Finite Group Databasebeta

← #10691954

Subgroups — #10691954

Order 768.

542 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
1#82307C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3C3×#1804C211yesmaximal, characteristic, complemented
4#70426C211yesmaximal, characteristic, complemented
5#69545C211yesmaximal, characteristic, complemented
6#67707C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9305—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932—21no—
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#102—21no—
14C3×#382C2211yescharacteristic
15C2×#830—22no—
16#5019C412yescomplemented
17#5016—22no—
18#5015C412yescomplemented
19D4×C3⋊C8—22no—
20D4×C3⋊C8—22no—
21#4966—21no—
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C3×#340—21no—
26#4483—22no—
27#4451—22no—
28#3036—31no—
29#2972—31no—
30#1804D311yescharacteristic, complemented
31#1517—31no—
32#1480—31no—
33#1301—31no—
34#1286—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C2×#235—21no—
38C22×C3×C8—21no—
39C3×C4×D4—21no—
40C3×C4×D4—21no—
41C3×C4×D4C2×C412yes—
42C3×C4×D4D412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#214—22no—
48#932—42no—
49#932—42no—
50C2×#213—22no—
51#930—21no—
52#930—21no—
53C2×#208—21no—
54C22×C3⋊C8—24no—
55#875—21no—
56#874C2×C411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#869C2×C411yescharacteristic
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×#102—42no—
63C3×#95—21no—
64C3×#95—21no—
65#830—24no—
66#830—21no—
67#830D412yes—
68#810—22no—
69#809—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)C2×C412yes—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C2×AGL(1,ℤ/8)—61no—
77#532—31no—
78C2×C4×D4C2×D311yescharacteristic
79#463—62no—
80#461—32no—
81#459—32no—
82C8×D4—62no—
83C8×D4—62no—
84#431—31no—
85#431—31no—
86#423—31no—
87C2×C4⋊C8C2×D311yescharacteristic
88#417—61no—
89C2×#102—21no—
90C2×C22⋊C8—62no—
91#382C2×D311yescharacteristic
92#340—21no—
93#327—62no—
94D4⋊C8—62no—
95C24×C3—21no—
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4C22⋊C412yes—
100C2×C3×D4D812yes—
101C2×C3×D4SD1612yes—
102C22×C3×C4—21no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—41no—
106C22×C3×C4—82no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—41no—
111#235—41no—
112#235—41no—
113#235—41no—
114#235—41no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4C22⋊C412yes—
127C3×C22⋊C4—22no—
128C3×C22⋊C4—22no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133#214—44no—
134#213—44no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139#208—41no—
140#208—41no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—84no—
145C2×#41—61no—
146C22×D4#6711yescharacteristic
147C23×C4—21no—
148AGL(1,ℤ/8)—122no—
149AGL(1,ℤ/8)—122no—
150C2×SD16—62no—
151C2×D8—62no—
152C2×M16—61no—
153C22×C8—21no—
154C22×C8—64no—
155C4⋊Q8—31no—
156#127—31no—
157#124—61no—
158#121—61no—
159#121—61no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—61no—
163C4×D4—61no—
164C4×D4C4×D312yes—
165C4×D4D1212yes—
166C2×C4⋊C4—21no—
167C2×C4⋊C4#6711yescharacteristic
168C2×C22⋊C4—21no—
169C2×C42C22×D311yescharacteristic
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—32no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176#103—62no—
177#102—42no—
178#102—62no—
179C22⋊C8—32no—
180C22⋊C8—64no—
181C22⋊C8—61no—
182C4×C8—32no—
183C4×C8—61no—
184#95—21no—
185#95—21no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—81no—
190C23×C3—81no—
191C23×C3#12111yescharacteristic
192C22×D3—41no—
193C3×D4—22no—
194C3×D4—22no—
195C3×D4—22no—
196C3×D4#10214yes—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—44no—
201C2×C3×C4—82no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4C2×D811yescharacteristic
213C2×C3×C4C2×SD1611yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4#13511yescharacteristic
216#67—81no—
217#67—81no—
218C2×Dic3—41no—
219D12—42no—
220D12—41no—
221D12—41no—
222D12—81no—
223C4×D3—81no—
224C4×D3—81no—
225Dic6—41no—
226Dic6—41no—
227C3×C8—81no—
228C3⋊C8—42no—
229C3⋊C8—42no—
230C3⋊C8—42no—
231C24—21no—
232#41—121no—
233#41—121no—
234#41—121no—
235#41—121no—
236C2×Q8—61no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—121no—
244C2×D4(C2×D3)⋊C412yes—
245C2×D4#21312yes—
246C2×D4#21512yes—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—82no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4#23711yescharacteristic
255C22×C4C2×#6711yescharacteristic
256SD16—124no—
257D8—124no—
258M16—121no—
259M16—121no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—244no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4(C2×D3)⋊C412yes—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C42C2×C4×D311yescharacteristic
278C42C2×D1211yescharacteristic
279C42D3×D411yescharacteristic
280C42#23711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—24no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3#38611yescharacteristic
289C22×C3#46511yescharacteristic
290C22×C3#46611yescharacteristic
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—42no—
299C3×C4—161no—
300C3×C4C2×#10211yescharacteristic
301C3×C4#41611yescharacteristic
302C3×C4#46511yescharacteristic
303C3×C4#46611yescharacteristic
304Dic3—81no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—81no—
309C23—81no—
310C23—121no—
311C23#93911yescharacteristic
312Q8—121no—
313Q8—121no—
314D4—22no—
315D4—22no—
316D4—22no—
317D4—122no—
318D4—121no—
319D4—121no—
320D4—241no—
321D4—241no—
322D4—241no—
323D4#80914yes—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—121no—
334C2×C4—241no—
335C2×C4—241no—
336C2×C4#67⋊C411yescharacteristic
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88211yescharacteristic
339C2×C4#88611yescharacteristic
340C2×C4C2×(C2×D3)⋊C411yescharacteristic
341C2×C4C2×#21311yescharacteristic
342C2×C4#93911yescharacteristic
343C2×C4C2×#21511yescharacteristic
344C2×C4#95011yescharacteristic
345C2×C4#95011yescharacteristic
346C8—81no—
347C8—122no—
348C8—122no—
349C8—122no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—44no—
353C2×C3#188111yescharacteristic
354C2×C3#189311yescharacteristic
355C2×C3#190411yescharacteristic
356D3—161no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—241no—
365C22—241no—
366C22—241no—
367C22#495811yescharacteristic
368C22#503811yescharacteristic
369C22#504811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—161no—
376C4—241no—
377C4C2×#80911yescharacteristic
378C4#500211yescharacteristic
379C4#503811yescharacteristic
380C4#504811yescharacteristic
381C3#930511yescharacteristic, complemented, Sylow
382C2—21no—
383C2—21no—
384C2—44no—
385C2—481no—
386C2#6951511yescharacteristic
387C2#7029311yescharacteristic
388C2#7031411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1804C2
socleC22×C3#386
prime radical#82307C2
cosocleC2×C3×C42C23
hypercenter#1804D3
hypocenterC3#9305
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. C3×#1804C2semidirect
4. #70426C2semidirect
5. #69545C2semidirect
6. #67707C2semidirect
7. #67631C2semidirect
16. #5019C4semidirect
18. #5015C4semidirect
30. #1804D3semidirect
381. C3#9305semidirect

Maximal subgroups

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

subgroupindex
#823072
#823012
C3×#18042
#704262
#695452
#677072
#676312
#93053