Finite Group Databasebeta

← #10690614

Subgroups — #10690614

Order 768.

564 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#442C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3#71601C211yesmaximal, characteristic, complemented
4#71597C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#67661C211yesmaximal, characteristic, complemented
7#67660C211yesmaximal, characteristic, complemented
8#9089—31nomaximal, Sylow
9C22×C3×D8C411yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#442—22no—
13C3×#442—22no—
14C2×C3×#106C411yescharacteristic
15C2×C3×#102C411yescharacteristic
16C2×C3×#102C411yescharacteristic
17C2×C3×#97C2211yescharacteristic
18C2×#830—22no—
19D4×C3⋊C8—22no—
20D4×C3⋊C8—22no—
21#4925C2211yescharacteristic
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25D4⋊(C3⋊C8)—22no—
26D4⋊(C3⋊C8)—22no—
27#4525—22no—
28#2972—31no—
29C2×#442D311yescharacteristic
30#1478—31no—
31#1478—31no—
32#1460—31no—
33#1281—31no—
34#1275—31no—
35C22×C3×D4C2×C411yescharacteristic
36C22×C3×D4C2×C411yescharacteristic
37C23×C3×C4—21no—
38C23×C3×C4—21no—
39C2×C3×D8—22no—
40C2×C3×D8—22no—
41C2×C3×D8C814yescomplemented
42C22×C3×C8C2×C411yescharacteristic
43C22×C3×C8C2×C411yescharacteristic
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4D412yes—
51C3×C4×D4D412yes—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C4⋊C4C2×C411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C22×C3⋊C8—24no—
58C3×#106—22no—
59C3×#102—22no—
60C3×#102—22no—
61C3×#97—21no—
62C3×#97—21no—
63#830—24no—
64#830—21no—
65#830—21no—
66#830—21no—
67#830D412yes—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C22×D8Dic311yescharacteristic
77C2×C4×D4C2×D311yescharacteristic
78C2×C4×D4C2×D311yescharacteristic
79#442—22no—
80#442—22no—
81C8×D4—62no—
82C8×D4—62no—
83#431—31no—
84#431—31no—
85#431—31no—
86#431—31no—
87C2×#106Dic311yescharacteristic
88C2×#102Dic311yescharacteristic
89C2×#102Dic311yescharacteristic
90C2×C22⋊C8—62no—
91C2×#97C2×D311yescharacteristic
92D4⋊C8—62no—
93D4⋊C8—62no—
94#323—62no—
95C24×C3—21no—
96C24×C3—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4C22⋊C412yes—
106C2×C3×D4C2×C812yes—
107C2×C3×D4C2×C812yescomplemented
108C2×C3×D4M1612yes—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—82no—
116C22×C3×C4—82no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C3×D8—28no—
121C2×C3×C8—21no—
122C2×C3×C8—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8C2×C812yescomplemented
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4C22⋊C412yes—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—22no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8—84no—
144C22×D4C2×Dic311yescharacteristic
145C22×D4C2×Dic311yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C2×D8—22no—
149C2×D8—22no—
150C2×D8C3⋊C814yescomplemented
151C22×C8—64no—
152C22×C8C2×Dic311yescharacteristic
153C22×C8C2×Dic311yescharacteristic
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4#6712yes—
161C4×D4#6712yes—
162C2×C4⋊C4C2×Dic311yescharacteristic
163C2×C4⋊C4C2×Dic311yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42C22×D311yescharacteristic
167#106—22no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#102—22no—
173#102—22no—
174C22⋊C8—32no—
175C22⋊C8—64no—
176C22⋊C8—61no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179#97—21no—
180#97—21no—
181C4×C8—32no—
182C4×C8—61no—
183C4×C8—61no—
184C4×C8—61no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—81no—
192C23×C3—81no—
193C23×C3—81no—
194C23×C3—81no—
195C23×C3C4×D411yescharacteristic
196C3×D4—22no—
197C3×D4—22no—
198C3×D4—22no—
199C3×D4—22no—
200C3×D4—22no—
201C3×D4—41no—
202C3×D4—41no—
203C3×D4C22⋊C814yes—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—44no—
208C2×C3×C4—41no—
209C2×C3×C4—82no—
210C2×C3×C4—82no—
211C2×C3×C4—82no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4C2×C22⋊C411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12311yescharacteristic
220C2×C3×C4C22×C811yescharacteristic
221C2×C3×C4C2×M1611yescharacteristic
222C2×C3×C4#13111yescharacteristic
223C2×C3×C4#13111yescharacteristic
224C3×C8—22no—
225C3×C8—41no—
226C3⋊C8—42no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229C3⋊C8—81no—
230C24—21no—
231C24—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4C2×C3⋊C812yes—
241C2×D4C2×C3⋊C812yescomplemented
242C2×D4#20812yes—
243C2×D4#21712yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4D3×D411yescharacteristic
253C22×C4#23711yescharacteristic
254C22×C4C22×Dic311yescharacteristic
255D8—28no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—121no—
264C2×C8—121no—
265C2×C8—244no—
266C2×C8C2×C3⋊C812yescomplemented
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4#21712yes—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42D3×D411yescharacteristic
276C42#23711yescharacteristic
277C42C2×#6711yescharacteristic
278C42C2×#6711yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—24no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#38611yescharacteristic
292C22×C3C8×D411yescharacteristic
293C22×C3#43511yescharacteristic
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—42no—
299C3×C4—81no—
300C3×C4C2×C22⋊C811yescharacteristic
301C3×C4#40811yescharacteristic
302C3×C4C8×D411yescharacteristic
303C3×C4#43511yescharacteristic
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—41no—
309C23—41no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23D4×Dic311yescharacteristic
315D4—22no—
316D4—22no—
317D4—22no—
318D4—22no—
319D4—22no—
320D4—41no—
321D4—41no—
322D4#83014yes—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—44no—
327C2×C4—41no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4C22×C3⋊C811yescharacteristic
334C2×C4C2×#20811yescharacteristic
335C2×C4D4×Dic311yescharacteristic
336C2×C4#93511yescharacteristic
337C2×C4#93711yescharacteristic
338C2×C4#93911yescharacteristic
339C2×C4#93911yescharacteristic
340C2×C4#94811yescharacteristic
341C2×C4#94811yescharacteristic
342C2×C4C2×#21711yescharacteristic
343C8—22no—
344C8—41no—
345C8—122no—
346C8—122no—
347C8—241no—
348C8—241no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—44no—
352C2×C3—82no—
353C2×C3#186911yescharacteristic
354C2×C3#190311yescharacteristic
355C2×C3#190411yescharacteristic
356C22—21no—
357C22—21no—
358C22—24no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—81no—
367C22—81no—
368C22D4×C3⋊C811yescharacteristic
369C22#501111yescharacteristic
370C22#522011yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4D4×C3⋊C811yescharacteristic
378C4#501111yescharacteristic
379C4C2×#83011yescharacteristic
380C4#523411yescharacteristic
381C3#908911yescharacteristic, complemented, Sylow
382C2—21no—
383C2—21no—
384C2—44no—
385C2—82no—
386C2#7161011yescharacteristic
387C2#7208911yescharacteristic
388C2#7209011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC2×C3×#442C2
socleC22×C3#386
prime radicalC22×C3×D8C4
cosocleC2×C3×C42C23
hypercenterC2×#442D3
hypocenterC3#9089
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. #71601C2semidirect
4. #71597C2semidirect
5. #71597C2semidirect
6. #67661C2semidirect
7. #67660C2semidirect
41. C2×C3×D8C8semidirect
107. C2×C3×D4C2×C8semidirect
126. C2×C3×C8C2×C8semidirect
150. C2×D8C3⋊C8semidirect
241. C2×D4C2×C3⋊C8semidirect
266. C2×C8C2×C3⋊C8semidirect
381. C3#9089semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4422
#823012
#716012
#715972
#676612
#676602
#90893