Finite Group Databasebeta

← #10690417

Subgroups — #10690417

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3167C211yesmaximal, characteristic
2#83330C211yesmaximal, characteristic, complemented
3#71835C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic
5#71800C211yesmaximal, characteristic
6#67656C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9054—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C3×#498C411yescharacteristic, complemented
12C3×#492C411yescharacteristic, complemented
13C3×#487C411yescharacteristic, complemented
14C3×#486C411yescharacteristic, complemented
15C3×#465—22no—
16C3×C4×SD16—22no—
17C2×C3×C4×C8C2211yescharacteristic
18C2×#837—21no—
19C2×#837—21no—
20#5024—22no—
21#5024—21no—
22Q8×C3⋊C8—21no—
23#4999—21no—
24#4997—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26#4925C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4538—22no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—21no—
32#3167D311yescharacteristic
33#2962—31no—
34#1980—31no—
35#1503—31no—
36#1480—31no—
37#1279—31no—
38#1278—31no—
39C2×C3×#41—21no—
40C2×C3×SD16—22no—
41C22×C3×C8—21no—
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#127C2×C411yescharacteristic
46C3×#125—21no—
47C3×#124—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8D412yes—
50C3×#121—21no—
51C3×#121—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8D412yes—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×#117—21no—
57C2×C3×C42C2311yescharacteristic
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—21no—
60C2×#208D411yescharacteristic
61C2×#208D411yescharacteristic
62C22×C3⋊C8—21no—
63C3×#106—22no—
64C3×#103—22no—
65C3×#102—22no—
66C3×C4×C8—21no—
67C3×C4×C8C2×C411yescharacteristic
68C3×C4×C8C2×C411yescharacteristic
69#837—42no—
70#837—42no—
71C3⋊#95—21no—
72#830—21no—
73C4⋊(C3⋊C8)—22no—
74C4⋊(C3⋊C8)—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D411yescharacteristic
78C4⋊(C3⋊C8)D411yescharacteristic
79#803—22no—
80#803—21no—
81C4×C3⋊C8—22no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533C2×D311yescharacteristic
85#532C2×D311yescharacteristic
86#498Dic311yescharacteristic, complemented
87#492Dic311yescharacteristic, complemented
88#487Dic311yescharacteristic, complemented
89#486Dic311yescharacteristic, complemented
90#465—22no—
91#446—61no—
92#446—62no—
93C8×Q8—61no—
94C4×SD16—22no—
95#431—31no—
96#424—61no—
97C2×C4⋊C8—31no—
98C2×#104—61no—
99C2×#104—61no—
100#405—61no—
101C4×M16—31no—
102C2×C4×C8C2×D311yescharacteristic
103C4×C4⋊C4—31no—
104#328—62no—
105D4⋊C8—61no—
106D4⋊C8—61no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8C22⋊C412yes—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—41no—
123C3×SD16—44no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—42no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4C22⋊C412yes—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—22no—
139C3×C42C22×C411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C4⋊Dic3—22no—
144C4⋊Dic3—41no—
145C4×Dic3—41no—
146C4×Dic3—41no—
147#208—21no—
148#208—21no—
149#208—21no—
150#208SD1612yescomplemented
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×#41—21no—
159C2×SD16—22no—
160C2×M16—31no—
161C2×M16—31no—
162C22×C8—21no—
163C22×C8—61no—
164C4⋊Q8C2×Dic311yescharacteristic
165C4⋊Q8C2×Dic311yescharacteristic
166C4⋊Q8C2×Dic311yescharacteristic
167#127C2×Dic311yescharacteristic
168#125—21no—
169#124—21no—
170C22⋊Q8—21no—
171C22⋊Q8#6712yes—
172#121—21no—
173#121—21no—
174C4×Q8—21no—
175C4×Q8#6712yes—
176C4×D4—21no—
177C4×D4—21no—
178#117—21no—
179C2×C4⋊C4—61no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#106—22no—
183C4⋊C8—31no—
184C4⋊C8—31no—
185C4⋊C8—62no—
186C4⋊C8—62no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#104—122no—
190#104—122no—
191#103—22no—
192#102—22no—
193C22⋊C8—61no—
194#97—62no—
195#97—61no—
196C4×C8—21no—
197C4×C8—62no—
198C4×C8—61no—
199C4×C8—61no—
200C4×C8C2×Dic311yescharacteristic
201C4×C8C2×Dic311yescharacteristic
202#95—61no—
203C23×C3—41no—
204C23×C3#12111yescharacteristic
205C3×Q8—42no—
206C3×Q8—41no—
207C3×Q8—41no—
208C3×Q8—41no—
209C3×Q8—41no—
210C3×D4—42no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—81no—
216C3×D4—81no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12311yescharacteristic
231C2×C3×C4C2×SD1611yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4#13711yescharacteristic
235C2×Dic3—42no—
236C2×Dic3—81no—
237C2×Dic3—81no—
238C3×C8—22no—
239C3×C8—41no—
240C3⋊C8—22no—
241C3⋊C8—41no—
242C3⋊C8—41no—
243C3⋊C8—41no—
244C3⋊C8—81no—
245#41—41no—
246#41—41no—
247#41—41no—
248#41—41no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8#21712yes—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C22×C4—41no—
257C22×C4—121no—
258C22×C4D3×D411yescharacteristic
259C22×C4#23711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261SD16—44no—
262M16—32no—
263M16—61no—
264M16—61no—
265M16—61no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—42no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—121no—
274C2×C8—121no—
275C2×C8—121no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—62no—
284C4⋊C4—121no—
285C4⋊C4#21712yes—
286C22⋊C4—22no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C42—22no—
290C42—121no—
291C42—121no—
292C42D3×D411yescharacteristic
293C42#23711yescharacteristic
294C42C22×Dic311yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3#38611yescharacteristic
302C22×C3#46011yescharacteristic
303C22×C3#46411yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—81no—
310C3×C4C4×SD1611yescharacteristic
311C3×C4#44011yescharacteristic
312C3×C4#44311yescharacteristic
313C3×C4#44411yescharacteristic
314C3×C4#44811yescharacteristic
315C3×C4#44911yescharacteristic
316Dic3—82no—
317C23—41no—
318C23#93911yescharacteristic
319Q8—42no—
320Q8—41no—
321Q8—41no—
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—122no—
340C2×C4—241no—
341C2×C4—241no—
342C2×C4D3×SD1611yescharacteristic
343C2×C4#91411yescharacteristic
344C2×C4#91511yescharacteristic
345C2×C4#91611yescharacteristic
346C2×C4D4×Dic311yescharacteristic
347C2×C4D4×Dic311yescharacteristic
348C2×C4#93511yescharacteristic
349C2×C4#93711yescharacteristic
350C2×C4#93911yescharacteristic
351C2×C4C2×#21711yescharacteristic
352C8—22no—
353C8—41no—
354C8—62no—
355C8—121no—
356C8—121no—
357C8—121no—
358C8—241no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—161no—
362C2×C3#187611yescharacteristic
363C2×C3#188011yescharacteristic
364C2×C3#188411yescharacteristic
365C22—21no—
366C22—21no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#516311yescharacteristic
371C22#516611yescharacteristic
372C22#522011yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—42no—
378C4—81no—
379C4—242no—
380C4Dic3×SD1611yescharacteristic
381C4#516411yescharacteristic
382C4#517011yescharacteristic
383C4#517211yescharacteristic
384C4#519611yescharacteristic
385C4#519711yescharacteristic
386C3#905411yescharacteristic, complemented, Sylow
387C2—21no—
388C2—21no—
389C2—161no—
390C2#7181711yescharacteristic
391C2#7204811yescharacteristic
392C2#7208711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3167C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3167D3
hypocenterC3#9054
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
2. #83330C2semidirect
3. #71835C2semidirect
6. #67656C2semidirect
7. #67655C2semidirect
11. C3×#498C4semidirect
12. C3×#492C4semidirect
13. C3×#487C4semidirect
14. C3×#486C4semidirect
86. #498Dic3semidirect
87. #492Dic3semidirect
88. #487Dic3semidirect
89. #486Dic3semidirect
150. #208SD16semidirect
386. C3#9054semidirect

Maximal subgroups

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

subgroupindex
C3×#31672
#833302
#718352
#718182
#718002
#676562
#676552
#90543