Finite Group Databasebeta

← #10691874

Subgroups — #10691874

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3205C211yesmaximal, characteristic
2#83276C211yesmaximal, characteristic, complemented
3#71814C211yesmaximal, characteristic, complemented
4#71765C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic, complemented
6#67740C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9293—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×#948—21no—
12C3×#493C412yescomplemented
13C3×#486C412yescomplemented
14C3×#459—22no—
15C3×#447—22no—
16C3×#431C2211yescharacteristic
17#5234—21no—
18C2×#832—21no—
19C2×#832—21no—
20#5011—22no—
21#5011—22no—
22#4936C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C2×#803C2211yescharacteristic
26#4544—22no—
27#4537—22no—
28#3205D311yescharacteristic
29#2932—31no—
30#1961—31no—
31#1545—31no—
32#1517—31no—
33#1286—31no—
34#1275—31no—
35C2×C3×#41—21no—
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C2×C3×D8—24no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127—21no—
42C3×#127—21no—
43C3×#127—21no—
44C3×#127C2×C412yes—
45C3×#127C2×C411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121D412yes—
49C3×C4×D4—21no—
50C3×C4×D4D412yes—
51C2×C3×C42C2311yescharacteristic
52#948—42no—
53#948—42no—
54C2×C4⋊Dic3—21no—
55C2×#208—22no—
56C2×#208—21no—
57C22×C3⋊C8—22no—
58C22×C3⋊C8D411yescharacteristic
59C22×C3⋊C8D411yescharacteristic
60C3×C4⋊C8—21no—
61C3×#102—24no—
62C3×C22⋊C8—21no—
63C3×C4×C8C2×C412yes—
64#832—42no—
65#832—42no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68#830—24no—
69#830—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)D412yes—
73#803—21no—
74#803—21no—
75C4×C3⋊C8—21no—
76C2×#131—61no—
77#532C2×D311yescharacteristic
78C2×#127C2×D311yescharacteristic
79#493Dic312yescomplemented
80#486Dic312yescomplemented
81#459—22no—
82#447—22no—
83#435—62no—
84#435—62no—
85#431—31no—
86#431C2×D311yescharacteristic
87#423—31no—
88C2×#102—61no—
89C2×#102—61no—
90#408—61no—
91C2×#97—31no—
92#382—31no—
93#331—62no—
94#327—62no—
95C24×C3—41no—
96C3×#41—41no—
97C3×#41—41no—
98C3×#41—41no—
99C3×#41—41no—
100C2×C3×Q8—21no—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—21no—
104C2×C3×D4—22no—
105C2×C3×D4—21no—
106C2×C3×D4—22no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—82no—
115C2×C3×D4—82no—
116C2×C3×D4C22⋊C412yes—
117C22×C3×C4—22no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—41no—
122C3×D8—84no—
123C2×C3×C8—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4C22⋊C412yes—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—22no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134#208—44no—
135#208—41no—
136#208—41no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—44no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8D812yescomplemented
146C2×#41—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C2×D8—24no—
151#131—122no—
152#131—122no—
153C2×M16—62no—
154C2×M16—61no—
155C22×C8—31no—
156C22×C8—31no—
157C22×C8—62no—
158C4⋊Q8C2×Dic311yescharacteristic
159#127—21no—
160#127—21no—
161#127—21no—
162#127C2×Dic312yes—
163#127C2×Dic311yescharacteristic
164#124—21no—
165#121—21no—
166#121#6712yes—
167C4×D4—21no—
168C4×D4#6712yes—
169C2×C4⋊C4—61no—
170C2×C42C22×D311yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—32no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175#102—24no—
176#102—122no—
177#102—122no—
178C22⋊C8—21no—
179C22⋊C8—64no—
180C22⋊C8—61no—
181#97—61no—
182#97—61no—
183C4×C8—61no—
184C4×C8C2×Dic312yes—
185#95—61no—
186#95—61no—
187C23×C3—22no—
188C23×C3—42no—
189C23×C3—82no—
190C23×C3—82no—
191C23×C3—81no—
192C23×C3—81no—
193C23×C3#12111yescharacteristic
194C3×Q8—41no—
195C3×Q8—41no—
196C3×D4—42no—
197C3×D4—44no—
198C3×D4—42no—
199C3×D4—42no—
200C3×D4—41no—
201C3×D4—42no—
202C3×D4—41no—
203C3×D4—42no—
204C3×D4—81no—
205C3×D4—82no—
206C3×D4—82no—
207C3×D4—82no—
208C3×D4—81no—
209C3×D4—81no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×C3×C4C2×D811yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228C3×C8—42no—
229C3⋊C8—22no—
230C3⋊C8—42no—
231C3⋊C8—41no—
232C3⋊C8—81no—
233C24—41no—
234#41—41no—
235#41—41no—
236#41—41no—
237#41—41no—
238C2×Q8—21no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4#21712yes—
255C22×C4—22no—
256C22×C4—121no—
257C22×C4D3×D411yescharacteristic
258C22×C4#23711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260D8—84no—
261M16—124no—
262M16—121no—
263M16—121no—
264C2×C8—22no—
265C2×C8—32no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—62no—
272C2×C8—124no—
273C2×C8—121no—
274C4⋊C4—21no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4#21712yes—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C42D3×D411yescharacteristic
281C42#23711yescharacteristic
282C42C22×Dic311yescharacteristic
283C42C2×#6711yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—42no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—82no—
292C22×C3—162no—
293C22×C3—162no—
294C22×C3#38611yescharacteristic
295C22×C3#45911yescharacteristic
296C22×C3#46311yescharacteristic
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—42no—
302C3×C4C4×D811yescharacteristic
303C3×C4#44211yescharacteristic
304C3×C4#44711yescharacteristic
305C3×C4#44911yescharacteristic
306Dic3—161no—
307C23—22no—
308C23—42no—
309C23—82no—
310C23—82no—
311C23—81no—
312C23—81no—
313C23#93911yescharacteristic
314Q8—41no—
315Q8—41no—
316D4—42no—
317D4—44no—
318D4—42no—
319D4—42no—
320D4—41no—
321D4—42no—
322D4—41no—
323D4—42no—
324D4—81no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—81no—
329D4—81no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—241no—
336C2×C4—241no—
337C2×C4—241no—
338C2×C4D3×D811yescharacteristic
339C2×C4#91111yescharacteristic
340C2×C4#91111yescharacteristic
341C2×C4#91211yescharacteristic
342C2×C4D4×Dic311yescharacteristic
343C2×C4D4×Dic311yescharacteristic
344C2×C4#93511yescharacteristic
345C2×C4#93711yescharacteristic
346C2×C4#93911yescharacteristic
347C2×C4C2×#21711yescharacteristic
348C8—42no—
349C8—62no—
350C8—122no—
351C8—121no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—82no—
356C2×C3—162no—
357C2×C3#187911yescharacteristic
358C2×C3#190311yescharacteristic
359C2×C3#191011yescharacteristic
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—162no—
369C22—162no—
370C22#515011yescharacteristic
371C22#515311yescharacteristic
372C22#522011yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—481no—
379C4Dic3×D811yescharacteristic
380C4#515211yescharacteristic
381C4#515611yescharacteristic
382C4#515811yescharacteristic
383C3#929311yescharacteristic, complemented, Sylow
384C2—21no—
385C2—21no—
386C2—82no—
387C2—162no—
388C2#7185411yescharacteristic
389C2#7203711yescharacteristic
390C2#7209011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3205C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3205D3
hypocenterC3#9293
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
2. #83276C2semidirect
3. #71814C2semidirect
4. #71765C2semidirect
5. #71759C2semidirect
6. #67740C2semidirect
7. #67738C2semidirect
12. C3×#493C4semidirect
13. C3×#486C4semidirect
79. #493Dic3semidirect
80. #486Dic3semidirect
145. C2×C3⋊C8D8semidirect
383. C3#9293semidirect

Maximal subgroups

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

subgroupindex
C3×#32052
#832762
#718142
#717652
#717592
#677402
#677382
#92933