Finite Group Databasebeta

← #10877164

Subgroups — #10877164

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#107C211yesmaximal, characteristic, complemented
2#78892C212yesmaximal
3#78655C211yesmaximal, characteristic
4C3×#2078C211yesmaximal, characteristic, complemented
5#70626C211yesmaximal, characteristic, complemented
6#70300C212yesmaximal, complemented
7#69994C211yesmaximal, characteristic, complemented
8#68990C212yesmaximal, complemented
9#68985C212yesmaximal, complemented
10#68939C211yesmaximal, characteristic, complemented
11#68923C211yesmaximal, characteristic, complemented
12#27675—31nomaximal, Sylow
13#5828C2211yescharacteristic
14C2×#948C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16C2×D3×C4⋊C4C2212yes—
17C2×C3×#107C2211yescharacteristic
18C3×#417C2212yes—
19C3×#408C2211yescharacteristic
20#5232C2212yes—
21#5126C2212yes—
22#5126C2211yescharacteristic
23C2×#822C2212yes—
24#5122C2211yescharacteristic
25#5121C2212yes—
26C2×#818C2211yescharacteristic
27#5102C2211yescharacteristic
28#4964C2212yes—
29C2×#807C2212yes—
30D3×#107—21no—
31D3×#107—21no—
32D3×#107—21no—
33D3×#107—21no—
34#4811—22no—
35#4805—22no—
36#4777—22no—
37#4774—22no—
38#4728—21no—
39#4728—21no—
40#67⋊C8—21no—
41#67⋊C8—21no—
42C3×#343C2212yes—
43C3×#341C2211yescharacteristic
44#4554C2212yes—
45#4554C2212yes—
46#4552C2211yescharacteristic
47#4530C2212yes—
48#3092—32no—
49#2929—31no—
50C22×#107—31no—
51#2089—31no—
52#2089—31no—
53#2078—31no—
54#2078D311yescharacteristic, complemented
55#1952—32no—
56#1821—32no—
57#1816—32no—
58#1810—31no—
59C2×C3×#41C2311yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3C2311yescharacteristic
62C2×C3×M16C2312yes—
63C2×C3×M16C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C2×C3×C4⋊C4C2312yes—
66#948—24no—
67#946D411yescharacteristic
68#946D411yescharacteristic
69#938D411yescharacteristic
70#938D411yescharacteristic
71#937—21no—
72#935—21no—
73C4×#67—21no—
74C4×#67—21no—
75#926C2311yescharacteristic
76C2×C4⋊Dic3C2312yes—
77C2×Dic3⋊C4—22no—
78C2×#208—21no—
79C2×#208C2312yes—
80C2×#208C2311yescharacteristic
81C22×C3⋊C8—21no—
82C22×C3⋊C8C2311yescharacteristic
83C2×D3×C8—24no—
84C2×D3×C8D411yescharacteristic
85C2×D3×C8D411yescharacteristic
86C2×D3×C8D411yescharacteristic
87C2×D3×C8D411yescharacteristic
88D3×C4⋊C4—22no—
89D3×C4⋊C4—22no—
90D3×C4⋊C4—22no—
91D3×C4⋊C4—22no—
92C3×#107—21no—
93C3×#107—21no—
94C3×#103—22no—
95C3×#102—22no—
96C3×C22⋊C8—21no—
97C3×C22⋊C8—21no—
98#835—22no—
99#832—22no—
100#822—24no—
101(C2×D3)⋊C8—21no—
102(C2×D3)⋊C8—21no—
103(C2×D3)⋊C8—22no—
104(C2×D3)⋊C8—22no—
105#818—21no—
106#818—21no—
107#818—22no—
108#817—22no—
109#815—21no—
110Dic3⋊C8—21no—
111Dic3⋊C8—21no—
112C8×Dic3—21no—
113#808—22no—
114#807—22no—
115#807—22no—
116#807—22no—
117C2×#131—31no—
118C23×C8—31no—
119#525—31no—
120C22×C4⋊C4—32no—
121#484—62no—
122#483—62no—
123#481—62no—
124#480—62no—
125#435—61no—
126#435—61no—
127C8×D4—61no—
128C8×D4—61no—
129C2×#108—32no—
130#428—32no—
131#428—32no—
132C2×#107—31no—
133C2×#107—32no—
134C2×#107—61no—
135C2×#107—61no—
136C2×#107—61no—
137C2×#107—61no—
138C2×#107C2×D311yescharacteristic
139#424—31no—
140#417—32no—
141#417C2×D312yes—
142#408C2×D311yescharacteristic
143#407—32no—
144#407—31no—
145#405—31no—
146#343—32no—
147#343—32no—
148#343—32no—
149#343C2×D312yes—
150#341—31no—
151#341C2×D311yescharacteristic
152C23×D3#4111yescharacteristic
153C3×#41—22no—
154C3×#41—22no—
155C2×C3×Q8C2×D411yescharacteristic
156C2×C3×D4—21no—
157C2×C3×D4C2×D411yescharacteristic
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4C2411yescharacteristic
161C2×#67—21no—
162C2×#67—21no—
163C22×Dic3—22no—
164C22×Dic3#4111yescharacteristic
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—42no—
168C2×C4×D3—42no—
169C2×C4×D3—84no—
170C2×C4×D3C2×D411yescharacteristic
171C2×C4×D3C2×D411yescharacteristic
172C2×C4×D3C2×D411yescharacteristic
173C2×C4×D3C2×D411yescharacteristic
174C3×M16—22no—
175C3×M16—22no—
176C3×M16—22no—
177C2×C3×C8—22no—
178C2×C3×C8—21no—
179C2×C3×C8—22no—
180C2×C3×C8C2×D411yescharacteristic
181C2×C3×C8C2×D411yescharacteristic
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184#217—21no—
185#217—21no—
186#217—21no—
187(C2×D3)⋊C4—21no—
188(C2×D3)⋊C4—21no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3C2×D411yescharacteristic
192C4⋊Dic3C2×D411yescharacteristic
193Dic3⋊C4—21no—
194Dic3⋊C4—21no—
195Dic3⋊C4—42no—
196Dic3⋊C4—42no—
197C4×Dic3—21no—
198#208—22no—
199#208—22no—
200#208—22no—
201#208—42no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—42no—
205C2×C3⋊C8C2×D411yescharacteristic
206C2×C3⋊C8C2×D411yescharacteristic
207C2×C3⋊C8#4114yes—
208D3×C8—22no—
209D3×C8—22no—
210D3×C8—22no—
211D3×C8—22no—
212C2×#41C22×D311yescharacteristic
213C23×C4—31no—
214C23×C4—62no—
215#131—64no—
216C2×M16—31no—
217C2×M16—32no—
218C2×M16—61no—
219C2×M16C22×D312yes—
220C2×M16C22×D311yescharacteristic
221C22×C8—31no—
222C22×C8—31no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8—31no—
226C22×C8—64no—
227C22×C8—61no—
228C22×C8C22×D311yescharacteristic
229#123—61no—
230C22⋊Q8—31no—
231C22⋊Q8—31no—
232#121—31no—
233#121—31no—
234#120—61no—
235C4×D4—61no—
236C4×D4—61no—
237#117—31no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4C22×D312yes—
245#108—64no—
246#107—21no—
247#107—21no—
248#107—62no—
249#107—62no—
250#107—62no—
251#107—62no—
252#107—61no—
253#107—61no—
254#106—62no—
255#106—62no—
256C4⋊C8—61no—
257C4⋊C8—61no—
258#103—22no—
259#103—62no—
260#102—22no—
261#102—62no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—61no—
265C22⋊C8—61no—
266C22⋊C8—62no—
267C22⋊C8—62no—
268#97—61no—
269C4×C8—61no—
270C23×C3—21no—
271C23×C3C2×#4111yescharacteristic
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3#12111yescharacteristic
276C22×D3#12111yescharacteristic
277C3×Q8—22no—
278C3×D4—22no—
279C3×D4—42no—
280C2×C3×C4—21no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—82no—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C2×#4112yes—
288C2×C3×C42−1+412yes—
289#67—42no—
290#67—42no—
291C2×Dic3—22no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—42no—
295C2×Dic3—82no—
296C2×Dic3#12111yescharacteristic
297C2×Dic3#12111yescharacteristic
298C4×D3—21no—
299C4×D3—21no—
300C4×D3—21no—
301C4×D3—21no—
302C4×D3—22no—
303C4×D3—22no—
304C4×D3—84no—
305C4×D3—84no—
306C3×C8—21no—
307C3×C8—21no—
308C3×C8—41no—
309C3×C8—42no—
310C3⋊C8—41no—
311C3⋊C8—42no—
312C3⋊C8#12312yes—
313C3⋊C8#12312yes—
314C24—31no—
315#41—22no—
316#41—22no—
317C2×Q8D3×D411yescharacteristic
318C2×D4—21no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4D3×D411yescharacteristic
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—122no—
333C22×C4—122no—
334C22×C4—244no—
335C22×C4C23×D311yescharacteristic
336M16—22no—
337M16—22no—
338M16—22no—
339M16—62no—
340M16—62no—
341M16—62no—
342M16—122no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—31no—
347C2×C8—31no—
348C2×C8—34no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—62no—
352C2×C8—62no—
353C2×C8—62no—
354C2×C8—61no—
355C2×C8—122no—
356C2×C8D3×D411yescharacteristic
357C2×C8D3×D411yescharacteristic
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—31no—
361C4⋊C4—31no—
362C4⋊C4—61no—
363C4⋊C4—61no—
364C4⋊C4—62no—
365C4⋊C4—62no—
366C4⋊C4—122no—
367C4⋊C4—122no—
368C22⋊C4—61no—
369C22⋊C4—61no—
370C22⋊C4—61no—
371C22⋊C4—61no—
372C22⋊C4—61no—
373C42—61no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3C2×#12111yescharacteristic
380C22×C3#55212yes—
381C2×D3—28no—
382C2×D3—21no—
383C2×D3—21no—
384C2×D3—21no—
385C2×D3—21no—
386C2×D3#47212yes—
387C2×D3#47312yes—
388C3×C4—41no—
389C3×C4—82no—
390C3×C4C2×#12311yescharacteristic
391C3×C4#53611yescharacteristic
392C3×C4#54712yes—
393Dic3—21no—
394Dic3—21no—
395Dic3—41no—
396Dic3—41no—
397Dic3—82no—
398C23—21no—
399C23—31no—
400C23—31no—
401C23—62no—
402C23—62no—
403C23—62no—
404C23D3×#4111yescharacteristic
405Q8—22no—
406D4—22no—
407D4—42no—
408D4—122no—
409D4—122no—
410C2×C4—21no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—62no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—82no—
426C2×C4—122no—
427C2×C4—244no—
428C2×C4—244no—
429C2×C4—242no—
430C2×C4C2×D3×D411yescharacteristic
431C2×C4C2×D3×D411yescharacteristic
432C2×C4D3×#4112yes—
433C2×C4#101012yes—
434C8—21no—
435C8—21no—
436C8—32no—
437C8—32no—
438C8—42no—
439C8—41no—
440C8—121no—
441C8—122no—
442C2×C3—22no—
443C2×C3—81no—
444C2×C3#248611yescharacteristic
445C2×C3C2×#47211yescharacteristic
446C2×C3C2×#47311yescharacteristic
447D3—22no—
448D3—22no—
449C22—21no—
450C22—21no—
451C22—32no—
452C22—32no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—68no—
457C22—61no—
458C22—61no—
459C22—61no—
460C22—61no—
461C22D3×#12111yescharacteristic
462C22#570812yes—
463C4—41no—
464C4—61no—
465C4—61no—
466C4—82no—
467C4—121no—
468C4—121no—
469C4—242no—
470C4D3×#12311yescharacteristic
471C4#565512yes—
472C4#567511yescharacteristic
473C3#2767511yescharacteristic, complemented, Sylow
474C2—22no—
475C2—62no—
476C2—62no—
477C2—81no—
478C2#7748511yescharacteristic
479C2D3×#47211yescharacteristic
480C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2078C2
socleC22×C3C2×#121
prime radical#5828C22
hypercenter#2078D3
hypocenterC3#27675
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2078

Complements

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

NH ≅ G/Ntype
1. C2×D3×#107C2semidirect
4. C3×#2078C2semidirect
5. #70626C2semidirect
6. #70300C2semidirect
7. #69994C2semidirect
8. #68990C2semidirect
9. #68985C2semidirect
10. #68939C2semidirect
11. #68923C2semidirect
54. #2078D3semidirect
473. C3#27675semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1072
#788922
#786552
C3×#20782
#706262
#703002
#699942
#689902
#689852
#689392
#689232
#276753