Finite Group Databasebeta

← #10691161

Subgroups — #10691161

Order 768.

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

normal subgroups only

This group has 582 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×#2635C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3#71673C212yesmaximal, complemented
4C4×#834C211yesmaximal, characteristic, complemented
5#67628C212yesmaximal, complemented
6#9185—31nomaximal, Sylow
7C2×C3×#121C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#400C411yescharacteristic, complemented
10C3×#399C411yescharacteristic, complemented
11C3×#394—21no—
12C3×#394C411yescharacteristic, complemented
13C3×#392—21no—
14C3×#392C411yescharacteristic, complemented
15C3×#388—21no—
16C3×#386—21no—
17C3×#381C2211yescharacteristic
18C2×#834—21no—
19C4×#217C2212yes—
20#5207C2212yes—
21C2×#830—21no—
22#5011—24no—
23#4926—21no—
24#4537—24no—
25#4526—21no—
26#2973—31no—
27#2635D311yescharacteristic
28C4×#99—31no—
29#1507—32no—
30#1284—32no—
31C22×C3×D4—21no—
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C22×C3×D4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C3×#121—22no—
38C3×#121—22no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
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⋊C4—21no—
50C2×C3×C22⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4D411yescharacteristic
52C2×C3×C42D411yescharacteristic
53C2×#217—22no—
54C2×C4×Dic3—22no—
55C2×#208—22no—
56C22×C3⋊C8—22no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95C2×C411yescharacteristic
61C3×#95C2×C411yescharacteristic
62#834—24no—
63C3⋊#95—22no—
64#830—22no—
65#830—22no—
66#830D414yescomplemented
67C4⋊(C3⋊C8)—22no—
68#803—22no—
69C2×#121C2×D311yescharacteristic
70C2×C4×D4C2×D311yescharacteristic
71#435—64no—
72#432—61no—
73C2×#99—61no—
74#407—32no—
75C2×C22⋊C8—61no—
76#400Dic311yescharacteristic, complemented
77#399Dic311yescharacteristic, complemented
78#394—21no—
79#394Dic311yescharacteristic, complemented
80#392—21no—
81#392Dic311yescharacteristic, complemented
82#388—21no—
83#386—21no—
84#381C2×D311yescharacteristic
85C4×C22⋊C4—32no—
86#342—61no—
87#327—64no—
88C24×C3—21no—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C2×D411yescharacteristic
91C2×C3×D4—22no—
92C2×C3×D4—22no—
93C2×C3×D4—22no—
94C2×C3×D4—41no—
95C2×C3×D4—42no—
96C2×C3×D4—42no—
97C2×C3×D4—81no—
98C2×C3×D4—81no—
99C2×C3×D4—81no—
100C2×C3×D4—81no—
101C2×C3×D4—81no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—81no—
108C22×C3×C4—81no—
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C22⋊C411yescharacteristic
111C22×C3×C4C22⋊C411yescharacteristic
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4112yes—
116C22×Dic3—42no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—81no—
120C3×C22⋊C4—22no—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—42no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—81no—
126C3×C22⋊C4—81no—
127C3×C22⋊C4—81no—
128C3×C22⋊C4—81no—
129C3×C42—21no—
130C3×C42—21no—
131#217—24no—
132#217—42no—
133C4×Dic3—82no—
134#208—82no—
135C2×C3⋊C8—24no—
136C2×C3⋊C8—82no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C2×Dic311yescharacteristic
140C22×D4#6711yescharacteristic
141C23×C4#6711yescharacteristic
142C23×C4C22×D311yescharacteristic
143C2×M16—62no—
144C22×C8—62no—
145#121—22no—
146#121—22no—
147C4×D4—22no—
148C4×D4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4C2×Dic311yescharacteristic
151C2×C4⋊C4C2×Dic311yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—62no—
159C2×C22⋊C4C2×Dic311yescharacteristic
160C2×C22⋊C4#6711yescharacteristic
161C2×C42—62no—
162C2×C42#6711yescharacteristic
163C4⋊C8—62no—
164#99—64no—
165C22⋊C8—34no—
166C22⋊C8—62no—
167C22⋊C8—62no—
168#97—62no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—62no—
173#95C2×Dic311yescharacteristic
174#95C2×Dic311yescharacteristic
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
192C23×C3#12112yes—
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—21no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
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—41no—
217C2×C3×C4—41no—
218C2×C3×C4—81no—
219C2×C3×C4C4×D412yes—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12311yescharacteristic
222C2×Dic3—44no—
223C2×Dic3—82no—
224C2×Dic3—82no—
225C3⋊C8—44no—
226C24—21no—
227C24#21711yescharacteristic
228C24C2×#6711yescharacteristic
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—81no—
246C22×C4—81no—
247C22×C4—122no—
248C22×C4#21711yescharacteristic
249C22×C4#21711yescharacteristic
250C22×C4#21711yescharacteristic
251C22×C4D3×D412yes—
252C22×C4#23712yes—
253C22×C4C22×Dic311yescharacteristic
254C22×C4C2×#6711yescharacteristic
255M16—242no—
256C2×C8—64no—
257C2×C8—242no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—81no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—64no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—122no—
272C42—21no—
273C42—21no—
274C42—242no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—42no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3#38611yescharacteristic
295C22×C3C2×#10411yescharacteristic
296C22×C3#42111yescharacteristic
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—22no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4—81no—
306Dic3—84no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23#83712yes—
323C23#93912yes—
324C23C2×#21711yescharacteristic
325D4—44no—
326D4—82no—
327D4—84no—
328D4—84no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—124no—
352C2×C4—242no—
353C2×C4—242no—
354C2×C4D4×Dic312yes—
355C2×C4#93511yescharacteristic
356C2×C4#93711yescharacteristic
357C8—124no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—42no—
361C2×C3—42no—
362C2×C3—161no—
363C2×C3#187111yescharacteristic
364C2×C3#191011yescharacteristic
365C2×C3#191311yescharacteristic
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#522011yescharacteristic
386C22C2×#83711yescharacteristic
387C22#523611yescharacteristic
388C4—21no—
389C4—21no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—244no—
398C3#918511yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—161no—
404C2#7171311yescharacteristic
405C2#7184811yescharacteristic
406C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2635C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2635D3
hypocenterC3#9185
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2635

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. #71673C2semidirect
4. C4×#834C2semidirect
5. #67628C2semidirect
9. C3×#400C4semidirect
10. C3×#399C4semidirect
12. C3×#394C4semidirect
14. C3×#392C4semidirect
66. #830D4semidirect
76. #400Dic3semidirect
77. #399Dic3semidirect
79. #394Dic3semidirect
81. #392Dic3semidirect
398. C3#9185semidirect

Maximal subgroups

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

subgroupindex
C3×#26352
#823022
#716732
C4×#8342
#676282
#91853