Finite Group Databasebeta

← #10854161

Subgroups — #10854161

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#74324C211yesmaximal, characteristic
2#73725C211yesmaximal, characteristic, complemented
3C3×#1948C211yesmaximal, characteristic, complemented
4#70586C212yesmaximal
5#70564C212yesmaximal, complemented
6#70479C211yesmaximal, characteristic
7#70468C211yesmaximal, characteristic, complemented
8#69577C211yesmaximal, characteristic, complemented
9C4×#809C211yesmaximal, characteristic
10#68901C212yesmaximal
11#68790C212yesmaximal, complemented
12#26087—31nomaximal, Sylow
13C2×#892—21no—
14C2×C4×D12C2211yescharacteristic
15C2×C3×#106C2212yes—
16C2×C3×C4⋊C8C2211yescharacteristic
17C3×#384C2211yescharacteristic
18C3×C4×C4⋊C4C2211yescharacteristic
19C2×#821C2212yes—
20C2×Dic3⋊C8C2212yes—
21#4990—22no—
22#4988C2212yes—
23#4987—21no—
24#4977C2212yes—
25Dic3×C4⋊C4C2212yes—
26#4972C2212yes—
27C2×#809C2211yescharacteristic
28C2×#809C2211yescharacteristic
29C2×#807C2212yes—
30C2×#807C2212yes—
31#4936—21no—
32#4935C2211yescharacteristic
33C4×C4⋊Dic3C2211yescharacteristic
34#4928—21no—
35C2×C4×C3⋊C8C2211yescharacteristic
36#4660—22no—
37C3×#340C2212yes—
38#4551C2212yes—
39#4550C2212yes—
40#4529C2211yescharacteristic
41#4529C2211yescharacteristic
42#2571—31no—
43#2354—31no—
44#2092—32no—
45#2067—32no—
46#1949—32no—
47#1948D311yescharacteristic, complemented
48#1939—31no—
49#1921—32no—
50#1845—31no—
51#1802—31no—
52C4×#102—31no—
53C22×D12C2311yescharacteristic
54C22×C4×D3—21no—
55C22×C3×C8C2312yes—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C42—21no—
62C2×C3×C42C2311yescharacteristic
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2311yescharacteristic
72C2×Dic3⋊C4—22no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3C2312yes—
75C22×C3⋊C8C2311yescharacteristic
76C22×C3⋊C8C2311yescharacteristic
77#892—44no—
78C4×D12—22no—
79C4×D12Q814yes—
80C3×#106—42no—
81C3×C4⋊C8—22no—
82C3×#95—21no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95D412yes—
88#821—42no—
89Dic3⋊C8—24no—
90#809—22no—
91#809—22no—
92#807—24no—
93#807—42no—
94C4×C3⋊C8—22no—
95C2×C4×D4—31no—
96C2×#117—61no—
97C2×#107—32no—
98C2×#107—32no—
99C2×#106C2×D312yes—
100C2×C4⋊C8—32no—
101C2×C4⋊C8C2×D311yescharacteristic
102C2×#102—31no—
103C2×#102—31no—
104C2×#102—32no—
105C2×C4×C8—31no—
106#400—62no—
107#390—32no—
108#389—32no—
109#389—61no—
110#388—62no—
111#384—31no—
112#384C2×D311yescharacteristic
113#382—32no—
114#382—61no—
115#380—61no—
116C4×C4⋊C4—31no—
117C4×C4⋊C4—32no—
118C4×C4⋊C4C2×D311yescharacteristic
119#340—32no—
120#340—31no—
121#340—31no—
122#340C2×D312yes—
123#336—32no—
124C23×D3—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2411yescharacteristic
131C22×Dic3—21no—
132C22×Dic3—22no—
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—22no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—42no—
154C3×C4⋊C4—81no—
155C3×C42—42no—
156C3×C42C2×Q812yes—
157C3×C42#4112yes—
158(C2×D3)⋊C4—24no—
159(C2×D3)⋊C4—44no—
160(C2×D3)⋊C4—82no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—21no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—42no—
167C4⋊Dic3—81no—
168C4⋊Dic3C2×Q812yes—
169Dic3⋊C4—82no—
170C4×Dic3—22no—
171C4×Dic3—22no—
172C4×Dic3—44no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8#4112yes—
178C2×C3⋊C8#4112yes—
179C22×D4—31no—
180C23×C4—61no—
181C22×C8—31no—
182C22×C8—31no—
183C22×C8C22×D312yes—
184C4×D4—34no—
185C4×D4—62no—
186#117—124no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—32no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4C22×D311yescharacteristic
197C2×C4⋊C4C22×D311yescharacteristic
198C2×C4⋊C4C22×D312yes—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C42—21no—
203C2×C42—32no—
204C2×C42—62no—
205C2×C42C22×D311yescharacteristic
206#107—64no—
207#107—122no—
208#106—42no—
209C4⋊C8—22no—
210C4⋊C8—64no—
211#102—62no—
212#102—62no—
213#102—122no—
214C4×C8—62no—
215#95—21no—
216#95—32no—
217#95—62no—
218#95—62no—
219#95—62no—
220#95—62no—
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—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4C22⋊Q812yes—
241C2×C3×C4C22⋊Q812yes—
242C2×C3×C4#13512yes—
243C2×C3×C4AGL(1,ℤ/8)12yes—
244C2×C3×C4C22×Q811yescharacteristic
245C2×C3×C4C2×#4111yescharacteristic
246C2×C3×C4C2×#4111yescharacteristic
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4H(5,𝔽2)11yescharacteristic
249C2×C3×C42−1+411yescharacteristic
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262D12—22no—
263D12—22no—
264D12—41no—
265D12—41no—
266C4×D3—48no—
267C3×C8—42no—
268C3⋊C8—22no—
269C3⋊C8—22no—
270C24—61no—
271C2×D4—32no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—32no—
281C22×C4—32no—
282C22×C4—61no—
283C22×C4—64no—
284C22×C4—62no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4C23×D311yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—32no—
294C2×C8—32no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—61no—
298C2×C8—61no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—32no—
306C4⋊C4—42no—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—61no—
310C4⋊C4—62no—
311C4⋊C4—62no—
312C4⋊C4—81no—
313C4⋊C4—122no—
314C4⋊C4—241no—
315C4⋊C4—242no—
316C22⋊C4—64no—
317C22⋊C4—124no—
318C22⋊C4—242no—
319C42—42no—
320C42—62no—
321C42—62no—
322C42—124no—
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—81no—
341C3×C4—81no—
342C3×C4—82no—
343C3×C4#47814yes—
344C3×C4#53311yescharacteristic
345C3×C4#54411yescharacteristic
346C3×C4#55211yescharacteristic
347C3×C4#55311yescharacteristic
348Dic3—42no—
349Dic3—42no—
350Dic3—42no—
351Dic3—82no—
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—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—128no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4#89612yes—
389C2×C4#89612yes—
390C2×C4#91412yes—
391C2×C4#91612yes—
392C2×C4C2×#23511yescharacteristic
393C2×C4#100411yescharacteristic
394C2×C4C2×D3×Q811yescharacteristic
395C2×C4#100711yescharacteristic
396C2×C4D3×#4111yescharacteristic
397C2×C4D3×#4111yescharacteristic
398C8—42no—
399C8—62no—
400C8—62no—
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#559911yescharacteristic
418C22#563111yescharacteristic
419C22C2×#91411yescharacteristic
420C22C2×#91611yescharacteristic
421C22#577411yescharacteristic
422C22#577911yescharacteristic
423C4—22no—
424C4—81no—
425C4—81no—
426C4—82no—
427C4—122no—
428C4—122no—
429C4—122no—
430C4—242no—
431C4#487114yes—
432C4#555811yescharacteristic
433C4#558111yescharacteristic
434C4#568811yescharacteristic
435C4#572911yescharacteristic
436C3#2608711yescharacteristic, complemented, Sylow
437C2—242no—
438C2#7622411yescharacteristic
439C2C2×#487111yescharacteristic
440C2#7924311yescharacteristic
441C2#8037911yescharacteristic
442C2#8042111yescharacteristic
443C2#8103211yescharacteristic
444C2#8108211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1948D3
hypocenterC3#26087
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. #73725C2semidirect
3. C3×#1948C2semidirect
5. #70564C2semidirect
7. #70468C2semidirect
8. #69577C2semidirect
11. #68790C2semidirect
47. #1948D3semidirect
436. C3#26087semidirect

Maximal subgroups

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

subgroupindex
#743242
#737252
C3×#19482
#705862
#705642
#704792
#704682
#695772
C4×#8092
#689012
#687902
#260873