Finite Group Databasebeta

← #10692847

Subgroups — #10692847

Order 768.

568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#2544C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3#71719C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic, complemented
5#67621C212yesmaximal, complemented
6#9447—31nomaximal, Sylow
7C2×C3×#127C411yescharacteristic
8C2×C3×#124C411yescharacteristic
9C2×C3×#124C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C3×#394—21no—
12C3×#394—21no—
13C3×#391C411yescharacteristic
14C3×#390C411yescharacteristic
15C3×#386—21no—
16C3×#386—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5207—21no—
19C2×#830—21no—
20#5011—24no—
21#4926C2212yes—
22#4926C2212yes—
23#4540—24no—
24#4525—21no—
25#4525—21no—
26#2973—31no—
27#2544D311yescharacteristic
28#1533—32no—
29#1466—31no—
30#1285—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4—21no—
33C22×C3×D4—21no—
34C22×C3×D4C2×C411yescharacteristic
35C22×C3×D4C2×C411yescharacteristic
36C23×C3×C4D411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C3×#127—42no—
39C3×#124—22no—
40C3×#124—22no—
41C3×#124—42no—
42C3×C4×D4—22no—
43C3×C4×D4—22no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C22⋊C4D411yescharacteristic
54C2×C3×C42C2×C411yescharacteristic
55C2×C3×C42C2×C411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×#208—22no—
58C22×C3⋊C8—22no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62#830—22no—
63#830—22no—
64#830—22no—
65#830—22no—
66C4⋊(C3⋊C8)—22no—
67C4⋊(C3⋊C8)—22no—
68#803—22no—
69#803D414yescomplemented
70C2×#127Dic311yescharacteristic
71C2×#124Dic311yescharacteristic
72C2×#124C2×D311yescharacteristic
73C2×C4×D4C2×D311yescharacteristic
74#435—64no—
75#432—32no—
76#432—32no—
77#407—61no—
78C2×C22⋊C8—61no—
79#394—21no—
80#394—21no—
81#391Dic311yescharacteristic
82#390Dic311yescharacteristic
83#386—21no—
84#386—21no—
85C4×C4⋊C4C2×D311yescharacteristic
86#329—64no—
87#323—61no—
88#323—61no—
89C24×C3—21no—
90C24×C3C22⋊C411yescharacteristic
91C24×C3C22⋊C411yescharacteristic
92C2×C3×Q8—22no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×D4—22no—
96C2×C3×D4—22no—
97C2×C3×D4—22no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—42no—
102C2×C3×D4—42no—
103C2×C3×D4—42no—
104C2×C3×D4—42no—
105C2×C3×D4—81no—
106C2×C3×D4—81no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—81no—
115C22×C3×C4—81no—
116C22×C3×C4C22⋊C411yescharacteristic
117C22×C3×C4C22⋊C411yescharacteristic
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C3×C4⋊C4—22no—
122C3×C4⋊C4—42no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—42no—
126C3×C22⋊C4—42no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—42no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—81no—
131C3×C22⋊C4—81no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42C2×D412yes—
135C3×C42#4112yes—
136#208—82no—
137C2×C3⋊C8—24no—
138C2×C3⋊C8—42no—
139C2×C3⋊C8—82no—
140C22×Q8C2×Dic311yescharacteristic
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C2×Dic311yescharacteristic
144C22×D4C2×Dic311yescharacteristic
145C23×C4#6711yescharacteristic
146C23×C4#6711yescharacteristic
147C2×M16—62no—
148C22×C8—62no—
149#127—42no—
150#124—22no—
151#124—22no—
152#124—42no—
153C4×D4—22no—
154C4×D4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C2×Dic311yescharacteristic
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4#6711yescharacteristic
164C2×C22⋊C4#6711yescharacteristic
165C2×C42C2×Dic311yescharacteristic
166C2×C42C2×Dic311yescharacteristic
167C2×C42C22×D311yescharacteristic
168C4⋊C8—62no—
169C4⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—62no—
172C22⋊C8—62no—
173C22⋊C8—62no—
174#97—34no—
175#97—62no—
176#95—21no—
177#95—21no—
178#95—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3#10012yes—
196C23×C3C2×C22⋊C411yescharacteristic
197C3×Q8—82no—
198C3×D4—44no—
199C3×D4—82no—
200C3×D4—84no—
201C3×D4—84no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4C4×D412yes—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12112yes—
226C2×C3×C4#12311yescharacteristic
227C3⋊C8—44no—
228C3⋊C8—82no—
229C24—21no—
230C24#21711yescharacteristic
231C24#21711yescharacteristic
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—81no—
246C2×D4—81no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4#21711yescharacteristic
257C22×C4#21711yescharacteristic
258C22×C4C22×Dic311yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261M16—242no—
262C2×C8—64no—
263C2×C8—122no—
264C2×C8—242no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—41no—
277C42—41no—
278C42D3×D412yes—
279C42#23712yes—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3#38611yescharacteristic
300C22×C3C2×#10011yescharacteristic
301C22×C3#41311yescharacteristic
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—42no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—81no—
310C3×C4—81no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23#83312yes—
328C23C2×#21711yescharacteristic
329Q8—82no—
330D4—44no—
331D4—82no—
332D4—84no—
333D4—84no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4D4×Dic312yes—
356C2×C4#93511yescharacteristic
357C2×C4#93711yescharacteristic
358C2×C4#93912yes—
359C8—124no—
360C8—242no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—42no—
364C2×C3—81no—
365C2×C3—161no—
366C2×C3#188411yescharacteristic
367C2×C3#190011yescharacteristic
368C2×C3#191111yescharacteristic
369C22—21no—
370C22—21no—
371C22—22no—
372C22—42no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22C2×#83311yescharacteristic
389C22#522011yescharacteristic
390C22#523711yescharacteristic
391C4—22no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C3#944711yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—42no—
404C2—81no—
405C2—161no—
406C2#7168311yescharacteristic
407C2#7181711yescharacteristic
408C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2544C2
socleC22×C3#386
prime radicalC2×C3×#127C4
cosocleC2×C3×C42C23
hypercenter#2544D3
hypocenterC3#9447
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2544

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. #71719C2semidirect
4. #71603C2semidirect
5. #67621C2semidirect
69. #803D4semidirect
400. C3#9447semidirect

Maximal subgroups

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

subgroupindex
C3×#25442
#823022
#717192
#716032
#676212
#94473