Finite Group Databasebeta

← #10690005

Subgroups — #10690005

Order 768.

680 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4841C211yesmaximal, characteristic
2#74360C211yesmaximal, characteristic
3#70460C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic
5#69582C211yesmaximal, characteristic, complemented
6C3×#1296C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#8985—31nomaximal, Sylow
9C22×#204—21no—
10C2×#893—21no—
11C2×D3×C4⋊C4—21no—
12C2×C4×D12C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C3×#384C2211yescharacteristic
17C2×#816—21no—
18#4989—22no—
19#4987—21no—
20#4972—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4841—22no—
24#4841C414yescomplemented
25#4551—21no—
26#4551—21no—
27#4529—22no—
28#4483—22no—
29C2×#461—31no—
30D4×C4⋊C4—31no—
31#1990—31no—
32#1988—31no—
33C2×D4⋊C8—31no—
34#1296D311yescharacteristic, complemented
35#1279—31no—
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C22×C4×D3—22no—
39C22×Dic6—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—22no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4—22no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56C2×#204—24no—
57C2×#204—42no—
58C2×#204—42no—
59#893—42no—
60#893—42no—
61D3×C4⋊C4—42no—
62D3×C4⋊C4—42no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12C2×C412yes—
66C4×D12D412yes—
67#869—21no—
68#869—21no—
69#869—21no—
70#869C2×C412yescomplemented
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8C2×C412yescomplemented
75C3⋊#95—22no—
76C3⋊#95—21no—
77#816—24no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C22×SD16—61no—
81C2×C4⋊Q8—31no—
82C2×C4×D4—31no—
83C2×C4×D4—61no—
84C22×C4⋊C4—61no—
85#461—34no—
86#461—62no—
87C2×C4⋊C8C2×D311yescharacteristic
88C2×C4⋊C8C2×D311yescharacteristic
89C2×#103—61no—
90C2×C4×C8—31no—
91#389—61no—
92#385—62no—
93#384C2×D311yescharacteristic
94#380—61no—
95C4×C4⋊C4—31no—
96#340—61no—
97#340—61no—
98#340—62no—
99D4⋊C8—62no—
100C23×D3—21no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—41no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12C22⋊C412yes—
113C2×D12D812yes—
114C2×D12SD1612yes—
115C2×C4×D3—22no—
116C2×C4×D3—22no—
117C2×C4×D3—41no—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3—44no—
122C2×C4×D3—44no—
123C2×C4×D3—82no—
124C2×Dic6—22no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145(C2×D3)⋊C4—22no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—44no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—22no—
151C4⋊Dic3—22no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156C4⋊Dic3C22⋊C412yes—
157C4⋊Dic3SD1612yescomplemented
158Dic3⋊C4—44no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164#204—48no—
165C22×Q8—61no—
166C22×D4—31no—
167C23×C4—62no—
168C23×C4—61no—
169C2×SD16—64no—
170C2×SD16—122no—
171C2×SD16—122no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—61no—
175C4⋊Q8—32no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179C4×D4—32no—
180C4×D4—32no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—122no—
184C4×D4—122no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—122no—
194C2×C4⋊C4—122no—
195C2×C4⋊C4D1212yes—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—62no—
198C2×C42—61no—
199C2×C42C22×D311yescharacteristic
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8C4×D312yescomplemented
204#103—64no—
205C4×C8—61no—
206C4×C8—61no—
207#95—62no—
208#95—61no—
209C23×C3#12011yescharacteristic
210C22×D3—22no—
211C22×D3—22no—
212C22×D3—22no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—41no—
216C22×D3—41no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4#13711yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248D12—22no—
249D12—22no—
250D12—22no—
251D12#10214yes—
252C4×D3—42no—
253C4×D3—42no—
254C4×D3—42no—
255C4×D3—42no—
256C4×D3—88no—
257Dic6—42no—
258Dic6—42no—
259Dic6—81no—
260Dic6—81no—
261C3×C8—42no—
262C3×C8—81no—
263C3⋊C8—42no—
264C24—61no—
265C2×Q8—62no—
266C2×Q8—121no—
267C2×Q8—121no—
268C2×Q8—121no—
269C2×Q8—121no—
270C2×Q8—121no—
271C2×D4—32no—
272C2×D4—32no—
273C2×D4—32no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C22×C4—22no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—61no—
283C22×C4—124no—
284C22×C4—124no—
285C22×C4—121no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4—121no—
290C22×C4—242no—
291C22×C4C2×D1211yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294SD16—128no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—62no—
302C2×C8—121no—
303C2×C8—121no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—124no—
318C4⋊C4—121no—
319C4⋊C4—121no—
320C22⋊C4—62no—
321C22⋊C4—62no—
322C22⋊C4—124no—
323C42—121no—
324C42—121no—
325C42C2×C4×D311yescharacteristic
326C42#23511yescharacteristic
327C42D3×D411yescharacteristic
328C42#23911yescharacteristic
329C22×C3#38611yescharacteristic
330C22×C3#44711yescharacteristic
331C22×C3#44811yescharacteristic
332C22×C3#45011yescharacteristic
333C22×C3#45211yescharacteristic
334C22×C3#45511yescharacteristic
335C22×C3#45611yescharacteristic
336C2×D3—24no—
337C2×D3—42no—
338C2×D3—42no—
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—42no—
342C2×D3—42no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—82no—
346C3×C4C2×#10211yescharacteristic
347C3×C4#41711yescharacteristic
348C3×C4C4×SD1611yescharacteristic
349C3×C4#44011yescharacteristic
350C3×C4#46111yescharacteristic
351C3×C4#46311yescharacteristic
352C3×C4#48011yescharacteristic
353C3×C4#48111yescharacteristic
354Dic3—42no—
355Dic3—42no—
356Dic3—82no—
357Dic3—81no—
358C23—62no—
359C23—62no—
360C23—62no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23#88211yescharacteristic
366Q8—122no—
367Q8—122no—
368Q8—241no—
369Q8—241no—
370D4—34no—
371D4—62no—
372D4—62no—
373D4—62no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—248no—
394C2×C4—241no—
395C2×C4—241no—
396C2×C4C4×D1211yescharacteristic
397C2×C4D3×C22⋊C411yescharacteristic
398C2×C4#88411yescharacteristic
399C2×C4#89311yescharacteristic
400C2×C4#89411yescharacteristic
401C2×C4#89511yescharacteristic
402C2×C4D3×D811yescharacteristic
403C2×C4#91111yescharacteristic
404C2×C4D3×SD1611yescharacteristic
405C2×C4D3×SD1611yescharacteristic
406C2×C4#91411yescharacteristic
407C2×C4#91611yescharacteristic
408C2×C4#91811yescharacteristic
409C2×C4#91811yescharacteristic
410C8—42no—
411C8—81no—
412C8—122no—
413C2×C3#163911yescharacteristic
414C2×C3#164311yescharacteristic
415C2×C3#164711yescharacteristic
416C2×C3#165111yescharacteristic
417C2×C3#187411yescharacteristic
418C2×C3#187711yescharacteristic
419C2×C3#188211yescharacteristic
420D3—44no—
421C22—64no—
422C22—122no—
423C22—122no—
424C22—122no—
425C22—122no—
426C22—122no—
427C22—122no—
428C22#466511yescharacteristic
429C22#474611yescharacteristic
430C22#475211yescharacteristic
431C22#477311yescharacteristic
432C22#477611yescharacteristic
433C22#480611yescharacteristic
434C22#480811yescharacteristic
435C4—21no—
436C4—21no—
437C4—82no—
438C4—122no—
439C4—122no—
440C4—242no—
441C4—241no—
442C4D3×#10211yescharacteristic
443C4#478711yescharacteristic
444C4#480311yescharacteristic
445C4#481811yescharacteristic
446C4#485211yescharacteristic
447C4#486311yescharacteristic
448C4#488211yescharacteristic
449C4#488911yescharacteristic
450C3#898511yescharacteristic, complemented, Sylow
451C2—124no—
452C2#6839211yescharacteristic
453C2#6843611yescharacteristic
454C2#6843811yescharacteristic
455C2#6849411yescharacteristic
456C2#6888011yescharacteristic
457C2#6889611yescharacteristic
458C2#6900411yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#120
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1296D3
hypocenterC3#8985
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #70460C2semidirect
5. #69582C2semidirect
6. C3×#1296C2semidirect
7. #67704C2semidirect
24. #4841C4semidirect
34. #1296D3semidirect
70. #869C2×C4semidirect
74. C3×C4⋊C8C2×C4semidirect
157. C4⋊Dic3SD16semidirect
203. C4⋊C8C4×D3semidirect
450. C3#8985semidirect

Maximal subgroups

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

subgroupindex
C2×#48412
#743602
#704602
C2×#44832
#695822
C3×#12962
#677042
#89853