Finite Group Databasebeta

← #10698732

Subgroups — #10698732

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#74369C211yesmaximal, characteristic
2#70485C211yesmaximal, characteristic, complemented
3#70479C211yesmaximal, characteristic
4C2×#4484C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic
6#69574C211yesmaximal, characteristic, complemented
7C3×#1296C211yesmaximal, characteristic, complemented
8#10436—31nomaximal, Sylow
9C2×#897—21no—
10C2×C4×D12C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×#384C2211yescharacteristic
15C2×#821—21no—
16C2×#816—21no—
17#4994—22no—
18#4988—21no—
19#4987—21no—
20#4984—21no—
21#4935C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4551—21no—
24#4551—21no—
25#4529—22no—
26#4484—22no—
27#4483—22no—
28#2538—31no—
29#1834—31no—
30#1803—31no—
31#1802—31no—
32C2×Q8⋊C8—31no—
33C2×D4⋊C8—31no—
34#1296D311yescharacteristic, complemented
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6D411yescharacteristic
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C42C2311yescharacteristic
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—22no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56#897—42no—
57#897—42no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3⋊#95—22no—
71#821—42no—
72#816—42no—
73C4×C3⋊C8—21no—
74C4×C3⋊C8D412yes—
75C2×C22⋊Q8—61no—
76C2×C4×Q8—31no—
77C2×C4×D4—31no—
78C2×C4⋊C8C2×D311yescharacteristic
79C2×C4⋊C8C2×D311yescharacteristic
80C2×#103—61no—
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—61no—
90#340—61no—
91#340—62no—
92Q8⋊C8—62no—
93D4⋊C8—62no—
94C23×D3—21no—
95C22×C3×C4—22no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—41no—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12Q1612yes—
109C2×C4×D3—41no—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×C4×D3—41no—
113C2×C4×D3—41no—
114C2×C4×D3—41no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6SD1612yes—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C3×C4⋊C4—42no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135C3×C42#4111yescharacteristic
136(C2×D3)⋊C4—44no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—82no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—44no—
145C4⋊Dic3—42no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C22×Q8—31no—
158C22×D4—31no—
159C23×C4—61no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—61no—
163C22⋊Q8—122no—
164C22⋊Q8—122no—
165C4×Q8—61no—
166C4×Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×D4—61no—
170C4×D4—61no—
171C4×D4—61no—
172C4×D4—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4D1212yes—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192#103—122no—
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—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
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×C4C2×Q1611yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×Dic3—22no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236D12—22no—
237D12—22no—
238D12—41no—
239D12—41no—
240C4×D3—81no—
241C4×D3—81no—
242C4×D3—81no—
243C4×D3—81no—
244Dic6—22no—
245Dic6—22no—
246Dic6—41no—
247Dic6—41no—
248C3×C8—81no—
249C3×C8—81no—
250C3⋊C8—42no—
251C24—61no—
252C2×Q8—32no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×D4—32no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C22×C4—22no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4C2×D1211yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—121no—
286C2×C8—121no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—122no—
297C4⋊C4—124no—
298C4⋊C4—121no—
299C4⋊C4—121no—
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—121no—
309C42—121no—
310C42#23511yescharacteristic
311C42#23511yescharacteristic
312C42D3×D411yescharacteristic
313C42#23911yescharacteristic
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—82no—
331C3×C4#46511yescharacteristic
332C3×C4#46711yescharacteristic
333C3×C4#46811yescharacteristic
334C3×C4#46911yescharacteristic
335C3×C4#48111yescharacteristic
336C3×C4#48211yescharacteristic
337C3×C4#48411yescharacteristic
338C3×C4#48511yescharacteristic
339Dic3—42no—
340Dic3—81no—
341Dic3—81no—
342Dic3—161no—
343C23—62no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23#88211yescharacteristic
351Q8—62no—
352Q8—62no—
353Q8—121no—
354Q8—121no—
355D4—62no—
356D4—62no—
357D4—121no—
358D4—121no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—62no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4—241no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4#87511yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4#88411yescharacteristic
383C2×C4#89411yescharacteristic
384C2×C4#89411yescharacteristic
385C2×C4#89511yescharacteristic
386C2×C4#91111yescharacteristic
387C2×C4#91111yescharacteristic
388C2×C4D3×SD1611yescharacteristic
389C2×C4#91411yescharacteristic
390C2×C4#91511yescharacteristic
391C2×C4#91611yescharacteristic
392C2×C4D3×Q1611yescharacteristic
393C2×C4#91911yescharacteristic
394C8—81no—
395C8—81no—
396C8—122no—
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#467211yescharacteristic
413C22#474611yescharacteristic
414C22#475311yescharacteristic
415C22#477611yescharacteristic
416C22#478111yescharacteristic
417C22#480911yescharacteristic
418C22#481011yescharacteristic
419C4—21no—
420C4—21no—
421C4—82no—
422C4—122no—
423C4—241no—
424C4—241no—
425C4—481no—
426C4#478011yescharacteristic
427C4#478211yescharacteristic
428C4#480711yescharacteristic
429C4#481311yescharacteristic
430C4#486311yescharacteristic
431C4#486811yescharacteristic
432C4#488411yescharacteristic
433C4#488511yescharacteristic
434C3#1043611yescharacteristic, complemented, Sylow
435C2—242no—
436C2#6839311yescharacteristic
437C2#6843311yescharacteristic
438C2#6845911yescharacteristic
439C2#6850611yescharacteristic
440C2#6882311yescharacteristic
441C2#6882511yescharacteristic
442C2#6897711yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1296D3
hypocenterC3#10436
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #70485C2semidirect
4. C2×#4484C2semidirect
6. #69574C2semidirect
7. C3×#1296C2semidirect
34. #1296D3semidirect
434. C3#10436semidirect

Maximal subgroups

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

subgroupindex
#743692
#704852
#704792
C2×#44842
C2×#44832
#695742
C3×#12962
#104363