Finite Group Databasebeta

← #10698727

Subgroups — #10698727

Order 768.

530 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82500C211yesmaximal, characteristic
2C3×#1803C211yesmaximal, characteristic, complemented
3C2×#4535C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic
5#69584C211yesmaximal, characteristic
6#69574C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10436—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#946—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#103—21no—
14C3×#384C2211yescharacteristic
15#4994—22no—
16#4987—21no—
17C2×#809—21no—
18#4939—21no—
19#4935C2211yescharacteristic
20#4928—21no—
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×#340—21no—
24#4535—22no—
25#4529—22no—
26#4529—21no—
27#4483—22no—
28#2538—31no—
29#1834—31no—
30#1803D311yescharacteristic, complemented
31#1802—31no—
32C2×Q8⋊C8—31no—
33C2×D4⋊C8—31no—
34#1296—31no—
35C22×C3×Q8D411yescharacteristic
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C22×C3×C8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50#946—42no—
51#946—42no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—22no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D412yes—
59C2×Dic3⋊C4—22no—
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×#103—42no—
69C3⋊#95—22no—
70#809—42no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4×C3⋊C8—21no—
74C4×C3⋊C8D412yes—
75C2×C22⋊Q8—61no—
76C2×C4×Q8C2×D311yescharacteristic
77C2×C4×D4—31no—
78C2×C4⋊C8—31no—
79C2×C4⋊C8C2×D311yescharacteristic
80C2×#103—21no—
81C2×#102—61no—
82C2×C4×C8—31no—
83#396—62no—
84#390—61no—
85#389—61no—
86#384—31no—
87#384C2×D311yescharacteristic
88#380—61no—
89#340—21no—
90#340—61no—
91#340—62no—
92Q8⋊C8—62no—
93D4⋊C8—62no—
94C23×D3—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8SD1612yes—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—22no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12Q1612yes—
115C2×C4×D3—41no—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—42no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C42—41no—
134C3×C42—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—44no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—82no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—42no—
150Dic3⋊C4—44no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C22×Q8#6711yescharacteristic
158C22×D4—31no—
159C23×C4—61no—
160C22×C8—21no—
161C22×C8—61no—
162C22×C8—61no—
163C22⋊Q8—122no—
164C22⋊Q8—122no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×D4—61no—
170C4×D4—61no—
171C4×D4—61no—
172C4×D4—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4#6711yescharacteristic
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192#103—42no—
193#102—122no—
194C4×C8—32no—
195C4×C8—61no—
196#95—62no—
197C23×C3#12011yescharacteristic
198C22×D3—22no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3—41no—
205C3×Q8—22no—
206C3×Q8—22no—
207C3×Q8—41no—
208C3×Q8—41no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4#12411yescharacteristic
226C2×C3×C4C2×SD1611yescharacteristic
227C2×C3×C4C2×Q1611yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4#13711yescharacteristic
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240D12—22no—
241D12—22no—
242D12—41no—
243D12—41no—
244C4×D3—81no—
245C4×D3—81no—
246C4×D3—81no—
247C4×D3—81no—
248C3×C8—81no—
249C3⋊C8—42no—
250C3⋊C8—81no—
251C24—61no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8#21512yes—
258C2×D4—32no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—61no—
268C22×C4—62no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4C2×#6711yescharacteristic
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—62no—
282C2×C8—121no—
283C2×C8—121no—
284C2×C8—121no—
285C2×C8—121no—
286C2×C8—121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—122no—
299C4⋊C4—124no—
300C4⋊C4—121no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C22⋊C4—124no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C22⋊C4—242no—
308C42—41no—
309C42—41no—
310C42#23511yescharacteristic
311C42#23511yescharacteristic
312C42D3×D411yescharacteristic
313C42#23711yescharacteristic
314C22×C3#39411yescharacteristic
315C22×C3#44911yescharacteristic
316C22×C3#44911yescharacteristic
317C22×C3#45011yescharacteristic
318C22×C3#45111yescharacteristic
319C22×C3#45411yescharacteristic
320C22×C3#45511yescharacteristic
321C2×D3—42no—
322C2×D3—42no—
323C2×D3—42no—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—81no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—42no—
331C3×C4—81no—
332C3×C4—161no—
333C3×C4#46511yescharacteristic
334C3×C4#46711yescharacteristic
335C3×C4#46811yescharacteristic
336C3×C4#46911yescharacteristic
337C3×C4#48111yescharacteristic
338C3×C4#48211yescharacteristic
339C3×C4#48411yescharacteristic
340C3×C4#48511yescharacteristic
341Dic3—82no—
342Dic3—81no—
343C23—62no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23#93711yescharacteristic
351Q8—22no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355D4—62no—
356D4—62no—
357D4—121no—
358D4—121no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—122no—
375C2×C4—121no—
376C2×C4—241no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4#88311yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4#88411yescharacteristic
383C2×C4#88511yescharacteristic
384C2×C4#91411yescharacteristic
385C2×C4#91611yescharacteristic
386C2×C4D3×Q1611yescharacteristic
387C2×C4#91811yescharacteristic
388C2×C4#92911yescharacteristic
389C2×C4#93211yescharacteristic
390C2×C4#93911yescharacteristic
391C2×C4C2×#21511yescharacteristic
392C2×C4#94911yescharacteristic
393C2×C4#95011yescharacteristic
394C8—81no—
395C8—122no—
396C8—241no—
397C2×C3#162811yescharacteristic
398C2×C3#163711yescharacteristic
399C2×C3#164211yescharacteristic
400C2×C3#164411yescharacteristic
401C2×C3#203411yescharacteristic
402C2×C3#203611yescharacteristic
403C2×C3#204011yescharacteristic
404D3—82no—
405C22—122no—
406C22—122no—
407C22—122no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22—241no—
412C22#495911yescharacteristic
413C22#503711yescharacteristic
414C22#504611yescharacteristic
415C22#517211yescharacteristic
416C22#518611yescharacteristic
417C22#520111yescharacteristic
418C22#520211yescharacteristic
419C4—21no—
420C4—21no—
421C4—42no—
422C4—81no—
423C4—161no—
424C4—241no—
425C4—242no—
426C4#481111yescharacteristic
427C4#481211yescharacteristic
428C4#481311yescharacteristic
429C4#481511yescharacteristic
430C4#496811yescharacteristic
431C4#500311yescharacteristic
432C4#503911yescharacteristic
433C4#504811yescharacteristic
434C3#1043611yescharacteristic, complemented, Sylow
435C2—242no—
436C2#6952411yescharacteristic
437C2#7027511yescharacteristic
438C2#7031211yescharacteristic
439C2#7102911yescharacteristic
440C2#7106211yescharacteristic
441C2#7112411yescharacteristic
442C2#7116311yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1803C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1803D3
hypocenterC3#10436
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C3×#1803C2semidirect
3. C2×#4535C2semidirect
6. #69574C2semidirect
7. #67702C2semidirect
30. #1803D3semidirect
434. C3#10436semidirect

Maximal subgroups

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

subgroupindex
#825002
C3×#18032
C2×#45352
C2×#44832
#695842
#695742
#677022
#104363