Finite Group Databasebeta

← #10854162

Subgroups — #10854162

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74342C211yesmaximal, characteristic, complemented
2#74324C211yesmaximal, characteristic
3C3×#1845C211yesmaximal, characteristic, complemented
4#70640C212yesmaximal
5#70639C212yesmaximal
6#70616C212yesmaximal, complemented
7#70610C212yesmaximal, complemented
8#69786C211yesmaximal, characteristic, complemented
9C4×#821C211yesmaximal, characteristic
10#69584C211yesmaximal, characteristic
11#69568C211yesmaximal, characteristic, complemented
12#26087—31nomaximal, Sylow
13C2×#892—21no—
14C2×C4×D12C2211yescharacteristic
15C2×C3×#107C2212yes—
16C2×C3×#107C2212yes—
17C2×C3×C4×C8C2211yescharacteristic
18C3×#384C2211yescharacteristic
19C3×#384C2211yescharacteristic
20C2×#821C2211yescharacteristic
21C2×#821C2211yescharacteristic
22C2×Dic3⋊C8C2212yes—
23#4990—22no—
24#4988C2212yes—
25#4987—21no—
26#4986—22no—
27#4984—21no—
28#4984C2212yes—
29#4977—21no—
30#4977C2212yes—
31Dic3×C4⋊C4C2212yes—
32C2×#809C2212yes—
33C2×#808C2212yes—
34C4×C4⋊Dic3C2211yescharacteristic
35C4×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4551C2211yescharacteristic
38#4551C2211yescharacteristic
39#4550C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#2571—31no—
43#2354—31no—
44#2092—32no—
45#2067—32no—
46#1949—32no—
47#1948—31no—
48#1939—31no—
49#1921—32no—
50#1845D311yescharacteristic, complemented
51#1802—31no—
52C4×#102—31no—
53C22×D12C2311yescharacteristic
54C22×C4×D3—21no—
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C42C2311yescharacteristic
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×C4⋊Dic3C2311yescharacteristic
71C2×C4⋊Dic3C2311yescharacteristic
72C2×C4⋊Dic3C2311yescharacteristic
73C2×C4×Dic3—21no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3C2312yes—
76C22×C3⋊C8C2312yes—
77#892—44no—
78C4×D12—22no—
79C4×D12Q814yes—
80C3×#107—24no—
81C3×#107—42no—
82C3×C4×C8—22no—
83C3⋊#95—21no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95D412yes—
89#821—22no—
90#821—22no—
91Dic3⋊C8—24no—
92#809—42no—
93#808—42no—
94C4⋊(C3⋊C8)—22no—
95C2×C4×D4—31no—
96C2×#117—61no—
97C2×#107C2×D312yes—
98C2×#107C2×D312yes—
99C2×#106—32no—
100C2×C4⋊C8—32no—
101C2×C4⋊C8—31no—
102C2×#102—31no—
103C2×#102—31no—
104C2×#102—32no—
105C2×C4×C8C2×D311yescharacteristic
106#400—62no—
107#390—32no—
108#389—32no—
109#389—61no—
110#388—62no—
111#384C2×D311yescharacteristic
112#384C2×D311yescharacteristic
113#382—32no—
114#382—61no—
115#380—61no—
116C4×C4⋊C4—31no—
117C4×C4⋊C4—32no—
118C4×C4⋊C4—31no—
119#340—31no—
120#340—31no—
121#340—32no—
122#340—32no—
123#336—32no—
124C23×D3—21no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2411yescharacteristic
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3C2×D412yes—
134C22×Dic3C2×D412yes—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12C2×Q812yes—
141C2×C4×D3—24no—
142C2×C4×D3—42no—
143C2×C4×D3—42no—
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8#4112yes—
149C2×C3×C8#4112yes—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—81no—
157C3×C4⋊C4—81no—
158C3×C42C2×Q812yes—
159C3×C42#4112yes—
160(C2×D3)⋊C4—24no—
161(C2×D3)⋊C4—44no—
162(C2×D3)⋊C4—82no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—82no—
171C4⋊Dic3C2×Q812yes—
172C4×Dic3—22no—
173C4×Dic3—22no—
174C4×Dic3—44no—
175C4×Dic3—42no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C22×D4—31no—
180C23×C4—61no—
181C22×C8—32no—
182C22×C8C22×D311yescharacteristic
183C22×C8C22×D311yescharacteristic
184C4×D4—34no—
185C4×D4—62no—
186#117—124no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4C22×D312yes—
198C2×C4⋊C4C22×D312yes—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C42—32no—
203C2×C42—62no—
204C2×C42—61no—
205C2×C42C22×D311yescharacteristic
206#107—24no—
207#107—42no—
208#106—122no—
209C4⋊C8—64no—
210C4⋊C8—62no—
211#102—62no—
212#102—62no—
213#102—122no—
214C4×C8—22no—
215#95—32no—
216#95—62no—
217#95—62no—
218#95—62no—
219#95—62no—
220#95—61no—
221C23×C3C22×D411yescharacteristic
222C22×D3—22no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4C22⋊Q812yes—
238C2×C3×C4C22⋊Q812yes—
239C2×C3×C4#13512yes—
240C2×C3×C4AGL(1,ℤ/8)12yes—
241C2×C3×C4C22×Q811yescharacteristic
242C2×C3×C4C2×#4111yescharacteristic
243C2×C3×C4C2×#4111yescharacteristic
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4H(5,𝔽2)11yescharacteristic
246C2×C3×C42−1+411yescharacteristic
247C2×Dic3—22no—
248C2×Dic3—22no—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262D12—22no—
263D12—22no—
264D12—41no—
265D12—41no—
266C4×D3—48no—
267C3×C8—22no—
268C3×C8—22no—
269C3⋊C8—42no—
270C24—61no—
271C2×D4—32no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—32no—
280C22×C4—32no—
281C22×C4—61no—
282C22×C4—64no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4C23×D311yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—62no—
295C2×C8—62no—
296C2×C8—62no—
297C2×C8#23712yes—
298C2×C8#23912yes—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—32no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—81no—
314C4⋊C4—81no—
315C4⋊C4—242no—
316C22⋊C4—64no—
317C22⋊C4—124no—
318C22⋊C4—242no—
319C42—62no—
320C42—62no—
321C42—124no—
322C42—122no—
323C42#23512yes—
324C42D3×Q812yes—
325C22×C3C2×C22⋊Q811yescharacteristic
326C22×C3#54711yescharacteristic
327C22×C3#55011yescharacteristic
328C22×C3C2×#13511yescharacteristic
329C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
330C22×C3#57611yescharacteristic
331C22×C3#57811yescharacteristic
332C2×D3—42no—
333C2×D3—42no—
334C2×D3—42no—
335C2×D3—81no—
336C2×D3—81no—
337C2×D3—81no—
338C2×D3—81no—
339C3×C4—22no—
340C3×C4—82no—
341C3×C4—82no—
342C3×C4#47814yes—
343C3×C4#53311yescharacteristic
344C3×C4#54411yescharacteristic
345C3×C4#55211yescharacteristic
346C3×C4#55311yescharacteristic
347Dic3—42no—
348Dic3—42no—
349Dic3—42no—
350Dic3—81no—
351Dic3—81no—
352C23—62no—
353C23—121no—
354C23—121no—
355C23—121no—
356C23—121no—
357C23—121no—
358C23—121no—
359C23C2×D3×D411yescharacteristic
360D4—62no—
361D4—62no—
362D4—121no—
363D4—121no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—128no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4#89612yes—
389C2×C4#89612yes—
390C2×C4#91112yes—
391C2×C4#91912yes—
392C2×C4C2×#23511yescharacteristic
393C2×C4C2×#23711yescharacteristic
394C2×C4#100411yescharacteristic
395C2×C4C2×D3×Q811yescharacteristic
396C2×C4C2×#23911yescharacteristic
397C2×C4#100711yescharacteristic
398C8—22no—
399C8—22no—
400C8—122no—
401C2×C3#244111yescharacteristic
402C2×C3C2×#47811yescharacteristic
403C2×C3#308311yescharacteristic
404C2×C3#334111yescharacteristic
405C2×C3#335311yescharacteristic
406C2×C3#339311yescharacteristic
407C2×C3#339911yescharacteristic
408D3—82no—
409C22—122no—
410C22—122no—
411C22—122no—
412C22—241no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22C2×#89611yescharacteristic
417C22#571811yescharacteristic
418C22#572611yescharacteristic
419C22C2×#91111yescharacteristic
420C22#575711yescharacteristic
421C22C2×#91911yescharacteristic
422C22#576611yescharacteristic
423C4—22no—
424C4—82no—
425C4—82no—
426C4—122no—
427C4—122no—
428C4—122no—
429C4—241no—
430C4—241no—
431C4#489114yes—
432C4#551311yescharacteristic
433C4#551411yescharacteristic
434C4#572811yescharacteristic
435C4#572911yescharacteristic
436C3#2608711yescharacteristic, complemented, Sylow
437C2—242no—
438C2#7432711yescharacteristic
439C2C2×#489111yescharacteristic
440C2#7916211yescharacteristic
441C2#8178011yescharacteristic
442C2#8179511yescharacteristic
443C2#8194811yescharacteristic
444C2#8195011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1845C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1845D3
hypocenterC3#26087
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
1. #74342C2semidirect
3. C3×#1845C2semidirect
6. #70616C2semidirect
7. #70610C2semidirect
8. #69786C2semidirect
11. #69568C2semidirect
50. #1845D3semidirect
436. C3#26087semidirect

Maximal subgroups

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

subgroupindex
#743422
#743242
C3×#18452
#706402
#706392
#706162
#706102
#697862
C4×#8212
#695842
#695682
#260873