Finite Group Databasebeta

← #10691171

Subgroups — #10691171

Order 768.

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

normal subgroups only

This group has 560 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×#2631C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3#71718C212yesmaximal, complemented
4C4×#833C211yesmaximal, characteristic, complemented
5#67633C212yesmaximal, complemented
6#9187—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#397—21no—
10C3×#393C411yescharacteristic
11C3×#393C411yescharacteristic
12C3×#392—21no—
13C3×#392C411yescharacteristic
14C3×#392C411yescharacteristic
15C3×#386—21no—
16C3×#386—21no—
17C3×#383C2211yescharacteristic
18C2×#833—21no—
19#5207—21no—
20C2×#830—21no—
21#5011—24no—
22#4926C2212yes—
23C4×#208C2212yes—
24#4540—24no—
25#4530—21no—
26#2973—31no—
27#2631D311yescharacteristic
28C4×#100—31no—
29#1527—32no—
30#1288—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4—21no—
33C22×C3×D4—21no—
34C22×C3×D4C2×C411yescharacteristic
35C22×C3×D4D411yescharacteristic
36C23×C3×C4D411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C3×#124—22no—
39C3×#124—22no—
40C3×C4×D4—22no—
41C3×C4×D4—22no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4C2×C411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×#208—22no—
53C2×#208—22no—
54C22×C3⋊C8—22no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95C2×C411yescharacteristic
60C3×#95C2×C411yescharacteristic
61#833—24no—
62#830—22no—
63#830—22no—
64#830—22no—
65C4⋊(C3⋊C8)—22no—
66#803—22no—
67#803D414yescomplemented
68C4×C3⋊C8—22no—
69C2×#124C2×D311yescharacteristic
70C2×C4×D4C2×D311yescharacteristic
71#435—64no—
72#432—32no—
73C2×#100—61no—
74#407—61no—
75C2×C22⋊C8—61no—
76C4×M16—32no—
77#397—21no—
78#393Dic311yescharacteristic
79#393Dic311yescharacteristic
80#392—21no—
81#392Dic311yescharacteristic
82#392Dic311yescharacteristic
83#386—21no—
84#386—21no—
85#383C2×D311yescharacteristic
86#343—61no—
87#329—64no—
88C24×C3—21no—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C22⋊C411yescharacteristic
91C2×C3×Q8—22no—
92C2×C3×Q8—81no—
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C2×C3×D4—22no—
96C2×C3×D4—41no—
97C2×C3×D4—42no—
98C2×C3×D4—42no—
99C2×C3×D4—81no—
100C2×C3×D4—81no—
101C2×C3×D4—81no—
102C2×C3×D4—81no—
103C2×C3×D4—81no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—81no—
112C22×C3×C4—81no—
113C22×C3×C4C22⋊C411yescharacteristic
114C22×C3×C4C22⋊C411yescharacteristic
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C3×C4⋊C4—22no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—42no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—81no—
125C3×C22⋊C4—81no—
126C3×C22⋊C4—81no—
127C3×C22⋊C4—81no—
128C3×C42C2×D412yes—
129C3×C42#4112yes—
130#208—24no—
131#208—42no—
132#208—82no—
133C2×C3⋊C8—24no—
134C2×C3⋊C8—42no—
135C2×C3⋊C8—82no—
136C22×Q8C2×Dic311yescharacteristic
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C2×Dic311yescharacteristic
140C22×D4#6711yescharacteristic
141C23×C4#6711yescharacteristic
142C23×C4#6711yescharacteristic
143C2×M16—62no—
144C2×M16—62no—
145C22×C8—62no—
146#124—22no—
147#124—22no—
148C4×D4—22no—
149C4×D4—22no—
150C2×C4⋊C4#6711yescharacteristic
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4C2×Dic311yescharacteristic
158C2×C22⋊C4C2×Dic311yescharacteristic
159C2×C42C22×D311yescharacteristic
160C4⋊C8—62no—
161#100—64no—
162C22⋊C8—62no—
163C22⋊C8—62no—
164C22⋊C8—62no—
165#97—34no—
166#97—62no—
167C4×C8—62no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95C2×Dic311yescharacteristic
173#95C2×Dic311yescharacteristic
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3—41no—
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×C3#10412yes—
191C23×C3C2×C22⋊C411yescharacteristic
192C3×Q8—82no—
193C3×D4—44no—
194C3×D4—82no—
195C3×D4—84no—
196C3×D4—84no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
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—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4C4×D412yes—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12112yes—
221C2×C3×C4#12311yescharacteristic
222C3⋊C8—44no—
223C3⋊C8—44no—
224C24—21no—
225C24#21711yescharacteristic
226C24#21711yescharacteristic
227C2×Q8—22no—
228C2×Q8—81no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—81no—
236C2×D4—81no—
237C2×D4—81no—
238C2×D4—81no—
239C2×D4—81no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4#21711yescharacteristic
250C22×C4#21711yescharacteristic
251C22×C4C22×Dic311yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254M16—64no—
255M16—122no—
256M16—242no—
257C2×C8—64no—
258C2×C8—122no—
259C2×C8—242no—
260C4⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—81no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42D3×D412yes—
271C42#23712yes—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—22no—
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#38611yescharacteristic
292C22×C3C2×#10411yescharacteristic
293C22×C3#42111yescharacteristic
294C3×C4—22no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—81no—
301C3×C4—81no—
302C3×C4—81no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23#83712yes—
320C23C2×#21711yescharacteristic
321Q8—82no—
322D4—44no—
323D4—82no—
324D4—84no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4D4×Dic312yes—
348C2×C4#93511yescharacteristic
349C2×C4#93711yescharacteristic
350C2×C4#93912yes—
351C8—124no—
352C8—124no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—42no—
356C2×C3—81no—
357C2×C3—161no—
358C2×C3#187111yescharacteristic
359C2×C3#191111yescharacteristic
360C2×C3#191411yescharacteristic
361C22—21no—
362C22—21no—
363C22—22no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22#522011yescharacteristic
381C22C2×#83711yescharacteristic
382C22#523611yescharacteristic
383C4—22no—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C3#918711yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—42no—
396C2—81no—
397C2—161no—
398C2#7171311yescharacteristic
399C2#7185311yescharacteristic
400C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2631C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#2631D3
hypocenterC3#9187
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2631

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. #71718C2semidirect
4. C4×#833C2semidirect
5. #67633C2semidirect
67. #803D4semidirect
392. C3#9187semidirect

Maximal subgroups

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

subgroupindex
C3×#26312
#823022
#717182
C4×#8332
#676332
#91873