Finite Group Databasebeta

← #10691872

Subgroups — #10691872

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×#2932C211yesmaximal, characteristic, complemented
2#83302C211yesmaximal, characteristic
3#71814C211yesmaximal, characteristic, complemented
4#70502C211yesmaximal, characteristic, complemented
5#69823C211yesmaximal, characteristic, complemented
6#67713C211yesmaximal, characteristic, complemented
7#67660C211yesmaximal, characteristic, complemented
8#9293—31nomaximal, Sylow
9C2×C3×#131—21no—
10C3×#532C2211yescharacteristic
11C2×#874C2211yescharacteristic
12C3×#435—22no—
13C3×#435—22no—
14C3×#423C2211yescharacteristic
15C3×#408—21no—
16C2×C3×#97C2211yescharacteristic
17C2×#821—21no—
18C2×#821—21no—
19#5088C412yescomplemented
20#5073C412yescomplemented
21#5036—22no—
22#5015—22no—
23#4936C2211yescharacteristic
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4537—22no—
27#4485—22no—
28#3205—31no—
29#2932D311yescharacteristic, complemented
30#1961—31no—
31#1545—31no—
32#1517—31no—
33#1286—31no—
34#1275—31no—
35C2×C3×#41—21no—
36C22×D12—21no—
37C22×D12—21no—
38C22×D12—21no—
39C3×#131—42no—
40C3×#131—42no—
41C2×C3×M16—22no—
42C2×C3×M16—21no—
43C22×C3×C8—22no—
44C22×C3×C8D411yescharacteristic
45C22×C3×C8D411yescharacteristic
46C3×C4⋊Q8C2×C411yescharacteristic
47C3×#127C2×C411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121D412yes—
51C3×C4×D4—21no—
52C3×C4×D4D412yes—
53C2×C3×C42C2311yescharacteristic
54C2×#213—24no—
55C2×C4⋊Dic3—21no—
56#874—21no—
57#874—21no—
58#874—21no—
59#874C2×C412yes—
60C3×C4⋊C8—21no—
61C3×C4⋊C8D412yes—
62C3×C22⋊C8—24no—
63C3×#97—21no—
64C3×#97—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67#830—21no—
68#830—21no—
69#821—42no—
70#821—42no—
71#809—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8C2×C412yes—
76C2×#131—21no—
77#532C2×D311yescharacteristic
78C2×#127—31no—
79#493—32no—
80#486—32no—
81#459—62no—
82#447—62no—
83#435—22no—
84#435—22no—
85#431—31no—
86#431—31no—
87#423C2×D311yescharacteristic
88C2×#102—61no—
89C2×#102—61no—
90#408—21no—
91C2×#97C2×D311yescharacteristic
92#382—31no—
93#331—62no—
94#327—62no—
95C23×D3—41no—
96C3×#41—41no—
97C3×#41—41no—
98C3×#41—41no—
99C3×#41—41no—
100C2×C3×Q8—21no—
101C2×C3×D4—22no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4C22⋊C412yes—
105C22×C3×C4—22no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—41no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—22no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—82no—
121C2×D12—82no—
122C3×M16—44no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—22no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—44no—
132C2×C3×C8D812yescomplemented
133C3×C4⋊C4—21no—
134C3×C4⋊C4C22⋊C412yes—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—22no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141#213—84no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C2×#41—21no—
147C22×D4—61no—
148C22×D4—61no—
149C22×D4—61no—
150C2×D8—64no—
151#131—42no—
152#131—42no—
153C2×M16—22no—
154C2×M16—21no—
155C22×C8—22no—
156C22×C8#6711yescharacteristic
157C22×C8#6711yescharacteristic
158C4⋊Q8C4×D311yescharacteristic
159#127—32no—
160#127—61no—
161#127—61no—
162#127—61no—
163#127C4×D311yescharacteristic
164#124—21no—
165#121—21no—
166#121D1212yes—
167C4×D4—21no—
168C4×D4#6712yes—
169C2×C4⋊C4—61no—
170C2×C42C22×D311yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C4⋊C8#6712yes—
175#102—64no—
176#102—122no—
177#102—122no—
178C22⋊C8—24no—
179C22⋊C8—61no—
180C22⋊C8—61no—
181#97—21no—
182#97—21no—
183C4×C8—32no—
184C4×C8—61no—
185#95—61no—
186#95—61no—
187C23×C3—22no—
188C23×C3#12111yescharacteristic
189C22×D3—42no—
190C22×D3—82no—
191C22×D3—82no—
192C22×D3—81no—
193C22×D3—81no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×D4—42no—
197C3×D4—42no—
198C3×D4—42no—
199C3×D4—41no—
200C3×D4—41no—
201C3×D4—81no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4C2×C22⋊C411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×C3×C4C2×D811yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220D12—44no—
221D12—42no—
222D12—42no—
223D12—82no—
224D12—82no—
225D12—82no—
226D12—81no—
227D12—81no—
228C3×C8—22no—
229C3×C8—42no—
230C3×C8—41no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24—121no—
234#41—41no—
235#41—41no—
236#41—41no—
237#41—41no—
238C2×Q8—21no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—62no—
243C2×D4—62no—
244C2×D4—62no—
245C2×D4—121no—
246C2×D4—121no—
247C2×D4—121no—
248C2×D4—121no—
249C2×D4—121no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—242no—
253C2×D4—242no—
254C2×D4(C2×D3)⋊C412yes—
255C22×C4—22no—
256C22×C4—121no—
257C22×C4C2×D1211yescharacteristic
258C22×C4#23511yescharacteristic
259C22×C4C2×#6711yescharacteristic
260D8—244no—
261M16—44no—
262M16—41no—
263M16—41no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—22no—
270C2×C8—44no—
271C2×C8—62no—
272C2×C8—121no—
273C2×C8#21312yescomplemented
274C4⋊C4—21no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4(C2×D3)⋊C412yes—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C42C2×C4×D311yescharacteristic
281C42#23511yescharacteristic
282C42C2×#6711yescharacteristic
283C42C2×#6711yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—42no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3#38611yescharacteristic
290C22×C3#45911yescharacteristic
291C22×C3#46311yescharacteristic
292C2×D3—82no—
293C2×D3—82no—
294C2×D3—82no—
295C2×D3—162no—
296C2×D3—162no—
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—122no—
309C23—242no—
310C23—242no—
311C23—241no—
312C23—241no—
313C23#93011yescharacteristic
314Q8—41no—
315Q8—41no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—124no—
323D4—122no—
324D4—122no—
325D4—242no—
326D4—242no—
327D4—242no—
328D4—241no—
329D4—241no—
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×C4C2×(C2×D3)⋊C411yescharacteristic
339C2×C4C4×#6711yescharacteristic
340C2×C4C4×#6711yescharacteristic
341C2×C4#92911yescharacteristic
342C2×C4#93011yescharacteristic
343C2×C4C2×#21311yescharacteristic
344C2×C4#93211yescharacteristic
345C2×C4#94911yescharacteristic
346C2×C4#95011yescharacteristic
347C2×C4#95311yescharacteristic
348C8—22no—
349C8—42no—
350C8—41no—
351C8—122no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—82no—
356C2×C3#187911yescharacteristic
357C2×C3#190311yescharacteristic
358C2×C3#191011yescharacteristic
359D3—162no—
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—242no—
366C22—242no—
367C22—242no—
368C22—482no—
369C22—482no—
370C22#501511yescharacteristic
371C22#501611yescharacteristic
372C22#521311yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—481no—
379C4C4×#21311yescharacteristic
380C4#501411yescharacteristic
381C4#521711yescharacteristic
382C4#523811yescharacteristic
383C3#929311yescharacteristic, complemented, Sylow
384C2—21no—
385C2—21no—
386C2—82no—
387C2—482no—
388C2#7161311yescharacteristic
389C2#7202311yescharacteristic
390C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2932C2
socleC22×C3#386
prime radical#83302C2
cosocleC2×C3×C42C23
hypercenter#2932D3
hypocenterC3#9293
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
1. C3×#2932C2semidirect
3. #71814C2semidirect
4. #70502C2semidirect
5. #69823C2semidirect
6. #67713C2semidirect
7. #67660C2semidirect
19. #5088C4semidirect
20. #5073C4semidirect
29. #2932D3semidirect
132. C2×C3×C8D8semidirect
273. C2×C8#213semidirect
383. C3#9293semidirect

Maximal subgroups

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

subgroupindex
C3×#29322
#833022
#718142
#705022
#698232
#677132
#676602
#92933