Finite Group Databasebeta

← #10698575

Subgroups — #10698575

Order 768.

516 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
1#73807C211yesmaximal, characteristic
2C3×#1948C211yesmaximal, characteristic, complemented
3C2×#4481C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic
5#69584C211yesmaximal, characteristic
6#69579C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#10412—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×C3×#106—21no—
12C2×C3×C4⋊C8C2211yescharacteristic
13C3×#384C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#4987—21no—
16#4987—21no—
17C2×#810—22no—
18C2×#809—21no—
19#4939—21no—
20#4936—21no—
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4673—22no—
24C3×#340—21no—
25#4529—21no—
26#4483—22no—
27#4481—22no—
28#2563—31no—
29#1956—31no—
30#1948D311yescharacteristic, complemented
31#1802—31no—
32C2×#333—31no—
33C2×D4⋊C8—31no—
34#1279—31no—
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6D412yes—
38C22×C3×C8—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—22no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C22×C3⋊C8—21no—
53C22×C3⋊C8—21no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62C3×#106—42no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8D412yes—
65C3×#95—21no—
66C3⋊#95—22no—
67C3⋊#95—22no—
68#810—44no—
69#809—42no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8—21no—
74C2×C4⋊Q8—31no—
75C2×C4×D4—31no—
76C2×#106—21no—
77C2×C4⋊C8—31no—
78C2×C4⋊C8C2×D311yescharacteristic
79C2×#103—62no—
80C2×#102—61no—
81C2×C4×C8—31no—
82#393—62no—
83#390—61no—
84#384C2×D311yescharacteristic
85#382—61no—
86#380—61no—
87#380—61no—
88C4×C4⋊C4C2×D311yescharacteristic
89#340—21no—
90#340—61no—
91#333—62no—
92D4⋊C8—62no—
93C23×D3—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—41no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—22no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12SD1612yes—
108C2×C4×D3—41no—
109C2×C4×D3—41no—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×C4×D3—41no—
113C2×C4×D3—41no—
114C2×Dic6—22no—
115C2×Dic6—22no—
116C2×Dic6—42no—
117C2×Dic6—42no—
118C2×C3×C8—22no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42—41no—
131C3×C42—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135C3×C42#4111yescharacteristic
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—82no—
139(C2×D3)⋊C4—82no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C22×Q8—32no—
153C22×D4—31no—
154C23×C4—61no—
155C22×C8—21no—
156C22×C8—61no—
157C22×C8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4×D4—61no—
163C4×D4—61no—
164C4×D4—61no—
165C4×D4—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4#6711yescharacteristic
173C2×C4⋊C4#6711yescharacteristic
174C2×C22⋊C4—61no—
175C2×C22⋊C4—62no—
176C2×C22⋊C4—62no—
177C2×C42—21no—
178C2×C42C22×D311yescharacteristic
179#106—42no—
180C4⋊C8—21no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8D1212yes—
184#103—124no—
185#102—122no—
186C4×C8—61no—
187C4×C8—61no—
188#95—21no—
189#95—62no—
190#95—62no—
191C23×C3#12011yescharacteristic
192C22×D3—22no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4#12311yescharacteristic
215C2×C3×C4#12411yescharacteristic
216C2×C3×C4C2×SD1611yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×C3×C4#13711yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230D12—22no—
231D12—22no—
232D12—41no—
233D12—41no—
234C4×D3—81no—
235C4×D3—81no—
236C4×D3—81no—
237C4×D3—81no—
238Dic6—44no—
239Dic6—82no—
240C3×C8—42no—
241C3⋊C8—42no—
242C3⋊C8—81no—
243C24—61no—
244C2×Q8—62no—
245C2×Q8—62no—
246C2×Q8—122no—
247C2×Q8—122no—
248C2×D4—32no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—61no—
258C22×C4—62no—
259C22×C4—121no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4C2×#6711yescharacteristic
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—62no—
272C2×C8—121no—
273C2×C8—121no—
274C2×C8—121no—
275C2×C8—121no—
276C2×C8—121no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C22⋊C4—242no—
295C22⋊C4—242no—
296C42—41no—
297C42—41no—
298C42C2×D1211yescharacteristic
299C42#23511yescharacteristic
300C42#23911yescharacteristic
301C42#23911yescharacteristic
302C22×C3#39411yescharacteristic
303C22×C3#44811yescharacteristic
304C22×C3#44911yescharacteristic
305C22×C3#45211yescharacteristic
306C22×C3#45211yescharacteristic
307C22×C3#45311yescharacteristic
308C22×C3#45511yescharacteristic
309C2×D3—42no—
310C2×D3—42no—
311C2×D3—42no—
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—81no—
319C3×C4—81no—
320C3×C4—161no—
321C3×C4#46311yescharacteristic
322C3×C4#46411yescharacteristic
323C3×C4#46511yescharacteristic
324C3×C4#47011yescharacteristic
325C3×C4#48311yescharacteristic
326C3×C4#48511yescharacteristic
327C3×C4#48811yescharacteristic
328C3×C4#48911yescharacteristic
329Dic3—82no—
330Dic3—81no—
331C23—62no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23—121no—
338C23#93711yescharacteristic
339Q8—124no—
340Q8—242no—
341D4—62no—
342D4—62no—
343D4—121no—
344D4—121no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—241no—
363C2×C4—241no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4#89411yescharacteristic
367C2×C4#89411yescharacteristic
368C2×C4#89511yescharacteristic
369C2×C4#89511yescharacteristic
370C2×C4D3×SD1611yescharacteristic
371C2×C4#91511yescharacteristic
372C2×C4#91511yescharacteristic
373C2×C4#91611yescharacteristic
374C2×C4#93011yescharacteristic
375C2×C4#93211yescharacteristic
376C2×C4#94211yescharacteristic
377C2×C4#94711yescharacteristic
378C2×C4#95011yescharacteristic
379C2×C4#95011yescharacteristic
380C8—42no—
381C8—122no—
382C8—241no—
383C2×C3#162211yescharacteristic
384C2×C3#163411yescharacteristic
385C2×C3#164311yescharacteristic
386C2×C3#164611yescharacteristic
387C2×C3#203511yescharacteristic
388C2×C3#203711yescharacteristic
389C2×C3#204211yescharacteristic
390D3—82no—
391C22—122no—
392C22—122no—
393C22—122no—
394C22—241no—
395C22—241no—
396C22—241no—
397C22—241no—
398C22#499111yescharacteristic
399C22#503711yescharacteristic
400C22#504711yescharacteristic
401C22#517011yescharacteristic
402C22#517311yescharacteristic
403C22#519811yescharacteristic
404C22#520211yescharacteristic
405C4—21no—
406C4—21no—
407C4—81no—
408C4—81no—
409C4—161no—
410C4—242no—
411C4—241no—
412C4#486211yescharacteristic
413C4#486411yescharacteristic
414C4#486711yescharacteristic
415C4#486811yescharacteristic
416C4#501611yescharacteristic
417C4#502811yescharacteristic
418C4#506811yescharacteristic
419C4#508911yescharacteristic
420C3#1041211yescharacteristic, complemented, Sylow
421C2—242no—
422C2#6962811yescharacteristic
423C2#7058211yescharacteristic
424C2#7067511yescharacteristic
425C2#7102811yescharacteristic
426C2#7104711yescharacteristic
427C2#7112511yescharacteristic
428C2#7114511yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1948C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1948D3
hypocenterC3#10412
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. C3×#1948C2semidirect
3. C2×#4481C2semidirect
6. #69579C2semidirect
7. #67619C2semidirect
30. #1948D3semidirect
420. C3#10412semidirect

Maximal subgroups

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

subgroupindex
#738072
C3×#19482
C2×#44812
C2×#44832
#695842
#695792
#676192
#104123