Finite Group Databasebeta

← #10691482

Subgroups — #10691482

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3100C211yesmaximal, characteristic
2#82319C211yesmaximal, characteristic, complemented
3#71815C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#9237—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#926—21no—
12C3×#480—21no—
13C3×#474C412yescomplemented
14C3×#459C412yescomplemented
15C3×#447—21no—
16C3×C4×D8—22no—
17C3×#431C2211yescharacteristic
18C2×#837—21no—
19C2×#837—21no—
20#5221—21no—
21C4×#217—21no—
22#4984—21no—
23#4951—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25C42×Dic3C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4537—22no—
29D4⋊(C3⋊C8)—22no—
30#3100D311yescharacteristic
31#2293—31no—
32#1962—31no—
33#1502—31no—
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C2×C3×D8—22no—
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#127C2×C411yescharacteristic
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4C2×C412yes—
51C3×C4×D4D412yes—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#217—22no—
56C2×#217—22no—
57#926—42no—
58#926—42no—
59C2×C4⋊Dic3—22no—
60C2×C4⋊Dic3—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3D411yescharacteristic
65C2×C4×Dic3D411yescharacteristic
66C2×#208—21no—
67C3×#107—22no—
68C3×C4⋊C8C2×C412yes—
69C3×#102—22no—
70C3×#102—22no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#837—42no—
74#837—42no—
75C3⋊#95—22no—
76C3⋊#95D412yes—
77#830—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4×C3⋊C8—21no—
82#532C2×D311yescharacteristic
83C2×C4×D4C2×D311yescharacteristic
84C2×#117—61no—
85#480—21no—
86#474Dic312yescomplemented
87#459Dic312yescomplemented
88#447—21no—
89C4×D8—22no—
90#431—31no—
91#431C2×D311yescharacteristic
92#423—31no—
93C2×#104—61no—
94C2×#104—61no—
95#390—61no—
96#389—61no—
97#388—62no—
98C4×C4⋊C4—31no—
99C4×C22⋊C4—61no—
100C43—31no—
101#327—62no—
102D4⋊C8—62no—
103C24×C3—21no—
104C3×#41—41no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—42no—
114C2×C3×D4—41no—
115C2×C3×D4C22⋊C412yes—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—22no—
130C3×D8—44no—
131C2×C3×C8—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4C22⋊C412yes—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144#217—44no—
145#217—44no—
146C4⋊Dic3—44no—
147C4⋊Dic3—42no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—42no—
152C4×Dic3—42no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C4×Dic3D812yescomplemented
156#208—41no—
157#208—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×#41—21no—
161C22×D4#6711yescharacteristic
162C23×C4—21no—
163C2×D8—22no—
164C2×M16—61no—
165C4⋊Q8C2×Dic311yescharacteristic
166#127C2×Dic311yescharacteristic
167#124—21no—
168#121—21no—
169#121—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4C2×Dic312yes—
175C4×D4#6712yes—
176#117—122no—
177#117—122no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4#6711yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C42—31no—
185C2×C42—31no—
186C2×C42—61no—
187C2×C42—61no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—22no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8C2×Dic312yes—
195#104—122no—
196#104—122no—
197#102—22no—
198#102—22no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C4×C8—21no—
202C4×C8—61no—
203#95—32no—
204#95—62no—
205C23×C3—22no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3#12011yescharacteristic
212C3×Q8—41no—
213C3×Q8—41no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—41no—
219C3×D4—41no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4—81no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12311yescharacteristic
241C2×C3×C4C2×D811yescharacteristic
242C2×C3×C4#13511yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C3×C8—42no—
255C3⋊C8—81no—
256C3⋊C8—81no—
257C24—21no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4#21712yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—62no—
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4C2×#6711yescharacteristic
284D8—44no—
285M16—121no—
286M16—121no—
287C2×C8—22no—
288C2×C8—121no—
289C2×C8—121no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—122no—
294C4⋊C4—124no—
295C4⋊C4#21712yes—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—124no—
301C22⋊C4—124no—
302C42—32no—
303C42—61no—
304C42—61no—
305C42—61no—
306C42—122no—
307C42—122no—
308C42—121no—
309C42—121no—
310C42#23711yescharacteristic
311C42#23711yescharacteristic
312C42C22×Dic311yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3#38611yescharacteristic
325C22×C3#44711yescharacteristic
326C22×C3#44911yescharacteristic
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—42no—
331C3×C4—81no—
332C3×C4#42112yes—
333C3×C4C4×D811yescharacteristic
334C3×C4#44211yescharacteristic
335C3×C4#45911yescharacteristic
336C3×C4#46311yescharacteristic
337Dic3—42no—
338Dic3—42no—
339Dic3—82no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23#93711yescharacteristic
347Q8—41no—
348Q8—41no—
349D4—22no—
350D4—22no—
351D4—42no—
352D4—41no—
353D4—41no—
354D4—42no—
355D4—81no—
356D4—81no—
357D4—81no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4D3×D811yescharacteristic
380C2×C4#91111yescharacteristic
381C2×C4#91111yescharacteristic
382C2×C4#91211yescharacteristic
383C2×C4D4×Dic311yescharacteristic
384C2×C4D4×Dic311yescharacteristic
385C2×C4#93511yescharacteristic
386C2×C4#93911yescharacteristic
387C2×C4#93911yescharacteristic
388C2×C4C2×#21711yescharacteristic
389C8—42no—
390C8—241no—
391C8—241no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—82no—
395C2×C3—161no—
396C2×C3#187911yescharacteristic
397C2×C3#189711yescharacteristic
398C2×C3#190611yescharacteristic
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22#515611yescharacteristic
410C22#515811yescharacteristic
411C22#522011yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—122no—
418C4—122no—
419C4—242no—
420C4Dic3×D811yescharacteristic
421C4#515011yescharacteristic
422C4#515211yescharacteristic
423C4#515311yescharacteristic
424C4#523612yes—
425C3#923711yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2#7171211yescharacteristic
431C2#7203711yescharacteristic
432C2#7211411yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3100C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3100D3
hypocenterC3#9237
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
2. #82319C2semidirect
3. #71815C2semidirect
4. #71811C2semidirect
5. #71598C2semidirect
6. #67746C2semidirect
7. #67736C2semidirect
13. C3×#474C4semidirect
14. C3×#459C4semidirect
86. #474Dic3semidirect
87. #459Dic3semidirect
155. C4×Dic3D8semidirect
425. C3#9237semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
#823192
#718152
#718112
#715982
#677462
#677362
#92373