Finite Group Databasebeta

← #10705665

Subgroups — #10705665

Order 768.

486 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 71 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#76696C211yesmaximal, characteristic
2#68951C211yesmaximal, characteristic, complemented
3#68102C211yesmaximal, characteristic, complemented
4C3×#1343C211yesmaximal, characteristic
5#67645C211yesmaximal, characteristic
6#67490C211yesmaximal, characteristic, complemented
7#67449C211yesmaximal, characteristic, complemented
8#11758—31nomaximal, Sylow
9C2×#936C411yescharacteristic, complemented
10C2×#877—21no—
11C2×#877C2211yescharacteristic
12C3×#392C2211yescharacteristic
13#5222C411yescharacteristic, complemented
14#5221—21no—
15#5220—21no—
16#4951—21no—
17#4947C2211yescharacteristic
18#4944—21no—
19#4719—21no—
20#4656C411yescharacteristic, complemented
21#4648C411yescharacteristic, complemented
22C3×#342C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4527—21no—
25#4526C2211yescharacteristic
26#4525C2211yescharacteristic
27#4469—21no—
28#2610—31no—
29#1740—31no—
30#1464—31no—
31#1351—31no—
32#1351—31no—
33#1343—31no—
34#1343D311yescharacteristic
35C22×C3×D4—21no—
36C22×C3×D4C2×C411yescharacteristic
37C23×Dic3D411yescharacteristic
38C22×Dic6C2×C411yescharacteristic
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4C2311yescharacteristic
42C2×#217—21no—
43C2×#217—21no—
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47C2×#217—21no—
48C2×#217D411yescharacteristic
49#936—41no—
50#936—41no—
51C2×C4⋊Dic3—21no—
52C2×Dic3⋊C4—41no—
53C2×Dic3⋊C4C2×C411yescharacteristic
54C2×Dic3⋊C4D411yescharacteristic
55C2×C4×Dic3—21no—
56C2×C4×Dic3—41no—
57C2×C4×Dic3C2×C411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59#877—24no—
60#877—41no—
61#877—41no—
62#877—42no—
63C3×C22⋊C8—21no—
64C3×#95C2×C411yescharacteristic
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95C2×C411yescharacteristic
69#830—21no—
70#815—41no—
71Dic3⋊C8—41no—
72C2×#124—31no—
73C2×#117—31no—
74C2×#117—61no—
75#432—61no—
76#395—31no—
77#394—31no—
78#393—31no—
79#392C2×D311yescharacteristic
80#390—61no—
81#388—61no—
82#388—61no—
83#386—61no—
84#376—31no—
85#342—31no—
86#342—61no—
87#342C2×D311yescharacteristic
88#324—61no—
89#323—31no—
90#323C2×D311yescharacteristic
91C24×C3—21no—
92C24×C3C2×D411yescharacteristic
93C2×C3×D4—41no—
94C2×C3×D4—42no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—81no—
98C2×C3×D4—81no—
99C22×C3×C4—21no—
100C22×C3×C4—41no—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—22no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—41no—
111C22×Dic3C22⋊C411yescharacteristic
112C22×Dic3C22⋊C411yescharacteristic
113C22×Dic3C22⋊C411yescharacteristic
114C22×Dic3C22⋊C411yescharacteristic
115C2×Dic6—41no—
116C2×Dic6—81no—
117C2×C3×C8—41no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—41no—
122#217—22no—
123#217—41no—
124#217—42no—
125#217—41no—
126#217—41no—
127#217—41no—
128#217—42no—
129#217—81no—
130#217—81no—
131C4⋊Dic3—81no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—44no—
135C4×Dic3—21no—
136C4×Dic3—21no—
137C4×Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—44no—
140C4×Dic3—42no—
141C2×C3⋊C8—41no—
142C22×Q8—31no—
143C22×D4—21no—
144C22×D4C4×D311yescharacteristic
145C23×C4—31no—
146#124—121no—
147#124—121no—
148#117—64no—
149#117—121no—
150#117—121no—
151#117—122no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—121no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4C22×D311yescharacteristic
166C2×C42—31no—
167C2×C42—31no—
168C2×C42—61no—
169C2×C42—121no—
170C4⋊C8—121no—
171C22⋊C8—21no—
172C22⋊C8—61no—
173#97—121no—
174#95—31no—
175#95—61no—
176#95—61no—
177#95—61no—
178#95C4×D311yescharacteristic
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3C2×C22⋊C411yescharacteristic
191C23×C3AGL(1,ℤ/8)11yescharacteristic
192C23×C3#13711yescharacteristic
193C3×D4—82no—
194C3×D4—82no—
195C3×D4—82no—
196C2×C3×C4—22no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—22no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3#9911yescharacteristic
228C2×Dic3#9911yescharacteristic
229C2×Dic3#10412yes—
230Dic6—82no—
231C3×C8—81no—
232C3⋊C8—81no—
233C24—21no—
234C24D3×D411yescharacteristic
235C2×Q8—121no—
236C2×Q8—241no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—81no—
242C2×D4—81no—
243C22×C4—21no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—41no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—121no—
257C22×C4C2×C4×D311yescharacteristic
258C22×C4D3×D411yescharacteristic
259C2×C8—41no—
260C2×C8—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—124no—
264C4⋊C4—241no—
265C22⋊C4—22no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—41no—
269C22⋊C4—62no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—122no—
273C22⋊C4—121no—
274C22⋊C4—122no—
275C22⋊C4—121no—
276C22⋊C4—241no—
277C22⋊C4—241no—
278C42—61no—
279C42—61no—
280C42—61no—
281C42—61no—
282C42—122no—
283C42—124no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3C2×#9911yescharacteristic
301C22×C3#41711yescharacteristic
302C22×C3C2×#10411yescharacteristic
303C3×C4—42no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—82no—
307Dic3—21no—
308Dic3—21no—
309Dic3—22no—
310Dic3—41no—
311Dic3—41no—
312Dic3—81no—
313Dic3—82no—
314Dic3—81no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23D3×C22⋊C411yescharacteristic
327C23#91111yescharacteristic
328C23#91511yescharacteristic
329Q8—242no—
330D4—82no—
331D4—82no—
332D4—82no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—32no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—122no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—241no—
366C2×C4—241no—
367C8—81no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—161no—
374C2×C3#150411yescharacteristic
375C2×C3#212211yescharacteristic
376C2×C3#212611yescharacteristic
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22D3×#9911yescharacteristic
394C22#477111yescharacteristic
395C22D3×#10411yescharacteristic
396C4—42no—
397C4—41no—
398C4—41no—
399C4—61no—
400C4—61no—
401C4—62no—
402C4—82no—
403C4—121no—
404C4—121no—
405C4—241no—
406C4—242no—
407C4—241no—
408C3#1175811yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—41no—
412C2—41no—
413C2—161no—
414C2#6830211yescharacteristic
415C2#6913711yescharacteristic
416C2#6915111yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1343C2
socleC22×C3C2×#99
prime radicalC3×#392C22
cosocleC2×C3×C22⋊C4C23
hypercenter#1343D3
hypocenterC3#11758
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1343

Complements

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

NH ≅ G/Ntype
2. #68951C2semidirect
3. #68102C2semidirect
6. #67490C2semidirect
7. #67449C2semidirect
9. C2×#936C4semidirect
13. #5222C4semidirect
20. #4656C4semidirect
21. #4648C4semidirect
408. C3#11758semidirect

Maximal subgroups

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

subgroupindex
#766962
#689512
#681022
C3×#13432
#676452
#674902
#674492
#117583