Finite Group Databasebeta

← #10691122

Subgroups — #10691122

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2349C211yesmaximal, characteristic, complemented
2#82307C211yesmaximal, characteristic
3#71597C211yesmaximal, characteristic, complemented
4#69545C211yesmaximal, characteristic, complemented
5#69469C211yesmaximal, characteristic, complemented
6#67631C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9179—31nomaximal, Sylow
9C2×C3×C4×D4—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C2×#932—21no—
12#5477C2211yescharacteristic
13C3×#389—21no—
14C3×#385—22no—
15C3×#382C2211yescharacteristic
16C3×#380—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18C2×C3×#95—21no—
19#5019C412yescomplemented
20#5016—22no—
21#5015C412yescomplemented
22#4966—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26D4⋊(C3⋊C8)—22no—
27#4451—21no—
28#4451—21no—
29#4451—22no—
30#3036—31no—
31#2349D311yescharacteristic, complemented
32#1517—31no—
33#1502—31no—
34#1478—31no—
35#1303—31no—
36#1286—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4—22no—
40C2×#235—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4—42no—
44C3×C4×D4—42no—
45C3×C4×D4C2×C412yes—
46C3×C4×D4D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#214—22no—
56#932—42no—
57#932—42no—
58C2×#213—22no—
59#930—21no—
60#930—21no—
61C2×#208—21no—
62#875—21no—
63#874C2×C411yescharacteristic, complemented
64C4×D12—21no—
65C4×D12—21no—
66#869C2×C411yescharacteristic, complemented
67C3×#95—22no—
68C3×#95—21no—
69C3×#95D411yescharacteristic
70C3×#95D411yescharacteristic
71C3×#95D412yes—
72#830—21no—
73#830—21no—
74#810—22no—
75#809—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)C2×C412yescomplemented
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C2×AGL(1,ℤ/8)—61no—
83#532—31no—
84C2×C4×D4—21no—
85C2×C4×D4C2×D311yescharacteristic
86#463—62no—
87#461—32no—
88#459—32no—
89#431—31no—
90#431—31no—
91#423—31no—
92#417—61no—
93#389—21no—
94#385—22no—
95#382C2×D311yescharacteristic
96#380—21no—
97C4×C4⋊C4C2×D311yescharacteristic
98C2×#95—21no—
99#327—61no—
100#327—61no—
101#327—62no—
102D4⋊C8—62no—
103C24×C3—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4C22⋊C412yes—
108C2×C3×D4D812yes—
109C2×C3×D4SD1612yes—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—21no—
116C22×C3×C4—44no—
117C22×C3×C4—44no—
118C22×C3×C4—82no—
119C22×C3×C4—82no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—41no—
124#235—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×C4×D3—41no—
132C2×Dic6—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—42no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—82no—
138C3×C4⋊C4C22⋊C412yes—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—44no—
142C3×C42—41no—
143C3×C42—41no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147C3×C42#4111yescharacteristic
148#214—44no—
149#213—44no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154#208—41no—
155#208—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×#41—61no—
160C22×D4#6711yescharacteristic
161C23×C4—21no—
162C23×C4—22no—
163AGL(1,ℤ/8)—122no—
164AGL(1,ℤ/8)—122no—
165C2×SD16—62no—
166C2×D8—62no—
167C2×M16—61no—
168C4⋊Q8—31no—
169#127—31no—
170#124—61no—
171#121—61no—
172#121—61no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4C4×D312yes—
180C4×D4D1212yes—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4#6711yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C42—21no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—32no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—62no—
194#102—62no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C4×C8—61no—
198C4×C8—61no—
199#95—22no—
200#95—21no—
201#95#6711yescharacteristic
202#95#6711yescharacteristic
203#95#6712yes—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3#12011yescharacteristic
210C22×D3—41no—
211C3×D4—22no—
212C3×D4—22no—
213C3×D4—22no—
214C3×D4#10214yes—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—82no—
229C2×C3×C4—82no—
230C2×C3×C4—88no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12311yescharacteristic
237C2×C3×C4C2×D811yescharacteristic
238C2×C3×C4C2×SD1611yescharacteristic
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240C2×C3×C4#13711yescharacteristic
241#67—81no—
242#67—81no—
243C2×Dic3—41no—
244D12—41no—
245D12—41no—
246D12—81no—
247D12—81no—
248C4×D3—81no—
249C4×D3—81no—
250Dic6—41no—
251Dic6—41no—
252C3⋊C8—42no—
253C3⋊C8—81no—
254C3⋊C8—81no—
255C24—21no—
256#41—121no—
257#41—121no—
258#41—121no—
259#41—121no—
260C2×Q8—61no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—121no—
268C2×D4(C2×D3)⋊C412yes—
269C2×D4#21312yes—
270C2×D4#21512yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—121no—
282C22×C4C2×#6711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284C22×C4C2×#6711yescharacteristic
285SD16—124no—
286D8—124no—
287M16—121no—
288M16—121no—
289C2×C8—62no—
290C2×C8—121no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—82no—
299C4⋊C4(C2×D3)⋊C412yes—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—44no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C42—41no—
306C42—41no—
307C42C2×C4×D311yescharacteristic
308C42C2×D1211yescharacteristic
309C42#23511yescharacteristic
310C42#23511yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—24no—
314C22×C3—44no—
315C22×C3—44no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3#38611yescharacteristic
319C22×C3#44711yescharacteristic
320C22×C3#44811yescharacteristic
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—81no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—42no—
329C3×C4—42no—
330C3×C4—81no—
331C3×C4—82no—
332C3×C4C2×#10211yescharacteristic
333C3×C4#41711yescharacteristic
334C3×C4#48011yescharacteristic
335C3×C4#48111yescharacteristic
336Dic3—81no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—121no—
343C23#95311yescharacteristic
344Q8—121no—
345Q8—121no—
346D4—22no—
347D4—22no—
348D4—22no—
349D4—121no—
350D4—121no—
351D4—241no—
352D4—241no—
353D4—241no—
354D4—241no—
355D4#80914yes—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—88no—
372C2×C4—121no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4C2×(C2×D3)⋊C411yescharacteristic
376C2×C4C4×#6711yescharacteristic
377C2×C4C4×#6711yescharacteristic
378C2×C4#92911yescharacteristic
379C2×C4#93011yescharacteristic
380C2×C4#93011yescharacteristic
381C2×C4C2×#21311yescharacteristic
382C2×C4#93211yescharacteristic
383C2×C4C2×#21511yescharacteristic
384C2×C4#94211yescharacteristic
385C8—122no—
386C8—241no—
387C8—241no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—44no—
391C2×C3#187411yescharacteristic
392C2×C3#190411yescharacteristic
393C2×C3#190611yescharacteristic
394D3—161no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—42no—
401C22—42no—
402C22—241no—
403C22—241no—
404C22—241no—
405C22#521311yescharacteristic
406C22#521711yescharacteristic
407C22#522711yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—42no—
413C4—42no—
414C4—82no—
415C4—81no—
416C4—241no—
417C4C2×#80911yescharacteristic
418C4#496611yescharacteristic
419C4#496711yescharacteristic
420C4#496811yescharacteristic
421C3#917911yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—44no—
425C2—481no—
426C2#7156011yescharacteristic
427C2#7164511yescharacteristic
428C2#7165111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2349C2
socleC22×C3#386
prime radical#82307C2
cosocleC2×C3×C42C23
hypercenter#2349D3
hypocenterC3#9179
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C3×#2349C2semidirect
3. #71597C2semidirect
4. #69545C2semidirect
5. #69469C2semidirect
6. #67631C2semidirect
7. #67620C2semidirect
19. #5019C4semidirect
21. #5015C4semidirect
31. #2349D3semidirect
63. #874C2×C4semidirect
66. #869C2×C4semidirect
79. C4⋊(C3⋊C8)C2×C4semidirect
421. C3#9179semidirect

Maximal subgroups

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

subgroupindex
C3×#23492
#823072
#715972
#695452
#694692
#676312
#676202
#91793