Finite Group Databasebeta

← #10854151

Subgroups — #10854151

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74336C211yesmaximal, characteristic, complemented
2#74324C211yesmaximal, characteristic
3C3×#1845C211yesmaximal, characteristic, complemented
4#70641C212yesmaximal
5#70639C212yesmaximal
6#70615C212yesmaximal, complemented
7#70609C212yesmaximal, complemented
8#69786C211yesmaximal, characteristic, complemented
9C4×#821C211yesmaximal, characteristic
10#69584C211yesmaximal, characteristic
11#69567C211yesmaximal, characteristic, complemented
12#26086—31nomaximal, Sylow
13C2×#892—21no—
14C2×#890—21no—
15C2×C4×D12C2211yescharacteristic
16C2×C3×#107C2212yes—
17C2×C3×#107C2212yes—
18C2×C3×C4×C8C2211yescharacteristic
19C3×#384C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C2×#821C2211yescharacteristic
22C2×#821C2211yescharacteristic
23C2×Dic3⋊C8C2212yes—
24#4990—22no—
25#4988C2212yes—
26#4987—21no—
27#4984C2212yes—
28#4982—22no—
29#4975C2212yes—
30Dic3×C4⋊C4C2212yes—
31#4972—21no—
32C2×#809C2212yes—
33C2×#807C2212yes—
34C4×C4⋊Dic3C2211yescharacteristic
35C4×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4551C2211yescharacteristic
38#4551C2211yescharacteristic
39#4550C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#2552—31no—
43#2354—31no—
44#2093—32no—
45#2067—32no—
46#1947—32no—
47#1947—31no—
48#1939—31no—
49#1918—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×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3C2312yes—
78C22×C3⋊C8C2312yes—
79#892—44no—
80#890—44no—
81C4×D12—22no—
82C4×D12Q814yes—
83C3×#107—24no—
84C3×#107—42no—
85C3×C4×C8—22no—
86C3⋊#95—21no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95D412yes—
90#821—22no—
91#821—22no—
92Dic3⋊C8—24no—
93#809—42no—
94#807—42no—
95C4⋊(C3⋊C8)—22no—
96C2×#125—61no—
97C2×C4×D4—31no—
98C2×#117—61no—
99C2×#107—32no—
100C2×#107C2×D312yes—
101C2×#107C2×D312yes—
102C2×C4⋊C8—32no—
103C2×C4⋊C8—31no—
104C2×#102—31no—
105C2×#102—31no—
106C2×#102—32no—
107C2×C4×C8C2×D311yescharacteristic
108#398—62no—
109#390—32no—
110#389—32no—
111#389—61no—
112#388—62no—
113#384—32no—
114#384C2×D311yescharacteristic
115#384C2×D311yescharacteristic
116#380—61no—
117C4×C4⋊C4—31no—
118C4×C4⋊C4—32no—
119C4×C4⋊C4—31no—
120#340—31no—
121#340—31no—
122#340—32no—
123#340—32no—
124#336—32no—
125C23×D3—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2411yescharacteristic
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3C2×D412yes—
135C22×Dic3C2×D412yes—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12C2×Q812yes—
142C2×C4×D3—24no—
143C2×C4×D3—42no—
144C2×C4×D3—42no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8#4112yes—
150C2×C3×C8#4112yes—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4—81no—
158C3×C4⋊C4—81no—
159C3×C42C2×Q812yes—
160C3×C42#4112yes—
161(C2×D3)⋊C4—24no—
162(C2×D3)⋊C4—44no—
163(C2×D3)⋊C4—82no—
164C4⋊Dic3—22no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—44no—
171C4⋊Dic3D814yes—
172C4⋊Dic3C2×Q812yes—
173Dic3⋊C4—44no—
174Dic3⋊C4—82no—
175C4×Dic3—22no—
176C4×Dic3—22no—
177C4×Dic3—42no—
178C4×Dic3—44no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C22×D4—31no—
183C23×C4—61no—
184C22×C8—32no—
185C22×C8C22×D311yescharacteristic
186C22×C8C22×D311yescharacteristic
187#125—124no—
188C4×D4—34no—
189C4×D4—62no—
190#117—124no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4C22×D312yes—
204C2×C4⋊C4C22×D312yes—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C42—32no—
209C2×C42—62no—
210C2×C42—61no—
211C2×C42C22×D311yescharacteristic
212#107—24no—
213#107—42no—
214#107—122no—
215C4⋊C8—64no—
216C4⋊C8—62no—
217#102—62no—
218#102—62no—
219#102—122no—
220C4×C8—22no—
221#95—32no—
222#95—62no—
223#95—62no—
224#95—61no—
225C23×C3C22×D411yescharacteristic
226C22×D3—22no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4C22⋊Q812yes—
242C2×C3×C4C22⋊Q812yes—
243C2×C3×C4C2×D812yes—
244C2×C3×C4#13712yes—
245C2×C3×C4C22×Q811yescharacteristic
246C2×C3×C4C2×#4111yescharacteristic
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4111yescharacteristic
249C2×C3×C4H(5,𝔽2)11yescharacteristic
250C2×C3×C42−1+411yescharacteristic
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—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3C2×D814yes—
266D12—22no—
267D12—22no—
268D12—41no—
269D12—41no—
270C4×D3—48no—
271C3×C8—22no—
272C3×C8—22no—
273C3⋊C8—42no—
274C24—61no—
275C2×D4—32no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—61no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—32no—
284C22×C4—32no—
285C22×C4—61no—
286C22×C4—64no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—122no—
290C22×C4—122no—
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4C23×D311yescharacteristic
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—62no—
301C2×C8#23712yes—
302C2×C8#23912yes—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—34no—
308C4⋊C4—32no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—62no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C4⋊C4—124no—
320C4⋊C4—124no—
321C4⋊C4—242no—
322C22⋊C4—64no—
323C22⋊C4—124no—
324C22⋊C4—242no—
325C42—62no—
326C42—62no—
327C42—124no—
328C42—122no—
329C42#23512yes—
330C42D3×Q812yes—
331C22×C3C2×C22⋊Q811yescharacteristic
332C22×C3#54711yescharacteristic
333C22×C3#55011yescharacteristic
334C22×C3C22×D811yescharacteristic
335C22×C3C2×#13711yescharacteristic
336C22×C3#57611yescharacteristic
337C22×C3#57811yescharacteristic
338C2×D3—42no—
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—81no—
342C2×D3—81no—
343C2×D3—81no—
344C2×D3—81no—
345C3×C4—22no—
346C3×C4—82no—
347C3×C4—82no—
348C3×C4#47414yes—
349C3×C4#53311yescharacteristic
350C3×C4#54411yescharacteristic
351C3×C4#55211yescharacteristic
352C3×C4#55311yescharacteristic
353Dic3—24no—
354Dic3—42no—
355Dic3—42no—
356Dic3—81no—
357Dic3—81no—
358C23—62no—
359C23—121no—
360C23—121no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23C2×D3×D411yescharacteristic
366D4—62no—
367D4—62no—
368D4—121no—
369D4—121no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—34no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—128no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4#89612yes—
395C2×C4#89612yes—
396C2×C4D3×D812yes—
397C2×C4#91812yes—
398C2×C4C2×#23511yescharacteristic
399C2×C4C2×#23711yescharacteristic
400C2×C4#100411yescharacteristic
401C2×C4C2×D3×Q811yescharacteristic
402C2×C4C2×#23911yescharacteristic
403C2×C4#100711yescharacteristic
404C8—22no—
405C8—22no—
406C8—122no—
407C2×C3#244111yescharacteristic
408C2×C3C2×#47411yescharacteristic
409C2×C3#308311yescharacteristic
410C2×C3#333411yescharacteristic
411C2×C3#335711yescharacteristic
412C2×C3#339111yescharacteristic
413C2×C3#339811yescharacteristic
414D3—82no—
415C22—122no—
416C22—122no—
417C22—122no—
418C22—241no—
419C22—241no—
420C22—241no—
421C22—241no—
422C22C2×#89611yescharacteristic
423C22#571811yescharacteristic
424C22#572611yescharacteristic
425C22C2×D3×D811yescharacteristic
426C22#575711yescharacteristic
427C22C2×#91811yescharacteristic
428C22#576611yescharacteristic
429C4—22no—
430C4—64no—
431C4—82no—
432C4—82no—
433C4—122no—
434C4—122no—
435C4—241no—
436C4—241no—
437C4#489014yes—
438C4#551311yescharacteristic
439C4#551411yescharacteristic
440C4#572811yescharacteristic
441C4#572911yescharacteristic
442C3#2608611yescharacteristic, complemented, Sylow
443C2—242no—
444C2#7432711yescharacteristic
445C2C2×#489011yescharacteristic
446C2#7916211yescharacteristic
447C2#8177811yescharacteristic
448C2#8179711yescharacteristic
449C2#8194611yescharacteristic
450C2#8195111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1845C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1845D3
hypocenterC3#26086
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. #74336C2semidirect
3. C3×#1845C2semidirect
6. #70615C2semidirect
7. #70609C2semidirect
8. #69786C2semidirect
11. #69567C2semidirect
50. #1845D3semidirect
442. C3#26086semidirect

Maximal subgroups

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

subgroupindex
#743362
#743242
C3×#18452
#706412
#706392
#706152
#706092
#697862
C4×#8212
#695842
#695672
#260863