Finite Group Databasebeta

← #10691886

Subgroups — #10691886

Order 768.

594 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
1C3×#2328C211yesmaximal, characteristic
2#83169C211yesmaximal, characteristic
3#71759C211yesmaximal, characteristic
4#69546C211yesmaximal, characteristic, complemented
5#69441C211yesmaximal, characteristic, complemented
6#67631C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9295—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×C3×C4×D4—21no—
11#5470C2211yescharacteristic
12C3×#390—21no—
13C3×#386—22no—
14C3×#386—22no—
15C3×#382C2211yescharacteristic
16C3×#376C2211yescharacteristic
17#5218—22no—
18#5076C412yescomplemented
19#5075—22no—
20#5070C412yescomplemented
21#4925C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4544—22no—
25#4523—21no—
26#4523—21no—
27#4452—22no—
28#3208—31no—
29#2328D311yescharacteristic
30#1995—31no—
31#1545—31no—
32#1518—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4D412yes—
38C23×C3×C4—22no—
39C3×#127—21no—
40C3×#127—21no—
41C3×#127C2×C412yes—
42C3×#127D412yes—
43C3×C4×D4—42no—
44C3×C4×D4—42no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×#214—24no—
53#926—21no—
54#924—21no—
55#924—21no—
56C2×#208—21no—
57#870—21no—
58#869C2×C411yescharacteristic, complemented
59#869C2×C411yescharacteristic, complemented
60C4×Dic6—21no—
61C4×Dic6—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3×#95D412yes—
66#832—24no—
67#830—21no—
68#830—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4×C3⋊C8—21no—
74C4×C3⋊C8C2×C412yescomplemented
75#533—31no—
76C2×#127C2×D311yescharacteristic
77C2×C4×D4—21no—
78#492—32no—
79#486—32no—
80#460—62no—
81#450—62no—
82#431—31no—
83#431—31no—
84#423—31no—
85#390—21no—
86#386—22no—
87#386—22no—
88#382C2×D311yescharacteristic
89#376C2×D311yescharacteristic
90#339—61no—
91#339—61no—
92#331—62no—
93#328—62no—
94C24×C3—22no—
95C2×C3×D4—22no—
96C2×C3×D4—22no—
97C2×C3×D4—22no—
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—82no—
105C2×C3×D4—82no—
106C2×C3×D4C22⋊C414yes—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4—44no—
111C22×C3×C4—82no—
112C22×C3×C4—82no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×Dic6—21no—
117C2×Dic6—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C22⋊C4—44no—
121C3×C22⋊C4—82no—
122C3×C22⋊C4—82no—
123C3×C42—21no—
124C3×C42—21no—
125C3×C42—21no—
126C3×C42SD1612yescomplemented
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131#217—41no—
132#214—84no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139C4×Dic3—41no—
140#208—41no—
141#208—41no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—41no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4#6712yes—
148C23×C4—22no—
149C2×SD16—64no—
150C2×M16—61no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153#127—21no—
154#127—21no—
155#127C4×D312yes—
156#127D1212yes—
157#125—61no—
158C22⋊Q8—61no—
159C22⋊Q8—61no—
160C4×Q8—61no—
161C4×Q8—61no—
162C4×D4—42no—
163C4×D4—42no—
164#117—61no—
165C2×C4⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C42#6711yescharacteristic
170C2×C42#6711yescharacteristic
171C2×C42C22×D311yescharacteristic
172C4⋊C8—61no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176#102—64no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179C4×C8—32no—
180C4×C8—61no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95#6712yes—
185C23×C3—24no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—42no—
192C23×C3#12011yescharacteristic
193C3×D4—44no—
194C3×D4—42no—
195C3×D4—42no—
196C3×D4—42no—
197C3×D4—42no—
198C3×D4—82no—
199C3×D4—82no—
200C3×D4—82no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—88no—
213C2×C3×C4C2×C22⋊C411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12311yescharacteristic
219C2×C3×C4C2×SD1611yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C3×C4#13711yescharacteristic
223C2×Dic3—41no—
224C2×Dic3—41no—
225Dic6—42no—
226Dic6—81no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229C3⋊C8—81no—
230C24—22no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—82no—
243C2×D4—82no—
244C2×D4(C2×D3)⋊C414yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4C2×#6711yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254SD16—244no—
255M16—121no—
256M16—121no—
257C2×C8—62no—
258C2×C8—121no—
259C2×C8—121no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C22⋊C4—44no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—121no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—121no—
276C42#21412yescomplemented
277C42C2×C4×D311yescharacteristic
278C42C2×D1211yescharacteristic
279C42#23511yescharacteristic
280C42#23511yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—44no—
284C22×C3—44no—
285C22×C3—44no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3—82no—
290C22×C3#38611yescharacteristic
291C22×C3#45011yescharacteristic
292C22×C3#45211yescharacteristic
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—42no—
298C3×C4—81no—
299C3×C4—82no—
300C3×C4C4×SD1611yescharacteristic
301C3×C4#44011yescharacteristic
302C3×C4#46111yescharacteristic
303C3×C4#46311yescharacteristic
304Dic3—81no—
305Dic3—81no—
306C23—24no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23#95311yescharacteristic
314Q8—122no—
315Q8—241no—
316D4—44no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—42no—
321D4—82no—
322D4—82no—
323D4—82no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—88no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4C2×(C2×D3)⋊C411yescharacteristic
339C2×C4C4×#6711yescharacteristic
340C2×C4C4×#6711yescharacteristic
341C2×C4#92911yescharacteristic
342C2×C4#93011yescharacteristic
343C2×C4#93011yescharacteristic
344C2×C4#93211yescharacteristic
345C2×C4C2×#21411yescharacteristic
346C2×C4#95011yescharacteristic
347C2×C4#95111yescharacteristic
348C8—122no—
349C8—241no—
350C8—241no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—84no—
354C2×C3#187711yescharacteristic
355C2×C3#190411yescharacteristic
356C2×C3#191111yescharacteristic
357C22—21no—
358C22—21no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22#521311yescharacteristic
367C22#521811yescharacteristic
368C22#523911yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—42no—
374C4—82no—
375C4—81no—
376C4—241no—
377C4—241no—
378C4#501611yescharacteristic
379C4C4×#21411yescharacteristic
380C4#501811yescharacteristic
381C4#501911yescharacteristic
382C3#929511yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—84no—
386C2#7161211yescharacteristic
387C2#7165111yescharacteristic
388C2#7165211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2328C2
socleC22×C3#386
prime radicalC2×C3×#127C22
cosocleC2×C3×C42C23
hypercenter#2328D3
hypocenterC3#9295
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
4. #69546C2semidirect
5. #69441C2semidirect
6. #67631C2semidirect
7. #67614C2semidirect
18. #5076C4semidirect
20. #5070C4semidirect
58. #869C2×C4semidirect
59. #869C2×C4semidirect
74. C4×C3⋊C8C2×C4semidirect
126. C3×C42SD16semidirect
276. C42#214semidirect
382. C3#9295semidirect

Maximal subgroups

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

subgroupindex
C3×#23282
#831692
#717592
#695462
#694412
#676312
#676142
#92953