Finite Group Databasebeta

← #10692897

Subgroups — #10692897

Order 768.

690 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2541C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3#71760C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic, complemented
5#67621C212yesmaximal, complemented
6#9457—31nomaximal, Sylow
7C2×C3×#127C412yes—
8C2×C3×#127C2211yescharacteristic
9C2×C3×#121—21no—
10C2×C3×#121—21no—
11C2×C3×C4×D4C2211yescharacteristic
12C3×#392—21no—
13C3×#392—21no—
14C3×#390C412yes—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5207—21no—
17C2×#830—21no—
18#5011—24no—
19#4926C2212yes—
20#4926C2212yes—
21#4544—24no—
22#4525—21no—
23#4525—21no—
24#2973—31no—
25#2541D311yescharacteristic
26#1555—32no—
27#1466—31no—
28#1285—32no—
29C22×C3×D4—22no—
30C22×C3×D4—22no—
31C22×C3×D4—21no—
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C22×C3×D4C2×C412yes—
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D411yescharacteristic
37C3×#127—22no—
38C3×#127—22no—
39C3×#127—44no—
40C3×#121—44no—
41C3×#121—44no—
42C3×C4×D4—22no—
43C3×C4×D4—22no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4D411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C42C2×C412yes—
51C2×C3×C42C2311yescharacteristic
52C2×#208—22no—
53C22×C3⋊C8—22no—
54C3×#95—21no—
55#830—22no—
56#830—22no—
57#830—22no—
58#830—22no—
59C4⋊(C3⋊C8)—22no—
60C4⋊(C3⋊C8)D414yescomplemented
61#803—22no—
62#803—22no—
63C2×#127Dic312yes—
64C2×#127C2×D311yescharacteristic
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4C2×D311yescharacteristic
68#435—64no—
69#432—32no—
70#432—32no—
71#407—61no—
72C2×C22⋊C8—61no—
73#392—21no—
74#392—21no—
75#390Dic312yes—
76C4×C4⋊C4C2×D311yescharacteristic
77#331—64no—
78#323—61no—
79#323—61no—
80C24×C3—22no—
81C24×C3C22⋊C411yescharacteristic
82C24×C3C22⋊C411yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—24no—
86C2×C3×D4—41no—
87C2×C3×D4—42no—
88C2×C3×D4—42no—
89C2×C3×D4—42no—
90C2×C3×D4—42no—
91C2×C3×D4—44no—
92C2×C3×D4—44no—
93C2×C3×D4—44no—
94C2×C3×D4—44no—
95C2×C3×D4—41no—
96C2×C3×D4—42no—
97C2×C3×D4—41no—
98C2×C3×D4—82no—
99C2×C3×D4—82no—
100C2×C3×D4—82no—
101C22×C3×C4—22no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—42no—
107C22×C3×C4—42no—
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C3×C4⋊C4—24no—
114C3×C4⋊C4—22no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—44no—
118C3×C22⋊C4—44no—
119C3×C42—42no—
120C3×C42C2×D412yes—
121C3×C42#4112yes—
122#208—82no—
123C2×C3⋊C8—24no—
124C2×C3⋊C8—42no—
125C2×C3⋊C8—82no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4C2×Dic311yescharacteristic
131C22×D4C2×Dic312yes—
132C23×C4#6711yescharacteristic
133C23×C4#6711yescharacteristic
134C2×M16—62no—
135C22×C8—62no—
136#127—22no—
137#127—22no—
138#127—44no—
139#121—44no—
140#121—44no—
141C4×D4—22no—
142C4×D4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4C2×Dic311yescharacteristic
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4#6711yescharacteristic
148C2×C22⋊C4#6711yescharacteristic
149C2×C42C2×Dic312yes—
150C2×C42C22×D311yescharacteristic
151C4⋊C8—34no—
152C4⋊C8—62no—
153C22⋊C8—62no—
154C22⋊C8—62no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157#97—62no—
158#97—62no—
159#95—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—42no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3#10012yes—
177C23×C3C2×C22⋊C411yescharacteristic
178C3×D4—44no—
179C3×D4—84no—
180C3×D4—82no—
181C3×D4—84no—
182C3×D4—88no—
183C3×D4—88no—
184C3×D4—82no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—22no—
188C2×C3×C4—24no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4C4×D412yes—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12112yes—
203C2×C3×C4#12311yescharacteristic
204C3⋊C8—44no—
205C3⋊C8—82no—
206C24—22no—
207C24#21711yescharacteristic
208C24#21711yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—82no—
225C2×D4—82no—
226C2×D4—82no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4#21711yescharacteristic
235C22×C4#21711yescharacteristic
236C22×C4C22×Dic311yescharacteristic
237C22×C4C2×#6711yescharacteristic
238C22×C4C2×#6711yescharacteristic
239M16—242no—
240C2×C8—64no—
241C2×C8—122no—
242C2×C8—242no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—42no—
250C42D3×D412yes—
251C42#23712yes—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3—82no—
265C22×C3—82no—
266C22×C3—82no—
267C22×C3—82no—
268C22×C3—82no—
269C22×C3—82no—
270C22×C3—82no—
271C22×C3#38611yescharacteristic
272C22×C3C2×#10011yescharacteristic
273C22×C3#41311yescharacteristic
274C3×C4—21no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—44no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4#45312yes—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23#83312yes—
298C23C2×#21711yescharacteristic
299D4—44no—
300D4—84no—
301D4—82no—
302D4—84no—
303D4—88no—
304D4—88no—
305D4—82no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4D4×Dic312yes—
322C2×C4#93511yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93912yes—
325C8—124no—
326C8—242no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—42no—
330C2×C3—81no—
331C2×C3—162no—
332C2×C3#188711yescharacteristic
333C2×C3#190011yescharacteristic
334C2×C3#191011yescharacteristic
335C22—21no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22C2×#83311yescharacteristic
355C22#522011yescharacteristic
356C22#523711yescharacteristic
357C4—22no—
358C4—21no—
359C4—22no—
360C4—44no—
361C4—41no—
362C4—41no—
363C4#507712yes—
364C3#945711yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—162no—
370C2#7168311yescharacteristic
371C2#7176311yescharacteristic
372C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2541C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#2541D3
hypocenterC3#9457
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. #71760C2semidirect
4. #71603C2semidirect
5. #67621C2semidirect
60. C4⋊(C3⋊C8)D4semidirect
364. C3#9457semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
#823022
#717602
#716032
#676212
#94573