Finite Group Databasebeta

← #10699014

Subgroups — #10699014

Order 768.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4841C211yesmaximal, characteristic
2C2×C3×#334C211yesmaximal, characteristic, complemented
3#70479C211yesmaximal, characteristic
4C2×#4482C211yesmaximal, characteristic, complemented
5#69787C211yesmaximal, characteristic
6#69777C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10477—31nomaximal, Sylow
9C22×#204—21no—
10C2×C4×D12C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C2×#821—21no—
16C2×(C2×D3)⋊C8—21no—
17C2×(C2×D3)⋊C8—21no—
18C2×#816—21no—
19#4935C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21#4841—22no—
22#4841—22no—
23C3×#334—22no—
24#4551—21no—
25#4551—21no—
26#4551—21no—
27#4482—22no—
28C2×#461—31no—
29#1990—31no—
30#1954—31no—
31#1802—31no—
32C2×#334D311yescharacteristic, complemented
33C2×#332—31no—
34#1296—31no—
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C22×C3×C8D411yescharacteristic
41C22×C3×C8D411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C22×C3⋊C8—21no—
51C2×#204—24no—
52C2×#204—42no—
53C2×#204—42no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D412yes—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8D412yes—
66C3×C4×C8—21no—
67C3×C4×C8—21no—
68#821—42no—
69(C2×D3)⋊C8—42no—
70(C2×D3)⋊C8—42no—
71#816—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C22×SD16—61no—
75C2×C4⋊Q8—31no—
76C2×C4×D4—31no—
77#461—62no—
78#461—62no—
79C2×C4⋊C8—31no—
80C2×C4⋊C8C2×D311yescharacteristic
81C2×C4⋊C8C2×D311yescharacteristic
82C2×#103—61no—
83C2×#102—61no—
84C2×C22⋊C8—61no—
85C2×C22⋊C8—61no—
86C2×C4×C8C2×D311yescharacteristic
87#384—31no—
88#340—61no—
89#340—61no—
90#340—61no—
91#334—22no—
92#332—62no—
93C23×D3—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12SD1612yes—
106C2×C4×D3—41no—
107C2×C4×D3—41no—
108C2×C4×D3—41no—
109C2×C4×D3—41no—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×Dic6—22no—
113C2×Dic6—41no—
114C2×Dic6—41no—
115C2×Dic6—41no—
116C2×Dic6—41no—
117C2×Dic6—41no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8—22no—
121C2×C3×C8—21no—
122C2×C3×C8—21no—
123C2×C3×C8—21no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8SD1612yescomplemented
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—22no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—81no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150#204—48no—
151C22×Q8—61no—
152C22×D4—31no—
153C23×C4—61no—
154C2×SD16—64no—
155C2×SD16—122no—
156C2×SD16—122no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—61no—
160C22×C8D1211yescharacteristic
161C22×C8D1211yescharacteristic
162C4⋊Q8—32no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4×D4—61no—
167C4×D4—61no—
168C4×D4—61no—
169C4×D4—61no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8#6712yes—
184#103—64no—
185#102—122no—
186C22⋊C8—122no—
187C22⋊C8—122no—
188C4×C8—21no—
189C4×C8—21no—
190C23×C3#12111yescharacteristic
191C22×D3—22no—
192C22×D3—41no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×C3×C4#12411yescharacteristic
206C2×C3×C4C2×SD1611yescharacteristic
207C2×C3×C4C2×SD1611yescharacteristic
208C2×C3×C4#13511yescharacteristic
209C2×C3×C4#13511yescharacteristic
210C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
211C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C2×C3×C4#13711yescharacteristic
214C2×Dic3—42no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223D12—22no—
224D12—22no—
225D12—41no—
226D12—41no—
227C4×D3—81no—
228C4×D3—81no—
229C4×D3—81no—
230C4×D3—81no—
231Dic6—42no—
232Dic6—42no—
233Dic6—81no—
234Dic6—81no—
235C3×C8—22no—
236C3×C8—42no—
237C3×C8—41no—
238C3×C8—81no—
239C3⋊C8—81no—
240C24—61no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8—121no—
244C2×Q8—121no—
245C2×Q8—121no—
246C2×Q8—121no—
247C2×D4—32no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C22×C4—61no—
254C22×C4—121no—
255C22×C4—121no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4—121no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4C2×D1211yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4#23911yescharacteristic
265SD16—128no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—121no—
278C2×C8—121no—
279C2×C8—121no—
280C2×C8#20412yescomplemented
281C4⋊C4—61no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—241no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42#23511yescharacteristic
295C42D3×D411yescharacteristic
296C42#23911yescharacteristic
297C42C2×#6711yescharacteristic
298C22×C3#39411yescharacteristic
299C22×C3#46111yescharacteristic
300C22×C3#46311yescharacteristic
301C22×C3#46511yescharacteristic
302C22×C3#46911yescharacteristic
303C22×C3#47111yescharacteristic
304C22×C3#47311yescharacteristic
305C2×D3—42no—
306C2×D3—42no—
307C2×D3—42no—
308C2×D3—81no—
309C2×D3—81no—
310C2×D3—81no—
311C2×D3—81no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4#44911yescharacteristic
315C3×C4#45011yescharacteristic
316C3×C4#46011yescharacteristic
317C3×C4#46411yescharacteristic
318C3×C4#46511yescharacteristic
319C3×C4#46811yescharacteristic
320C3×C4#48111yescharacteristic
321C3×C4#48211yescharacteristic
322Dic3—81no—
323Dic3—82no—
324Dic3—161no—
325C23—62no—
326C23—121no—
327C23—121no—
328C23—121no—
329C23—121no—
330C23—121no—
331C23—121no—
332C23#89511yescharacteristic
333Q8—122no—
334Q8—122no—
335Q8—241no—
336Q8—241no—
337D4—62no—
338D4—62no—
339D4—121no—
340D4—121no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—241no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4—241no—
356C2×C4#89411yescharacteristic
357C2×C4#89411yescharacteristic
358C2×C4#89511yescharacteristic
359C2×C4C2×#20411yescharacteristic
360C2×C4#90411yescharacteristic
361C2×C4#90811yescharacteristic
362C2×C4#90911yescharacteristic
363C2×C4D3×SD1611yescharacteristic
364C2×C4#91411yescharacteristic
365C2×C4#91411yescharacteristic
366C2×C4#91611yescharacteristic
367C2×C4#93011yescharacteristic
368C2×C4#93711yescharacteristic
369C2×C4#94711yescharacteristic
370C8—22no—
371C8—41no—
372C8—42no—
373C8—81no—
374C8—241no—
375C2×C3#167111yescharacteristic
376C2×C3#167911yescharacteristic
377C2×C3#169311yescharacteristic
378C2×C3#170111yescharacteristic
379C2×C3#203411yescharacteristic
380C2×C3#205211yescharacteristic
381C2×C3#205511yescharacteristic
382D3—82no—
383C22—122no—
384C22—122no—
385C22—122no—
386C22—241no—
387C22—241no—
388C22—241no—
389C22—241no—
390C22#484111yescharacteristic
391C22#484411yescharacteristic
392C22#486411yescharacteristic
393C22#486511yescharacteristic
394C22#489611yescharacteristic
395C22#489811yescharacteristic
396C22#499111yescharacteristic
397C4—21no—
398C4—21no—
399C4—241no—
400C4—242no—
401C4—481no—
402C4#484511yescharacteristic
403C4#484811yescharacteristic
404C4#486311yescharacteristic
405C4#486611yescharacteristic
406C4#511411yescharacteristic
407C4#513411yescharacteristic
408C4#516911yescharacteristic
409C4#517211yescharacteristic
410C3#1047711yescharacteristic, complemented, Sylow
411C2—242no—
412C2#6853711yescharacteristic
413C2#6853911yescharacteristic
414C2#6854211yescharacteristic
415C2#6854411yescharacteristic
416C2#7048211yescharacteristic
417C2#7058111yescharacteristic
418C2#7071611yescharacteristic
subgroupG/H
centerC23#895
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#334C2
socleC23×C3#121
prime radicalC22×D12D4
hypercenterC2×#334D3
hypocenterC3#10477
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. C2×C3×#334C2semidirect
4. C2×#4482C2semidirect
6. #69777C2semidirect
7. #67702C2semidirect
32. C2×#334D3semidirect
129. C2×C3×C8SD16semidirect
280. C2×C8#204semidirect
410. C3#10477semidirect

Maximal subgroups

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

subgroupindex
C2×#48412
C2×C3×#3342
#704792
C2×#44822
#697872
#697772
#677022
#104773