Finite Group Databasebeta

← #10696398

Subgroups — #10696398

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83259C211yesmaximal, characteristic
2C2×#5080C211yesmaximal, characteristic, complemented
3C2×#4841C211yesmaximal, characteristic
4C2×C3×#332C211yesmaximal, characteristic, complemented
5#70460C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67704C211yesmaximal, characteristic, complemented
8#10068—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—22no—
11C22×#204—21no—
12C2×C4×D12C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×#816—21no—
17#5080—22no—
18#5080—22no—
19#4993—22no—
20#4988—21no—
21#4975—21no—
22C2×#808—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4841—22no—
26#4841—22no—
27C3×#332—22no—
28#4551—22no—
29#4483—22no—
30C2×#498—31no—
31C2×#461—31no—
32#2542—31no—
33#1988—31no—
34C2×#332D311yescharacteristic, complemented
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42C2311yescharacteristic
50#946—44no—
51#946—44no—
52C2×(C2×D3)⋊C4—24no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D412yes—
58C2×Dic3⋊C4—24no—
59C2×C4×Dic3—21no—
60C22×C3⋊C8—21no—
61C2×#204—24no—
62C2×#204—42no—
63C2×#204—42no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71#869D412yes—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3⋊#95—21no—
77#816—24no—
78#808—28no—
79C4×C3⋊C8—21no—
80C4×C3⋊C8D412yes—
81C22×SD16—61no—
82C2×C4⋊Q8—31no—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×C22⋊Q8—62no—
85C2×C4×D4—31no—
86#498—62no—
87#498—62no—
88#461—62no—
89#461—62no—
90C2×#106—62no—
91C2×C4⋊C8C2×D311yescharacteristic
92C2×C4⋊C8C2×D311yescharacteristic
93C2×#103—61no—
94C2×C4×C8—31no—
95#399—62no—
96#390—61no—
97#384—61no—
98C4×C4⋊C4—31no—
99#340—62no—
100#332—22no—
101D4⋊C8—62no—
102C23×D3—21no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—42no—
106C2×C3×Q8—42no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—22no—
112C22×Dic3—21no—
113C22×Dic3—41no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12SD1612yes—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×C4×D3—42no—
123C2×C4×D3—42no—
124C2×Dic6—22no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—42no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143(C2×D3)⋊C4—48no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—22no—
147C4⋊Dic3—22no—
148C4⋊Dic3—21no—
149C4⋊Dic3—22no—
150C4⋊Dic3—21no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4⋊Dic3SD1614yescomplemented
154Dic3⋊C4—48no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—24no—
158C2×C3⋊C8—22no—
159#204—48no—
160C22×Q8—21no—
161C22×Q8—61no—
162C22×D4—31no—
163C23×C4—61no—
164C2×SD16—64no—
165C2×SD16—122no—
166C2×SD16—122no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—32no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C22⋊Q8—124no—
179C22⋊Q8—124no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—64no—
190C2×C4⋊C4D1212yes—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#106—68no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8#6712yes—
200#103—64no—
201C4×C8—32no—
202C4×C8—61no—
203#95—61no—
204C23×C3#12011yescharacteristic
205C22×D3—22no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—42no—
209C22×D3—42no—
210C3×Q8—82no—
211C3×Q8—81no—
212C3×Q8—81no—
213C2×C3×C4—22no—
214C2×C3×C4—42no—
215C2×C3×C4—44no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×C3×C4C2×SD1611yescharacteristic
223C2×C3×C4C2×SD1612yes—
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)12yes—
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×Dic3—24no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236D12—22no—
237D12—22no—
238D12—41no—
239D12—41no—
240C4×D3—82no—
241C4×D3—81no—
242C4×D3—81no—
243Dic6—42no—
244Dic6—42no—
245Dic6—81no—
246Dic6—81no—
247C3×C8—42no—
248C3×C8—81no—
249C3⋊C8—24no—
250C24—61no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—62no—
256C2×Q8—121no—
257C2×Q8—121no—
258C2×Q8—121no—
259C2×Q8—121no—
260C2×Q8—121no—
261C2×D4—32no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C22×C4—22no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—122no—
273C22×C4—122no—
274C22×C4—121no—
275C22×C4C2×D1211yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278SD16—128no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—64no—
285C2×C8—62no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—34no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—61no—
294C4⋊C4—62no—
295C4⋊C4—61no—
296C4⋊C4—128no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—128no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—121no—
303C42—121no—
304C42#23511yescharacteristic
305C42D3×D411yescharacteristic
306C42D3×D411yescharacteristic
307C42C2×#6711yescharacteristic
308C22×C3#39211yescharacteristic
309C22×C3#44811yescharacteristic
310C22×C3#44911yescharacteristic
311C22×C3#45012yes—
312C22×C3#45612yes—
313C2×D3—44no—
314C2×D3—42no—
315C2×D3—82no—
316C2×D3—81no—
317C2×D3—81no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—82no—
321C3×C4#45011yescharacteristic
322C3×C4#45211yescharacteristic
323C3×C4#46012yes—
324C3×C4#46511yescharacteristic
325C3×C4#46811yescharacteristic
326C3×C4#49211yescharacteristic
327C3×C4#49611yescharacteristic
328Dic3—44no—
329Dic3—82no—
330Dic3—81no—
331C23—62no—
332C23—121no—
333C23—122no—
334C23—122no—
335C23—121no—
336C23#88211yescharacteristic
337Q8—81no—
338Q8—81no—
339Q8—82no—
340Q8—122no—
341Q8—122no—
342Q8—241no—
343Q8—241no—
344D4—62no—
345D4—62no—
346D4—121no—
347D4—121no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—64no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—241no—
358C2×C4—241no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4—242no—
362C2×C4#88211yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4#88411yescharacteristic
365C2×C4#91112yes—
366C2×C4D3×SD1611yescharacteristic
367C2×C4D3×SD1612yes—
368C2×C4#91411yescharacteristic
369C2×C4#91811yescharacteristic
370C2×C4#91911yescharacteristic
371C2×C4#93011yescharacteristic
372C2×C4#93711yescharacteristic
373C2×C4#94011yescharacteristic
374C8—42no—
375C8—64no—
376C8—81no—
377C2×C3#165112yes—
378C2×C3#165412yes—
379C2×C3#200011yescharacteristic
380C2×C3#200511yescharacteristic
381C2×C3#201411yescharacteristic
382D3—82no—
383C22—124no—
384C22—122no—
385C22—241no—
386C22—242no—
387C22—241no—
388C22#475212yes—
389C22#477612yes—
390C22#480611yescharacteristic
391C22#481011yescharacteristic
392C22#496011yescharacteristic
393C4—21no—
394C4—21no—
395C4—82no—
396C4—124no—
397C4—241no—
398C4—242no—
399C4#476212yes—
400C4#480711yescharacteristic
401C4#481211yescharacteristic
402C4#515411yescharacteristic
403C4#516911yescharacteristic
404C4#517511yescharacteristic
405C4#518711yescharacteristic
406C3#1006811yescharacteristic, complemented, Sylow
407C2—242no—
408C2#6843812yes—
409C2#6843912yes—
410C2#7011911yescharacteristic
411C2#7021011yescharacteristic
412C2#7027211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#332D3
hypocenterC3#10068
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. C2×#5080C2semidirect
4. C2×C3×#332C2semidirect
5. #70460C2semidirect
7. #67704C2semidirect
34. C2×#332D3semidirect
153. C4⋊Dic3SD16semidirect
406. C3#10068semidirect

Maximal subgroups

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

subgroupindex
#832592
C2×#50802
C2×#48412
C2×C3×#3322
#704602
C2×#44832
#677042
#100683