C22×C22⋊C4—31no—
74C22×C22⋊C4—61no—
75#407—61no—
76#399—31no—
77#399C2×D311yescharacteristic
78#397—31no—
79#397—31no—
80#388—61no—
81#388—61no—
82#385—61no—
83#381—31no—
84#379—61no—
85#342—31no—
86#342—61no—
87#342C2×D311yescharacteristic
88#323—31no—
89#323—61no—
90#323C2×D311yescharacteristic
91C24×C3C2×D411yescharacteristic
92C23×D3—21no—
93C23×D3—21no—
94C23×D3—21no—
95C23×D3—21no—
96C23×D3—22no—
97C23×D3—42no—
98C23×D3—81no—
99C23×D3C22⋊C411yescharacteristic
100C23×D3C22⋊C411yescharacteristic
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—41no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—41no—
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3—42no—
116C2×C4×D3—44no—
117C2×C4×D3—81no—
118C2×C4×D3—82no—
119C2×C4×D3—82no—
120C2×C3×C8—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—81no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—81no—
127#217—22no—
128#217—41no—
129#217—41no—
130#217—41no—
131#217—41no—
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—44no—
135(C2×D3)⋊C4—42no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—42no—
139(C2×D3)⋊C4—81no—
140C4⋊Dic3—41no—
141Dic3⋊C4—81no—
142C4×Dic3—42no—
143C2×C3⋊C8—41no—
144ΓL(1,𝔽25)—82no—
145C25—31no—
146C23×C4—31no—
147C23×C4—31no—
148C23×C4—121no—
149C2×M16—121no—
150#117—121no—
151#117—121no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4C4×D311yescharacteristic, complemented
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—64no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—121no—
169C2×C22⋊C4—121no—
170C2×C22⋊C4—122no—
171C2×C22⋊C4—121no—
172C2×C22⋊C4C22×D311yescharacteristic
173C2×C42—61no—
174C22⋊C8—21no—
175C22⋊C8—61no—
176C22⋊C8—121no—
177#95—31no—
178#95—61no—
179#95—61no—
180#95—61no—
181#95C4×D311yescharacteristic, complemented
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3C2×C22⋊C411yescharacteristic
187C23×C3AGL(1,ℤ/8)11yescharacteristic
188C23×C3#13711yescharacteristic
189C22×D3—22no—
190C22×D3—22no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—22no—
195C22×D3—22no—
196C22×D3—21no—
197C22×D3—22no—
198C22×D3—41no—
199C22×D3—42no—
200C22×D3—42no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—81no—
204C22×D3—81no—
205C22×D3—84no—
206C22×D3—82no—
207C22×D3—82no—
208C22×D3#9911yescharacteristic
209C22×D3#9911yescharacteristic
210C22×D3#10412yes—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237C4×D3—42no—
238C4×D3—42no—
239C4×D3—42no—
240C4×D3—81no—
241C4×D3—88no—
242C3×C8—81no—
243C3⋊C8—81no—
244C24—31no—
245C24—31no—
246C24—61no—
247C24—62no—
248C24—61no—
249C24—61no—
250C24—61no—
251C24—122no—
252C24—241no—
253C24D3×D411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—122no—
266C22×C4—124no—
267C22×C4—121no—
268C22×C4—241no—
269C22×C4—242no—
270C22×C4—242no—
271C22×C4C2×C4×D311yescharacteristic
272C22×C4D3×D411yescharacteristic
273M16—242no—
274C2×C8—41no—
275C2×C8—121no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C4⋊C4—121no—
279C4⋊C4—241no—
280C22⋊C4—22no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—62no—
284C22⋊C4—81no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—122no—
290C22⋊C4—124no—
291C22⋊C4—122no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C22⋊C4—241no—
297C42—122no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—81no—
307C22×C3C2×#9911yescharacteristic
308C22×C3#41711yescharacteristic
309C22×C3C2×#10411yescharacteristic
310C2×D3—21no—
311C2×D3—21no—
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—21no—
315C2×D3—22no—
316C2×D3—42no—
317C2×D3—42no—
318C2×D3—44no—
319C2×D3—44no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—42no—
323C2×D3—81no—
324C2×D3—82no—
325C2×D3—82no—
326C2×D3#35212yes—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—81no—
330C3×C4—82no—
331C3×C4—81no—
332Dic3—41no—
333Dic3—41no—
334Dic3—81no—
335Dic3—82no—
336Dic3—81no—
337C23—21no—
338C23—21no—
339C23—31no—
340C23—31no—
341C23—32no—
342C23—41no—
343C23—41no—
344C23—62no—
345C23—62no—
346C23—62no—
347C23—62no—
348C23—62no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23—121no—
354C23—121no—
355C23—121no—
356C23—122no—
357C23—122no—
358C23—241no—
359C23—241no—
360C23—242no—
361C23—242no—
362C23—244no—
363C23D3×C22⋊C411yescharacteristic
364C23#91411yescharacteristic
365C23#91811yescharacteristic
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—122no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—241no—
394C2×C4—248no—
395C2×C4—241no—
396C2×C4—241no—
397C8—81no—
398C8—241no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3#150411yescharacteristic
404C2×C3C2×#35211yescharacteristic
405C2×C3#212711yescharacteristic
406D3—22no—
407D3—41no—
408D3—42no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—32no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—61no—
419C22—61no—
420C22—61no—
421C22—61no—
422C22—61no—
423C22—62no—
424C22—81no—
425C22—122no—
426C22—122no—
427C22—122no—
428C22—124no—
429C22—124no—
430C22—121no—
431C22—121no—
432C22—241no—
433C22—242no—
434C22—242no—
435C22D3×#9911yescharacteristic
436C22#480311yescharacteristic
437C22D3×#10411yescharacteristic
438C4—41no—
439C4—41no—
440C4—81no—
441C4—82no—
442C4—81no—
443C4—121no—
444C4—121no—
445C4—241no—
446C4—242no—
447C4—241no—
448C3#1175511yescharacteristic, complemented, Sylow
449C2—21no—
450C2—21no—
451C2—41no—
452C2—41no—
453C2—62no—
454C2—121no—
455C2—122no—
456C2#6833411yescharacteristic
457C2D3×#35211yescharacteristic
458C2#6915011yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1352C2
socleC22×C3C2×#99
prime radicalC24×D3D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1352D3
hypocenterC3#11755
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1352

Complements

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

NH ≅ G/Ntype
4. C3×#1352C2semidirect
5. #67651C2semidirect
6. #67491C2semidirect
7. #67450C2semidirect
9. C2×#892C4semidirect
13. #4993C4semidirect
20. #4674C4semidirect
21. #4664C4semidirect
32. #1352D3semidirect
40. C2×C3×C4⋊C4C2×C4semidirect
54. C2×C4⋊Dic3C2×C4semidirect
65. C3×#95C2×C4semidirect
69. C3⋊#95C2×C4semidirect
155. C2×C4⋊C4C4×D3semidirect
181. #95C4×D3semidirect
448. C3#11755semidirect

Maximal subgroups

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

subgroupindex
#781052
D3×#3422
#681202
C3×#13522
#676512
#674912
#674502
#117553