Finite Group Databasebeta

← #10691737

Subgroups — #10691737

Order 768.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83355C211yesmaximal, characteristic
2#83330C211yesmaximal, characteristic, complemented
3C3×#1550C211yesmaximal, characteristic, complemented
4#70521C211yesmaximal, characteristic
5C2×#4485C211yesmaximal, characteristic
6#67714C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#9274—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#874C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×#823—21no—
14C2×#823—21no—
15#5088—21no—
16#5085—21no—
17#5046C414yes—
18#5027—22no—
19#5024—22no—
20#5024—21no—
21Q8×C3⋊C8—21no—
22#4999—21no—
23#4997—21no—
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C4×#208C2211yescharacteristic
27C3×#333—21no—
28C3×#332—21no—
29#4485—21no—
30#4485—21no—
31#3208—31no—
32#2962—31no—
33#1965—31no—
34#1550D311yescharacteristic, complemented
35C2×#331—31no—
36#1301—31no—
37#1283—31no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12C2×C412yes—
41C2×C3×M16—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8C2×C412yes—
48C3×C4×Q8—21no—
49C3×C4×Q8D412yes—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#215—24no—
53C2×#208D411yescharacteristic
54C2×#208D411yescharacteristic
55C22×C3⋊C8—21no—
56#874—21no—
57#874—21no—
58#874—21no—
59#874—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64#830C2×C412yes—
65#823—42no—
66#823—42no—
67#809—24no—
68C4⋊(C3⋊C8)—22no—
69C4⋊(C3⋊C8)—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)D411yescharacteristic
73C4⋊(C3⋊C8)D411yescharacteristic
74#803—22no—
75#803—21no—
76C4×C3⋊C8—22no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79#533C2×D311yescharacteristic
80C2×#127—31no—
81#492—61no—
82#486—61no—
83#460—62no—
84#450—34no—
85#446—61no—
86#446—62no—
87C8×Q8—61no—
88#431—31no—
89#424—61no—
90#423C2×D311yescharacteristic
91C2×C4⋊C8—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93C2×#100—61no—
94C2×#100—61no—
95#405—61no—
96C4×M16—31no—
97#333—21no—
98#332—21no—
99#331—61no—
100#331—61no—
101C23×D3—22no—
102C2×C3×Q8—21no—
103C2×C3×Q8C22⋊C412yes—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×D12—22no—
108C2×D12—22no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—82no—
118C2×D12—81no—
119C2×D12—81no—
120C3×M16—41no—
121C3×M16—41no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C2×C3×C8—81no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4C22⋊C412yes—
131C3×C22⋊C4—22no—
132C3×C42—22no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#215—84no—
138#208—21no—
139#208—21no—
140#208—21no—
141#208SD1612yescomplemented
142C2×C3⋊C8—21no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C22×D4—32no—
150C22×D4—61no—
151C22×D4—61no—
152C2×SD16—64no—
153C2×M16—21no—
154C2×M16—31no—
155C2×M16—31no—
156C22×C8—21no—
157C22×C8—61no—
158C4⋊Q8#6711yescharacteristic
159C4⋊Q8#6711yescharacteristic
160#127—61no—
161#127—61no—
162#127—61no—
163#127—61no—
164#125—21no—
165C22⋊Q8—21no—
166C22⋊Q8C4×D312yes—
167C4×Q8—21no—
168C4×Q8D1212yes—
169#117—21no—
170C2×C42C22×D311yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—31no—
176C4⋊C8—31no—
177C4⋊C8—62no—
178C4⋊C8—62no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181#102—64no—
182#100—122no—
183#100—122no—
184C22⋊C8—32no—
185#97—62no—
186#97—61no—
187C4×C8—62no—
188C4×C8—61no—
189C4×C8—61no—
190C23×C3C4×D411yescharacteristic
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—82no—
195C22×D3—82no—
196C3×Q8—42no—
197C3×Q8—41no—
198C3×Q8—41no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×C3×C4C2×SD1611yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C2×C3×C4#13711yescharacteristic
214D12—44no—
215D12—82no—
216D12—82no—
217D12—81no—
218D12—81no—
219D12—81no—
220D12—81no—
221D12—81no—
222D12—81no—
223C3×C8—81no—
224C3×C8—81no—
225C3⋊C8—22no—
226C3⋊C8—42no—
227C3⋊C8—41no—
228C3⋊C8—41no—
229C3⋊C8—41no—
230C24—62no—
231C2×Q8—21no—
232C2×Q8(C2×D3)⋊C412yes—
233C2×D4—62no—
234C2×D4—62no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—241no—
244C2×D4—241no—
245C2×D4—242no—
246C22×C4C2×C4×D311yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4#23711yescharacteristic
249SD16—244no—
250M16—32no—
251M16—41no—
252M16—41no—
253M16—61no—
254M16—61no—
255M16—61no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—62no—
263C2×C8—81no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4(C2×D3)⋊C412yes—
272C22⋊C4—22no—
273C42—22no—
274C42C2×D1211yescharacteristic
275C42D3×D411yescharacteristic
276C42#23711yescharacteristic
277C42C2×#6711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#38611yescharacteristic
281C22×C3C4×SD1611yescharacteristic
282C22×C3#44011yescharacteristic
283C2×D3—82no—
284C2×D3—82no—
285C2×D3—82no—
286C2×D3—162no—
287C2×D3—162no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—42no—
292C3×C4—42no—
293C3×C4#44811yescharacteristic
294C3×C4#44911yescharacteristic
295C3×C4#46011yescharacteristic
296C3×C4#46411yescharacteristic
297C3×C4#47111yescharacteristic
298C3×C4#47211yescharacteristic
299C23—122no—
300C23—122no—
301C23—122no—
302C23—242no—
303C23—242no—
304C23#67⋊C411yescharacteristic
305Q8—42no—
306Q8—41no—
307Q8—41no—
308D4—124no—
309D4—242no—
310D4—242no—
311D4—241no—
312D4—241no—
313D4—241no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4#67⋊C411yescharacteristic
323C2×C4#88211yescharacteristic
324C2×C4#88611yescharacteristic
325C2×C4D3×SD1611yescharacteristic
326C2×C4#91411yescharacteristic
327C2×C4#91811yescharacteristic
328C2×C4#91911yescharacteristic
329C2×C4C2×(C2×D3)⋊C411yescharacteristic
330C2×C4#93911yescharacteristic
331C2×C4#93911yescharacteristic
332C8—62no—
333C8—81no—
334C8—81no—
335C8—122no—
336C8—121no—
337C8—121no—
338C8—121no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#188011yescharacteristic
342C2×C3#190711yescharacteristic
343C2×C3#191411yescharacteristic
344D3—162no—
345C22—21no—
346C22—21no—
347C22—242no—
348C22—242no—
349C22—242no—
350C22—482no—
351C22—482no—
352C22#478811yescharacteristic
353C22#481711yescharacteristic
354C22#495811yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4#480611yescharacteristic
361C4#481011yescharacteristic
362C4#516311yescharacteristic
363C4#518511yescharacteristic
364C4#519911yescharacteristic
365C4#520311yescharacteristic
366C3#927411yescharacteristic, complemented, Sylow
367C2—21no—
368C2—21no—
369C2—482no—
370C2#7013711yescharacteristic
371C2#7024011yescharacteristic
372C2#7030311yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1550C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1550D3
hypocenterC3#9274
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
2. #83330C2semidirect
3. C3×#1550C2semidirect
6. #67714C2semidirect
7. #67706C2semidirect
34. #1550D3semidirect
141. #208SD16semidirect
366. C3#9274semidirect

Maximal subgroups

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

subgroupindex
#833552
#833302
C3×#15502
#705212
C2×#44852
#677142
#677062
#92743