Finite Group Databasebeta

← #10692928

Subgroups — #10692928

Order 768.

610 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 93 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#82664C211yesmaximal, characteristic
2#74054C211yesmaximal, characteristic, complemented
3C3×#1510C211yesmaximal, characteristic
4#71674C211yesmaximal, characteristic
5#69973C211yesmaximal, characteristic, complemented
6#67682C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9464—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#935—21no—
11C2×#935—21no—
12C22×C4⋊Dic3C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×#381C2211yescharacteristic
15#5221C412yescomplemented
16#5211C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5207C2211yescharacteristic
19#4975C412yescomplemented
20#4949—22no—
21#4942—21no—
22Dic3×C22⋊C4—21no—
23#4654—22no—
24#4653—22no—
25C3×#327—22no—
26#4554—22no—
27#4537—22no—
28#2501—31no—
29#2313—31no—
30#1821—31no—
31#1510—31no—
32#1510D311yescharacteristic
33#1284—31no—
34#1284—31no—
35C22×C3×D4—21no—
36C22×C3×D4C2×C411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3D411yescharacteristic
39C23×Dic3D411yescharacteristic
40C2×C3×M16—21no—
41C3×#121—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×#121—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—22no—
51C2×#217—22no—
52#935—44no—
53#935—44no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3C2×C412yes—
59C2×C4⋊Dic3D412yes—
60C2×Dic3⋊C4—22no—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4—22no—
63C2×C4×Dic3—22no—
64C2×C4×Dic3C2×C412yes—
65C2×#208—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3×#95D412yes—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77C2×#123—61no—
78C2×#123—61no—
79C2×#121C2×D311yescharacteristic
80C22×C4⋊C4—31no—
81#407—31no—
82#407C2×D311yescharacteristic
83#399—62no—
84#390—32no—
85#387—62no—
86#387—62no—
87#384—32no—
88#381—31no—
89#381C2×D311yescharacteristic
90#380—61no—
91C4×C22⋊C4—31no—
92C4×C22⋊C4—61no—
93#343—62no—
94#327—22no—
95#327—62no—
96C24×C3—21no—
97C24×C3C2×D411yescharacteristic
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—42no—
101C2×C3×D4—42no—
102C2×C3×D4—42no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—42no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—22no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—42no—
121C22×Dic3—42no—
122C22×Dic3C22⋊C412yes—
123C22×Dic3C22⋊C412yes—
124C3×M16—42no—
125C2×C3×C8—41no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C22⋊C4—42no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131#217—24no—
132#217—44no—
133#217—44no—
134#217—42no—
135C4⋊Dic3—22no—
136C4⋊Dic3—22no—
137C4⋊Dic3—22no—
138C4⋊Dic3—22no—
139Dic3⋊C4—44no—
140Dic3⋊C4—44no—
141Dic3⋊C4—82no—
142C4×Dic3—42no—
143C4×Dic3—82no—
144#208—42no—
145C2×C3⋊C8—41no—
146C22×D4—21no—
147C22×D4C4×D311yescharacteristic
148C23×C4—31no—
149C23×C4—31no—
150C23×C4C22×D311yescharacteristic
151C2×M16—21no—
152C2×M16—61no—
153#123—124no—
154#123—124no—
155#121—21no—
156#121—21no—
157#121—21no—
158#121—21no—
159C2×C4⋊C4—32no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—62no—
166C2×C4⋊C4—62no—
167C2×C4⋊C4—62no—
168C2×C4⋊C4C4×D311yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—61no—
175C2×C42—32no—
176C2×C42—62no—
177C22⋊C8—21no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180C22⋊C8—61no—
181#95—21no—
182#95—32no—
183#95—62no—
184#95—62no—
185#95—61no—
186#95—61no—
187#95#6712yes—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—42no—
194C23×C3—41no—
195C23×C3—42no—
196C23×C3C2×C22⋊C411yescharacteristic
197C23×C3#12011yescharacteristic
198C23×C3#12311yescharacteristic
199C3×D4—81no—
200C3×D4—81no—
201C3×D4—82no—
202C3×D4—82no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×Dic3—24no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C3×C8—81no—
234C3⋊C8—81no—
235C24—21no—
236C24D3×D411yescharacteristic
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—32no—
247C22×C4—32no—
248C22×C4—42no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—122no—
256C22×C4—122no—
257C22×C4C2×C4×D311yescharacteristic
258C22×C4#23511yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4#23711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263M16—42no—
264M16—122no—
265C2×C8—41no—
266C2×C8—121no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—124no—
274C4⋊C4—124no—
275C4⋊C4—242no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—64no—
280C22⋊C4—124no—
281C22⋊C4—124no—
282C22⋊C4—122no—
283C42—122no—
284C42—242no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—44no—
293C22×C3—82no—
294C22×C3—82no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3#38611yescharacteristic
299C22×C3#41012yes—
300C22×C3#45312yes—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—82no—
306C3×C4—81no—
307Dic3—44no—
308Dic3—42no—
309Dic3—42no—
310Dic3—82no—
311C23—22no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—41no—
318C23—42no—
319C23D3×C22⋊C411yescharacteristic
320C23#88311yescharacteristic
321C23#93711yescharacteristic
322D4—81no—
323D4—81no—
324D4—82no—
325D4—82no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—64no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4#67⋊C411yescharacteristic
353C2×C4#88411yescharacteristic
354C2×C4C4×#6711yescharacteristic
355C2×C4#93811yescharacteristic
356C8—81no—
357C8—241no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—41no—
361C2×C3—161no—
362C2×C3#188711yescharacteristic
363C2×C3#189911yescharacteristic
364C2×C3#191211yescharacteristic
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—44no—
373C22—82no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#474012yes—
379C22#495611yescharacteristic
380C22#519812yes—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—81no—
387C4—124no—
388C4—122no—
389C4—122no—
390C4—242no—
391C3#946411yescharacteristic, complemented, Sylow
392C2—22no—
393C2—22no—
394C2—41no—
395C2—161no—
396C2#7002911yescharacteristic
397C2#7032311yescharacteristic
398C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1510C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1510D3
hypocenterC3#9464
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #74054C2semidirect
5. #69973C2semidirect
6. #67682C2semidirect
7. #67628C2semidirect
15. #5221C4semidirect
19. #4975C4semidirect
391. C3#9464semidirect

Maximal subgroups

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

subgroupindex
#826642
#740542
C3×#15102
#716742
#699732
#676822
#676282
#94643