Finite Group Databasebeta

← #10691480

Subgroups — #10691480

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2293C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic
3#71598C211yesmaximal, characteristic, complemented
4#69469C211yesmaximal, characteristic, complemented
5#69395C211yesmaximal, characteristic, complemented
6#67620C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#9237—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×C3×#117—21no—
11#5477C2211yescharacteristic
12C3×#390—21no—
13C3×#389—21no—
14C3×#388—22no—
15C3×C4×C4⋊C4C2211yescharacteristic
16C3×C4×C22⋊C4—21no—
17C3×C43C2211yescharacteristic
18#5217—21no—
19#5015C412yescomplemented
20C4×#213—22no—
21#5007C412yescomplemented
22#4967—21no—
23C2×#805—21no—
24C2×#805—21no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28D4⋊(C3⋊C8)—22no—
29#4451—22no—
30#3100—31no—
31#2293D311yescharacteristic, complemented
32#1962—31no—
33#1502—31no—
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C2×#235—21no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C3×C4×D4C2×C412yes—
43C3×C4×D4D412yes—
44C3×#117—42no—
45C3×#117—42no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42—21no—
55C2×C3×C42D411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×#213—22no—
59#930—21no—
60#930—21no—
61C2×#208—21no—
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869C2×C411yescharacteristic
67C3×#95—22no—
68C3×#95D412yes—
69#832—22no—
70#830—21no—
71#830—21no—
72#809—22no—
73#807—22no—
74#805—42no—
75#805—42no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)C2×C412yes—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82#532—31no—
83C2×C4×D4C2×D311yescharacteristic
84C2×#117—21no—
85#480—61no—
86#474—32no—
87#459—32no—
88#447—61no—
89C4×D8—62no—
90#431—31no—
91#431—31no—
92#423—31no—
93C2×#104—61no—
94C2×#104—61no—
95#390—21no—
96#389—21no—
97#388—22no—
98C4×C4⋊C4C2×D311yescharacteristic
99C4×C22⋊C4—21no—
100C43C2×D311yescharacteristic
101#327—62no—
102D4⋊C8—62no—
103C24×C3—21no—
104C2×C3×D4—21no—
105C2×C3×D4—42no—
106C2×C3×D4C22⋊C412yes—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C2×#67—41no—
121#235—41no—
122#235—41no—
123#235—41no—
124#235—41no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4—44no—
133C3×C4⋊C4C22⋊C412yes—
134C3×C22⋊C4—22no—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—44no—
137C3×C22⋊C4—44no—
138C3×C42—21no—
139C3×C42—21no—
140C3×C42—21no—
141C3×C42—42no—
142C3×C42—42no—
143C3×C42—41no—
144C3×C42—41no—
145C3×C42D812yescomplemented
146C3×C42C22×C411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149C3×C42#4111yescharacteristic
150#213—44no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×#41—61no—
161C22×D4#6711yescharacteristic
162C23×C4—21no—
163C2×D8—62no—
164C2×M16—61no—
165C4⋊Q8—31no—
166#127—31no—
167#124—61no—
168#121—61no—
169#121—61no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4C4×D312yes—
175C4×D4D1212yes—
176#117—42no—
177#117—42no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4#6711yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42#6711yescharacteristic
188C2×C42#6711yescharacteristic
189C2×C42C22×D311yescharacteristic
190#107—62no—
191C4⋊C8—32no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195#104—122no—
196#104—122no—
197#102—62no—
198#102—62no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C4×C8—61no—
202C4×C8—61no—
203#95—22no—
204#95#6712yes—
205C23×C3—22no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C22×D3—41no—
212C3×D4—22no—
213C3×D4—22no—
214C3×D4—42no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4C2×C22⋊C411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239C2×C3×C4C2×D811yescharacteristic
240C2×C3×C4#13511yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243#67—81no—
244#67—81no—
245C2×Dic3—41no—
246D12—42no—
247D12—41no—
248D12—41no—
249D12—81no—
250C4×D3—81no—
251C4×D3—81no—
252Dic6—41no—
253Dic6—41no—
254C3⋊C8—42no—
255C3⋊C8—81no—
256C3⋊C8—81no—
257C24—21no—
258#41—121no—
259#41—121no—
260#41—121no—
261#41—121no—
262C2×Q8—61no—
263C2×D4—21no—
264C2×D4—42no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—121no—
269C2×D4(C2×D3)⋊C412yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—121no—
281C22×C4C2×#6711yescharacteristic
282C22×C4C2×#6711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284D8—124no—
285M16—121no—
286M16—121no—
287C2×C8—62no—
288C2×C8—121no—
289C2×C8—121no—
290C4⋊C4—21no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4(C2×D3)⋊C412yes—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—42no—
306C42—42no—
307C42—41no—
308C42—41no—
309C42#21312yescomplemented
310C42C2×C4×D311yescharacteristic
311C42C2×D1211yescharacteristic
312C42#23511yescharacteristic
313C42#23511yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3#38611yescharacteristic
322C22×C3#44711yescharacteristic
323C22×C3#44911yescharacteristic
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—42no—
331C3×C4—42no—
332C3×C4—42no—
333C3×C4—82no—
334C3×C4#42112yes—
335C3×C4C4×D811yescharacteristic
336C3×C4#44211yescharacteristic
337C3×C4#45911yescharacteristic
338C3×C4#46311yescharacteristic
339Dic3—81no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—121no—
346C23#95311yescharacteristic
347Q8—121no—
348Q8—121no—
349D4—22no—
350D4—22no—
351D4—42no—
352D4—122no—
353D4—121no—
354D4—121no—
355D4—241no—
356D4—241no—
357D4—241no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—121no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4C2×(C2×D3)⋊C411yescharacteristic
380C2×C4C4×#6711yescharacteristic
381C2×C4C4×#6711yescharacteristic
382C2×C4#92911yescharacteristic
383C2×C4#93011yescharacteristic
384C2×C4#93011yescharacteristic
385C2×C4C2×#21311yescharacteristic
386C2×C4#93211yescharacteristic
387C2×C4#94911yescharacteristic
388C2×C4#95011yescharacteristic
389C8—122no—
390C8—241no—
391C8—241no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—82no—
395C2×C3#187911yescharacteristic
396C2×C3#189711yescharacteristic
397C2×C3#190611yescharacteristic
398D3—161no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—82no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22#521311yescharacteristic
410C22#521711yescharacteristic
411C22#523811yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—82no—
419C4—241no—
420C4#500512yes—
421C4C4×#21311yescharacteristic
422C4#501411yescharacteristic
423C4#501511yescharacteristic
424C4#501611yescharacteristic
425C3#923711yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—481no—
430C2#7159111yescharacteristic
431C2#7161311yescharacteristic
432C2#7164511yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2293C2
socleC22×C3#386
prime radical#82308C2
cosocleC2×C3×C42C23
hypercenter#2293D3
hypocenterC3#9237
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. C3×#2293C2semidirect
3. #71598C2semidirect
4. #69469C2semidirect
5. #69395C2semidirect
6. #67620C2semidirect
7. #67611C2semidirect
19. #5015C4semidirect
21. #5007C4semidirect
31. #2293D3semidirect
145. C3×C42D8semidirect
309. C42#213semidirect
425. C3#9237semidirect

Maximal subgroups

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

subgroupindex
C3×#22932
#823082
#715982
#694692
#693952
#676202
#676112
#92373