Finite Group Databasebeta

← #10690032

Subgroups — #10690032

Order 768.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×M16C211yesmaximal, characteristic
2#82314C211yesmaximal, characteristic
3C4×#832C211yesmaximal, characteristic
4#70501C211yesmaximal, characteristic, complemented
5#69868C211yesmaximal, characteristic, complemented
6#67715C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#8989—31nomaximal, Sylow
9C22×C3×M16—21no—
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#436—21no—
13C3×#435—22no—
14C3×C8×D4—21no—
15C3×#423C2211yescharacteristic
16C3×#407—21no—
17C3×C4×M16C2211yescharacteristic
18#5218—21no—
19C2×#832—21no—
20#5019C411yescharacteristic, complemented
21#5019C411yescharacteristic, complemented
22C4×#214—21no—
23C4×#214—21no—
24#5008C411yescharacteristic, complemented
25#5008C411yescharacteristic, complemented
26#4971—21no—
27C4×C4⋊Dic3C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×C4×C3⋊C8C2211yescharacteristic
30#4555—21no—
31#4555—21no—
32#4529—21no—
33#4484—22no—
34#3101—31no—
35D4×M16D311yescharacteristic
36#1986—31no—
37C4×#102—31no—
38#1493—31no—
39#1303—31no—
40#1278—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×M16—22no—
44C2×C3×M16—21no—
45C2×C3×M16—42no—
46C2×C3×M16—42no—
47C2×C3×M16D411yescharacteristic
48C2×C3×M16D411yescharacteristic
49C22×C3×C8—22no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4C2×C411yescharacteristic
53C3×C4×D4C2×C411yescharacteristic
54C3×C4×D4D411yescharacteristic
55C3×C4×D4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#214—21no—
60C2×#214—21no—
61#926—21no—
62C2×C4⋊Dic3—21no—
63#924—21no—
64#924—21no—
65C2×C4×Dic3—21no—
66C22×C3⋊C8—21no—
67#870—21no—
68#869C2×C411yescharacteristic
69#869C2×C411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C22⋊C8—22no—
75C3×C22⋊C8—22no—
76C3×#97—21no—
77C3×C4×C8—21no—
78#832—24no—
79#832—21no—
80#832—21no—
81C3⋊#95—21no—
82#830—21no—
83#810—21no—
84#810—21no—
85#808—21no—
86#808—21no—
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)C2×C411yescharacteristic
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92C22×M16—21no—
93#533—31no—
94C2×C4×D4C2×D311yescharacteristic
95#483—61no—
96#476—31no—
97#476—31no—
98#461—31no—
99#461—31no—
100#450—61no—
101C4×SD16—61no—
102C4×SD16—61no—
103#436—21no—
104#435—22no—
105C8×D4—21no—
106#431—31no—
107#423C2×D311yescharacteristic
108C2×#102—61no—
109#407—21no—
110C4×M16C2×D311yescharacteristic
111C2×C4×C8—31no—
112C4×C4⋊C4—31no—
113#340—61no—
114#339—61no—
115#339—61no—
116Q8⋊C8—62no—
117C24×C3—21no—
118C2×C3×D4—22no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C2×C3×D4C22⋊C411yescharacteristic
122C2×C3×D4C22⋊C411yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×Dic3—41no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C3×M16—21no—
139C3×M16—21no—
140C3×M16—21no—
141C3×M16—44no—
142C3×M16—41no—
143C3×M16—41no—
144C3×M16SD1612yescomplemented
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—22no—
148C2×C3×C8—44no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4C22⋊C411yescharacteristic
151C3×C4⋊C4C22⋊C411yescharacteristic
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—21no—
156C3×C42C22×C411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160#217—41no—
161#214—42no—
162#214—42no—
163C4⋊Dic3—22no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—41no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C4×Dic3—41no—
172C4×Dic3—41no—
173C4×Dic3—41no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C22×D4D1211yescharacteristic
180C23×C4—21no—
181C2×SD16—61no—
182C2×SD16—61no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—42no—
186C2×M16—42no—
187C2×M16#6711yescharacteristic
188C2×M16#6711yescharacteristic
189C22×C8—22no—
190C22×C8—61no—
191C4⋊Q8—31no—
192C4⋊Q8—31no—
193#125—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C4×Q8—61no—
197C4×Q8—61no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4C4×D311yescharacteristic
201C4×D4C4×D311yescharacteristic
202C4×D4#6711yescharacteristic
203C4×D4#6711yescharacteristic
204#117—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4D1211yescharacteristic
207C2×C22⋊C4—21no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210#106—61no—
211#106—61no—
212C4⋊C8—21no—
213C4⋊C8—31no—
214C4⋊C8—31no—
215C4⋊C8#6712yes—
216#103—61no—
217#103—61no—
218#102—64no—
219#102—61no—
220#102—61no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—61no—
224#97—21no—
225C4×C8—21no—
226C4×C8—61no—
227C4×C8—61no—
228C4×C8—61no—
229#95—61no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—42no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—81no—
236C23×C3#12111yescharacteristic
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—42no—
241C3×D4#10412yes—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4C2×C22⋊C411yescharacteristic
258C2×C3×C4C4×D411yescharacteristic
259C2×C3×C4C4×D411yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12311yescharacteristic
263C2×C3×C4C2×SD1611yescharacteristic
264C2×C3×C4#13511yescharacteristic
265C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
266C2×C3×C4#13711yescharacteristic
267C2×Dic3—42no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—81no—
271C2×Dic3—81no—
272Dic6—42no—
273Dic6—41no—
274Dic6—41no—
275C3×C8—22no—
276C3×C8—42no—
277C3×C8—41no—
278C3⋊C8—42no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281C24—21no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4(C2×D3)⋊C411yescharacteristic
288C2×D4(C2×D3)⋊C411yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—121no—
299C22×C4C2×D1211yescharacteristic
300C22×C4#23511yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—122no—
303SD16—122no—
304M16—21no—
305M16—21no—
306M16—21no—
307M16—44no—
308M16—41no—
309M16—41no—
310M16#21412yescomplemented
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—44no—
315C2×C8—62no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—121no—
319C2×C8—121no—
320C4⋊C4—21no—
321C4⋊C4—62no—
322C4⋊C4—61no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—121no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C4⋊C4(C2×D3)⋊C411yescharacteristic
330C4⋊C4(C2×D3)⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—121no—
336C42—121no—
337C42—121no—
338C42—121no—
339C42C2×C4×D311yescharacteristic
340C42#23511yescharacteristic
341C42C2×#6711yescharacteristic
342C42C2×#6711yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3#38611yescharacteristic
354C22×C3#46111yescharacteristic
355C22×C3#46311yescharacteristic
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4C2×#10411yescharacteristic
362C3×C4#42111yescharacteristic
363C3×C4C4×SD1611yescharacteristic
364C3×C4#44011yescharacteristic
365C3×C4#45011yescharacteristic
366C3×C4#45211yescharacteristic
367Dic3—82no—
368Dic3—81no—
369Dic3—81no—
370C23—21no—
371C23—21no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—81no—
376C23#93011yescharacteristic
377Q8—122no—
378Q8—121no—
379Q8—121no—
380D4—21no—
381D4—21no—
382D4—21no—
383D4—42no—
384D4#82512yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—122no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—241no—
404C2×C4—241no—
405C2×C4C2×(C2×D3)⋊C411yescharacteristic
406C2×C4C4×#6711yescharacteristic
407C2×C4C4×#6711yescharacteristic
408C2×C4#92911yescharacteristic
409C2×C4#93011yescharacteristic
410C2×C4#93211yescharacteristic
411C2×C4C2×#21411yescharacteristic
412C2×C4#95011yescharacteristic
413C2×C4#95111yescharacteristic
414C2×C4#95311yescharacteristic
415C8—22no—
416C8—42no—
417C8—41no—
418C8—122no—
419C8—121no—
420C8—121no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—42no—
424C2×C3—81no—
425C2×C3#187111yescharacteristic
426C2×C3#187711yescharacteristic
427C2×C3#188111yescharacteristic
428C22—21no—
429C22—21no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—82no—
437C22—82no—
438C22#501611yescharacteristic
439C22#501911yescharacteristic
440C22#521311yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—41no—
445C4—41no—
446C4—242no—
447C4—241no—
448C4—241no—
449C4C4×#21411yescharacteristic
450C4#501811yescharacteristic
451C4C2×#82511yescharacteristic
452C4#513911yescharacteristic
453C4#521811yescharacteristic
454C4#523911yescharacteristic
455C3#898911yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#7161211yescharacteristic
461C2#7198411yescharacteristic
462C2#7202011yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×D4×M16C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenterD4×M16D3
hypocenterC3#8989
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
4. #70501C2semidirect
5. #69868C2semidirect
6. #67715C2semidirect
7. #67667C2semidirect
20. #5019C4semidirect
21. #5019C4semidirect
24. #5008C4semidirect
25. #5008C4semidirect
144. C3×M16SD16semidirect
310. M16#214semidirect
455. C3#8989semidirect

Maximal subgroups

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

subgroupindex
C3×D4×M162
#823142
C4×#8322
#705012
#698682
#677152
#676672
#89893