Finite Group Databasebeta

← #10696454

Subgroups — #10696454

Order 768.

654 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83259C211yesmaximal, characteristic
2C2×#5078C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic
4C2×C3×#332C211yesmaximal, characteristic, complemented
5#70456C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67704C211yesmaximal, characteristic, complemented
8#10076—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—22no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#821—21no—
16C2×#818—21no—
17C2×#816—22no—
18#5078—22no—
19#5078—22no—
20#4993—22no—
21#4988—21no—
22#4975—21no—
23C2×#807—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4846—22no—
27#4846—22no—
28C3×#332—22no—
29#4483—22no—
30C2×#500—31no—
31C2×#474—31no—
32#2542—31no—
33#1981—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—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869D412yes—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3⋊#95—21no—
74#821—24no—
75#818—24no—
76#816—44no—
77#807—28no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8D412yes—
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×C22⋊Q8—62no—
83C2×C4×D4—31no—
84#500—62no—
85#500—62no—
86#474—62no—
87#474—62no—
88C2×#107—61no—
89C2×#107—62no—
90C2×C4⋊C8C2×D311yescharacteristic
91C2×C4⋊C8C2×D311yescharacteristic
92C2×#103—62no—
93C2×#102—61no—
94C2×C4×C8—31no—
95#399—62no—
96#390—61no—
97#384—61no—
98C4×C4⋊C4—31no—
99#332—22no—
100D4⋊C8—62no—
101C23×D3—21no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C22×C3×C4—22no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—22no—
111C22×Dic3—21no—
112C22×Dic3—41no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12Q1612yes—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3—42no—
122C2×C4×D3—42no—
123C2×Dic6—22no—
124C2×Dic6—42no—
125C2×Dic6—42no—
126C2×Dic6—41no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—42no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140(C2×D3)⋊C4—48no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—22no—
147C4⋊Dic3—21no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3D814yes—
151Dic3⋊C4—48no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C2×C3⋊C8—24no—
155C2×C3⋊C8—22no—
156C22×Q8—21no—
157C22×Q8—61no—
158C22×D4—31no—
159C23×C4—61no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—61no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—32no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171C22⋊Q8—124no—
172C22⋊Q8—124no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4D1212yes—
184C2×C22⋊C4—64no—
185C2×C22⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—68no—
189#107—64no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8#6712yes—
194#103—124no—
195#102—64no—
196C4×C8—32no—
197C4×C8—61no—
198#95—61no—
199C23×C3#12011yescharacteristic
200C22×D3—22no—
201C22×D3—41no—
202C22×D3—42no—
203C22×D3—41no—
204C22×D3—42no—
205C3×Q8—82no—
206C3×Q8—81no—
207C3×Q8—81no—
208C2×C3×C4—22no—
209C2×C3×C4—42no—
210C2×C3×C4—44no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12711yescharacteristic
217C2×C3×C4C2×D812yes—
218C2×C3×C4C2×Q1611yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)12yes—
221C2×C3×C4#13711yescharacteristic
222C2×C3×C4#13711yescharacteristic
223C2×Dic3—24no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—82no—
230D12—22no—
231D12—22no—
232D12—41no—
233D12—41no—
234C4×D3—82no—
235C4×D3—81no—
236C4×D3—81no—
237Dic6—44no—
238Dic6—81no—
239Dic6—81no—
240C3×C8—42no—
241C3×C8—81no—
242C3⋊C8—24no—
243C24—61no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—62no—
249C2×Q8—121no—
250C2×Q8—122no—
251C2×Q8—122no—
252C2×D4—32no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C22×C4—22no—
259C22×C4—62no—
260C22×C4—61no—
261C22×C4—121no—
262C22×C4—121no—
263C22×C4—122no—
264C22×C4—122no—
265C22×C4—121no—
266C22×C4C2×D1211yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—64no—
275C2×C8—62no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—34no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—61no—
284C4⋊C4—62no—
285C4⋊C4—61no—
286C4⋊C4—128no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C22⋊C4—128no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C42—121no—
293C42—121no—
294C42#23511yescharacteristic
295C42D3×D411yescharacteristic
296C42D3×D411yescharacteristic
297C42C2×#6711yescharacteristic
298C22×C3#39211yescharacteristic
299C22×C3#44712yes—
300C22×C3#45111yescharacteristic
301C22×C3#45211yescharacteristic
302C22×C3#45612yes—
303C2×D3—44no—
304C2×D3—42no—
305C2×D3—82no—
306C2×D3—81no—
307C2×D3—81no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—82no—
311C3×C4#45111yescharacteristic
312C3×C4#45211yescharacteristic
313C3×C4#45912yes—
314C3×C4#46711yescharacteristic
315C3×C4#47011yescharacteristic
316C3×C4#49311yescharacteristic
317C3×C4#49611yescharacteristic
318Dic3—44no—
319Dic3—82no—
320Dic3—81no—
321C23—62no—
322C23—121no—
323C23—122no—
324C23—122no—
325C23—121no—
326C23#88211yescharacteristic
327Q8—81no—
328Q8—81no—
329Q8—82no—
330Q8—124no—
331Q8—241no—
332Q8—241no—
333D4—62no—
334D4—62no—
335D4—121no—
336D4—121no—
337C2×C4—22no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—64no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—242no—
347C2×C4—241no—
348C2×C4—241no—
349C2×C4—242no—
350C2×C4#88211yescharacteristic
351C2×C4#88411yescharacteristic
352C2×C4#88411yescharacteristic
353C2×C4D3×D812yes—
354C2×C4#91412yes—
355C2×C4#91511yescharacteristic
356C2×C4#91611yescharacteristic
357C2×C4D3×Q1611yescharacteristic
358C2×C4#91811yescharacteristic
359C2×C4#93011yescharacteristic
360C2×C4#93711yescharacteristic
361C2×C4#94011yescharacteristic
362C8—42no—
363C8—64no—
364C8—81no—
365C2×C3#164712yes—
366C2×C3#164912yes—
367C2×C3#200111yescharacteristic
368C2×C3#200511yescharacteristic
369C2×C3#201111yescharacteristic
370D3—82no—
371C22—124no—
372C22—122no—
373C22—241no—
374C22—242no—
375C22—241no—
376C22#475212yes—
377C22#477312yes—
378C22#480811yescharacteristic
379C22#480911yescharacteristic
380C22#496011yescharacteristic
381C4—21no—
382C4—21no—
383C4—82no—
384C4—124no—
385C4—241no—
386C4—242no—
387C4#478612yes—
388C4#481311yescharacteristic
389C4#481611yescharacteristic
390C4#515111yescharacteristic
391C4#517311yescharacteristic
392C4#517611yescharacteristic
393C4#518611yescharacteristic
394C3#1007611yescharacteristic, complemented, Sylow
395C2—242no—
396C2#6843612yes—
397C2#6846112yes—
398C2#7011911yescharacteristic
399C2#7020911yescharacteristic
400C2#7027311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#332D3
hypocenterC3#10076
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×#5078C2semidirect
4. C2×C3×#332C2semidirect
5. #70456C2semidirect
7. #67704C2semidirect
34. C2×#332D3semidirect
394. C3#10076semidirect

Maximal subgroups

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

subgroupindex
#832592
C2×#50782
C2×#48462
C2×C3×#3322
#704562
C2×#44832
#677042
#100763