Finite Group Databasebeta

← #10814411

Subgroups — #10814411

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D24C211yesmaximal, characteristic
2#73770C211yesmaximal, characteristic, complemented
3#73645C211yesmaximal, characteristic, complemented
4C3×#1770C211yesmaximal, characteristic, complemented
5#70480C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#69818C212yesmaximal, complemented
8#69812C212yesmaximal, complemented
9#69763C211yesmaximal, characteristic, complemented
10#68673C212yesmaximal
11#68621C212yesmaximal, complemented
12#22764—31nomaximal, Sylow
13C22×D24C2211yescharacteristic
14C2×C4×D12C2212yes—
15C2×C4×D12C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#97C2212yes—
19C2×C3×C4×C8C2211yescharacteristic
20C3×C4×C4⋊C4C2211yescharacteristic
21C2×#821C2212yes—
22C2×#821C2212yes—
23C2×#821C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#818C2211yescharacteristic
26#4990—22no—
27#4989—22no—
28#4939C2212yes—
29#4936C2211yescharacteristic
30#4936C2211yescharacteristic
31C4×C4⋊Dic3C2212yes—
32#4928C2212yes—
33C4×D24C418yescomplemented
34#4667—21no—
35#4666—21no—
36#4664—21no—
37D3×#95—21no—
38C3×#336C2212yes—
39#4551C2212yes—
40#4551C2212yes—
41#4551C2212yes—
42#4551C2212yes—
43C2×C4×D8—31no—
44#2357—31no—
45#2349—31no—
46#1936—32no—
47#1918—32no—
48#1879—32no—
49#1835—31no—
50#1804—31no—
51#1804—31no—
52#1770D311yescharacteristic, complemented
53#1762—32no—
54C22×D12C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C22×C4×D3—21no—
58C22×C4×D3—22no—
59C22×C3×C8C2312yes—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C2×C3×C4⋊C4D412yes—
63C2×C3×C42C2312yes—
64C2×C3×C42C2311yescharacteristic
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2312yes—
72C2×C4⋊Dic3C2311yescharacteristic
73C2×C4⋊Dic3C2311yescharacteristic
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76C2×D24—22no—
77C2×D24—22no—
78C2×D24C2×C414yes—
79C4×D12—22no—
80C4×D12—22no—
81C4×D12—24no—
82C4×D12—42no—
83C4×D12C2×C414yes—
84C4×D12C2×C414yescomplemented
85C3×C4⋊C8—22no—
86C3×#97—42no—
87C3×C4×C8C2×C414yescomplemented
88C3×#95D411yescharacteristic
89C3×#95D411yescharacteristic
90C3⋊#95—21no—
91C3⋊#95—22no—
92C3⋊#95—21no—
93C3⋊#95—22no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96#821—24no—
97#821—42no—
98#821C2×C414yes—
99#821C2×C414yescomplemented
100#818C2×C414yescomplemented
101C22×D8—31no—
102C2×C4×D4—31no—
103C2×C4×D4—31no—
104C2×C4×D4—32no—
105C4×D8—38no—
106C2×#107—31no—
107C2×C4⋊C8C2×D311yescharacteristic
108C2×#102—31no—
109C2×#102—31no—
110C2×#102—32no—
111C2×#102—32no—
112C2×#97C2×D312yes—
113C2×C4×C8C2×D311yescharacteristic
114#390—32no—
115#389—32no—
116#388—62no—
117#385—62no—
118#382—31no—
119#382—31no—
120#381—61no—
121#380—61no—
122C4×C4⋊C4—32no—
123C4×C4⋊C4C2×D311yescharacteristic
124C4×C22⋊C4—61no—
125C2×#95—61no—
126#340—32no—
127#340—32no—
128#340—32no—
129#340—32no—
130#336C2×D312yes—
131C23×D3—21no—
132C23×D3—21no—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12C22×C412yes—
151C2×D12C22×C412yes—
152C2×D12#4112yes—
153C2×D12#4112yes—
154C2×C4×D3—24no—
155C2×C4×D3—24no—
156C2×C4×D3—44no—
157C2×C4×D3—44no—
158C2×C4×D3—82no—
159C2×C4×D3—82no—
160C2×C4×D3—82no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8C22×C412yes—
169C2×C3×C8C22×C412yes—
170C3×C4⋊C4—42no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42C22×C412yes—
174C3×C42#4112yes—
175(C2×D3)⋊C4—24no—
176(C2×D3)⋊C4—24no—
177(C2×D3)⋊C4—44no—
178(C2×D3)⋊C4—44no—
179(C2×D3)⋊C4—82no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3C22×C412yes—
187C4⋊Dic3C22×C412yes—
188Dic3⋊C4—82no—
189C4×Dic3—82no—
190D24—28no—
191C22×D4—31no—
192C22×D4—31no—
193C23×C4—61no—
194C23×C4—61no—
195C23×C4—62no—
196C2×D8—34no—
197C2×D8—62no—
198C2×D8—62no—
199C22×C8C22×D312yes—
200C22×C8C22×D311yescharacteristic
201C22×C8C22×D311yescharacteristic
202C4×D4—34no—
203C4×D4—34no—
204C4×D4—64no—
205C4×D4—62no—
206C4×D4—62no—
207C4×D4—122no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—32no—
212C2×C4⋊C4—62no—
213C2×C4⋊C4D1212yes—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C42—62no—
220C2×C42C22×D312yes—
221C2×C42C22×D311yescharacteristic
222#107—34no—
223C4⋊C8—22no—
224#102—34no—
225#102—34no—
226#102—64no—
227#102—122no—
228#97—42no—
229C4×C8C4×D314yescomplemented
230#95—61no—
231#95—62no—
232#95—61no—
233#95—62no—
234#95—61no—
235#95—61no—
236#95D1211yescharacteristic
237#95D1211yescharacteristic
238C23×C3C22×D411yescharacteristic
239C22×D3—22no—
240C22×D3—22no—
241C22×D3—22no—
242C22×D3—22no—
243C22×D3—42no—
244C22×D3—42no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—42no—
248C22×D3—42no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4C4×D412yes—
258C2×C3×C4C4×D412yes—
259C2×C3×C4C23×C411yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C4H(5,𝔽2)11yescharacteristic
264C2×C3×C42−1+411yescharacteristic
265C2×Dic3—22no—
266C2×Dic3—22no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275D12—22no—
276D12—22no—
277D12—22no—
278D12—22no—
279D12—22no—
280D12—41no—
281D12—41no—
282D12#11714yes—
283C4×D3—84no—
284C4×D3—84no—
285C4×D3—88no—
286C3×C8—22no—
287C3×C8—22no—
288C3×C8—42no—
289C24—61no—
290C24—61no—
291C2×D4—32no—
292C2×D4—32no—
293C2×D4—32no—
294C2×D4—32no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—64no—
306C22×C4—64no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—124no—
310C22×C4—124no—
311C22×C4—242no—
312C22×C4—242no—
313C22×C4—242no—
314C22×C4C2×D1212yes—
315C22×C4C2×D1212yes—
316C22×C4C2×D1211yescharacteristic
317C22×C4C2×D1211yescharacteristic
318C22×C4C23×D311yescharacteristic
319D8—68no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8C2×C4×D312yes—
328C2×C8C2×C4×D312yes—
329C4⋊C4—32no—
330C4⋊C4—32no—
331C4⋊C4—42no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C4⋊C4—62no—
337C4⋊C4—62no—
338C4⋊C4—242no—
339C22⋊C4—64no—
340C22⋊C4—64no—
341C22⋊C4—124no—
342C22⋊C4—124no—
343C22⋊C4—242no—
344C42—22no—
345C42—22no—
346C42—242no—
347C42C2×C4×D312yes—
348C42#23512yes—
349C22×C3C2×C4×D411yescharacteristic
350C22×C3#54611yescharacteristic
351C22×C3#54811yescharacteristic
352C22×C3#57512yes—
353C22×C3#57612yes—
354C2×D3—24no—
355C2×D3—44no—
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—44no—
359C2×D3—42no—
360C2×D3—42no—
361C2×D3—42no—
362C2×D3—82no—
363C2×D3—82no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4C2×#11711yescharacteristic
369C3×C4#52011yescharacteristic
370C3×C4#55211yescharacteristic
371C3×C4#55311yescharacteristic
372Dic3—42no—
373Dic3—42no—
374Dic3—82no—
375Dic3—82no—
376C23—62no—
377C23—62no—
378C23—62no—
379C23—62no—
380C23—122no—
381C23—122no—
382C23—122no—
383C23—122no—
384C23—121no—
385C23—121no—
386C23C22×D1211yescharacteristic
387D4—34no—
388D4—62no—
389D4—62no—
390D4—62no—
391D4—62no—
392D4—62no—
393D4—121no—
394D4—121no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—244no—
414C2×C4—244no—
415C2×C4—248no—
416C2×C4C4×D1212yes—
417C2×C4C4×D1212yes—
418C2×C4C22×C4×D311yescharacteristic
419C2×C4C2×#23511yescharacteristic
420C2×C4#100411yescharacteristic
421C2×C4#100711yescharacteristic
422C2×C4D3×#4111yescharacteristic
423C2×C4D3×#4111yescharacteristic
424C8—22no—
425C8—22no—
426C8—42no—
427C2×C3#235211yescharacteristic
428C2×C3#294311yescharacteristic
429C2×C3#294811yescharacteristic
430C2×C3#332212yes—
431C2×C3#337012yes—
432D3—44no—
433D3—82no—
434C22—64no—
435C22—124no—
436C22—122no—
437C22—124no—
438C22—122no—
439C22—122no—
440C22—122no—
441C22—122no—
442C22—242no—
443C22—242no—
444C22C2×C4×D1211yescharacteristic
445C22#555411yescharacteristic
446C22#557611yescharacteristic
447C22#574812yes—
448C22#575212yes—
449C4—22no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—122no—
454C4—122no—
455C4—242no—
456C4—242no—
457C4#550411yescharacteristic
458C4D3×#11711yescharacteristic
459C4#555811yescharacteristic
460C4#558111yescharacteristic
461C3#2276411yescharacteristic, complemented, Sylow
462C2—124no—
463C2—242no—
464C2#7377111yescharacteristic
465C2#7849411yescharacteristic
466C2#7856111yescharacteristic
467C2#7955312yes—
468C2#7997112yes—
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1770C2
socleC23×C3C22×D4
prime radicalC22×D24C22
cosocleC22×C3×C4C24
hypercenter#1770D3
hypocenterC3#22764
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1770

Complements

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

NH ≅ G/Ntype
2. #73770C2semidirect
3. #73645C2semidirect
4. C3×#1770C2semidirect
5. #70480C2semidirect
6. #70480C2semidirect
7. #69818C2semidirect
8. #69812C2semidirect
9. #69763C2semidirect
11. #68621C2semidirect
33. C4×D24C4semidirect
52. #1770D3semidirect
84. C4×D12C2×C4semidirect
87. C3×C4×C8C2×C4semidirect
99. #821C2×C4semidirect
100. #818C2×C4semidirect
229. C4×C8C4×D3semidirect
461. C3#22764semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D242
#737702
#736452
C3×#17702
#704802
#698182
#698122
#697632
#686732
#686212
#227643