Finite Group Databasebeta

← #10692629

Subgroups — #10692629

Order 768.

840 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 97 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#82708C211yesmaximal, characteristic
2#73560C211yesmaximal, characteristic, complemented
3C3×#1507C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic, complemented
5#69967C211yesmaximal, characteristic, complemented
6#67683C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9409—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#937—21no—
11C2×#935—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C2×D3×C22⋊C4—21no—
14C3×#407C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5223C412yescomplemented
17C4×#217C2211yescharacteristic
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20#4993C412yescomplemented
21#4957—22no—
22Dic3×C22⋊C4—21no—
23C4×(C2×D3)⋊C4—22no—
24#4666—22no—
25C3×#327—22no—
26#4537—22no—
27#4468—22no—
28#2539—31no—
29#2273—31no—
30#1507—31no—
31#1507D311yescharacteristic, complemented
32#1462—31no—
33#1297—31no—
34#1297—31no—
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic, complemented
38C23×C3×C4C2311yescharacteristic
39C22×#67—22no—
40C23×Dic3D411yescharacteristic
41C22×C4×D3—22no—
42C2×C3×M16—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×#121—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—22no—
54#937—44no—
55#935—44no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4C2×C412yes—
63C2×(C2×D3)⋊C4D412yes—
64C2×Dic3⋊C4—22no—
65C2×C4×Dic3—22no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69D3×C22⋊C4—28no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95D412yescomplemented
73C3⋊#95—22no—
74C3⋊#95C2×C412yescomplemented
75C3⋊#95D412yescomplemented
76#830—21no—
77#830—21no—
78C2×#123—61no—
79C2×#121C2×D311yescharacteristic
80C2×#120—61no—
81C22×C22⋊C4—31no—
82C22×C22⋊C4—61no—
83#407—31no—
84#407C2×D311yescharacteristic
85#399—32no—
86#392—32no—
87#390—62no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4—31no—
90C4×C22⋊C4—62no—
91C4×C22⋊C4—61no—
92C4×C22⋊C4—62no—
93C4×C22⋊C4C2×D311yescharacteristic
94#327—22no—
95#327—62no—
96#323—62no—
97C24×C3—21no—
98C24×C3C2×D411yescharacteristic
99C23×D3—21no—
100C23×D3—21no—
101C23×D3—24no—
102C23×D3—42no—
103C23×D3—81no—
104C23×D3C22⋊C412yes—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—42no—
108C2×C3×D4—42no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—81no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—81no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—44no—
122C2×#67—44no—
123C2×#67—82no—
124C22×Dic3—22no—
125C22×Dic3—21no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—42no—
129C22×Dic3C22⋊C412yes—
130C2×C4×D3—28no—
131C2×C4×D3—82no—
132C2×C4×D3—82no—
133C3×M16—81no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—81no—
141#217—24no—
142#217—24no—
143#217—44no—
144(C2×D3)⋊C4—28no—
145(C2×D3)⋊C4—22no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—22no—
148(C2×D3)⋊C4—22no—
149(C2×D3)⋊C4—44no—
150Dic3⋊C4—44no—
151C4×Dic3—81no—
152C4×Dic3—81no—
153C4×Dic3—82no—
154#208—81no—
155C2×C3⋊C8—41no—
156C25—31no—
157C22×D4—21no—
158C22×D4—62no—
159C22×D4C4×D311yescharacteristic, complemented
160C23×C4—31no—
161C23×C4—62no—
162C23×C4C22×D311yescharacteristic
163C2×M16—21no—
164C2×M16—61no—
165#123—124no—
166#121—21no—
167#121—21no—
168#121—21no—
169#121—21no—
170#120—124no—
171C2×C4⋊C4—62no—
172C2×C4⋊C4C4×D311yescharacteristic, complemented
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—32no—
176C2×C22⋊C4—32no—
177C2×C22⋊C4—68no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C42—21no—
188C2×C42—62no—
189C2×C42—61no—
190C2×C42—61no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195#95—32no—
196#95—32no—
197#95—62no—
198#95#6712yescomplemented
199C23×C3—22no—
200C23×C3—22no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—42no—
205C23×C3—41no—
206C23×C3—42no—
207C23×C3C2×C22⋊C411yescharacteristic
208C23×C3#12011yescharacteristic
209C23×C3#12311yescharacteristic
210C22×D3—22no—
211C22×D3—22no—
212C22×D3—22no—
213C22×D3—24no—
214C22×D3—22no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—42no—
218C22×D3—42no—
219C22×D3—88no—
220C22×D3—81no—
221C22×D3—81no—
222C22×D3#10414yes—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—82no—
226C3×D4—82no—
227C2×C3×C4—24no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243#67—88no—
244C2×Dic3—24no—
245C2×Dic3—24no—
246C2×Dic3—22no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—81no—
254C4×D3—88no—
255C3×C8—81no—
256C3⋊C8—81no—
257C24—21no—
258C24—32no—
259C24—64no—
260C24—61no—
261C24—61no—
262C24—122no—
263C24—241no—
264C24D3×D411yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—124no—
273C2×D4—124no—
274C2×D4—242no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—32no—
278C22×C4—68no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—81no—
284C22×C4—122no—
285C22×C4—242no—
286C22×C4—242no—
287C22×C4C2×C4×D311yescharacteristic
288C22×C4#23511yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4#23711yescharacteristic
292C22×C4C2×#6711yescharacteristic
293M16—81no—
294M16—241no—
295C2×C8—41no—
296C2×C8—121no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—124no—
300C22⋊C4—24no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—68no—
304C22⋊C4—64no—
305C22⋊C4—64no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—62no—
310C22⋊C4—124no—
311C22⋊C4—124no—
312C42—81no—
313C42—241no—
314C42—241no—
315C42—242no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—44no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3#38611yescharacteristic
330C22×C3C2×#10412yes—
331C22×C3#45312yes—
332C2×D3—24no—
333C2×D3—44no—
334C2×D3—44no—
335C2×D3—42no—
336C2×D3—42no—
337C2×D3—44no—
338C2×D3—82no—
339C2×D3—84no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—44no—
345C3×C4—81no—
346Dic3—44no—
347Dic3—44no—
348Dic3—42no—
349C23—22no—
350C23—22no—
351C23—34no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—41no—
357C23—42no—
358C23—62no—
359C23—62no—
360C23—64no—
361C23—62no—
362C23—62no—
363C23—122no—
364C23—122no—
365C23—122no—
366C23—122no—
367C23—241no—
368C23—241no—
369C23—248no—
370C23D3×C22⋊C411yescharacteristic
371C23#88311yescharacteristic
372C23#93711yescharacteristic
373D4—81no—
374D4—81no—
375D4—82no—
376D4—82no—
377D4—248no—
378C2×C4—24no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—64no—
391C2×C4—64no—
392C2×C4—62no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—248no—
400C2×C4—241no—
401C2×C4#67⋊C411yescharacteristic
402C2×C4#88411yescharacteristic
403C2×C4C4×#6711yescharacteristic
404C2×C4#93811yescharacteristic
405C8—81no—
406C8—241no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—41no—
410C2×C3—161no—
411C2×C3#188711yescharacteristic
412C2×C3#189611yescharacteristic
413C2×C3#191311yescharacteristic
414D3—44no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—44no—
423C22—64no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—124no—
430C22—124no—
431C22—124no—
432C22—122no—
433C22—122no—
434C22—242no—
435C22—244no—
436C22D3×#10412yes—
437C22#495611yescharacteristic
438C22#519812yes—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—44no—
444C4—81no—
445C4—124no—
446C4—124no—
447C4—122no—
448C3#940911yescharacteristic, complemented, Sylow
449C2—22no—
450C2—22no—
451C2—41no—
452C2—124no—
453C2—161no—
454C2#7003411yescharacteristic
455C2#7032311yescharacteristic
456C2#7038211yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#386
prime radical#82708C2
cosocleC23×C3×C4C23
hypercenter#1507D3
hypocenterC3#9409
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #73560C2semidirect
3. C3×#1507C2semidirect
4. #71673C2semidirect
5. #69967C2semidirect
6. #67683C2semidirect
7. #67618C2semidirect
16. #5223C4semidirect
20. #4993C4semidirect
31. #1507D3semidirect
37. C22×C3×D4C2×C4semidirect
47. C2×C3×C4⋊C4C2×C4semidirect
72. C3×#95D4semidirect
74. C3⋊#95C2×C4semidirect
75. C3⋊#95D4semidirect
159. C22×D4C4×D3semidirect
172. C2×C4⋊C4C4×D3semidirect
198. #95#67semidirect
448. C3#9409semidirect

Maximal subgroups

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

subgroupindex
#827082
#735602
C3×#15072
#716732
#699672
#676832
#676182
#94093