Finite Group Databasebeta

← #10694005

Subgroups — #10694005

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#940C211yesmaximal, characteristic
2C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
3C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#70443C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic, complemented
6C4×#815C211yesmaximal, characteristic, complemented
7C2×#4459C211yesmaximal, characteristic, complemented
8#9648—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#940C411yescharacteristic
11C2×C4×#67—22no—
12C2×C4×D12C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5221C411yescharacteristic
16C2×#815—22no—
17#4988C411yescharacteristic
18#4975C411yescharacteristic
19#4957—22no—
20C42×Dic3C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×D4⋊C8—22no—
24#4550—22no—
25D4⋊(C3⋊C8)—22no—
26#4483—22no—
27#4459—22no—
28C4×#127—31no—
29#1831—31no—
30C4×#97—31no—
31C2×D4⋊C8—31no—
32C2×D4⋊C8—31no—
33C2×D4⋊C8—31no—
34C2×D4⋊C8D311yescharacteristic, complemented
35C22×C3×D4C2×C411yescharacteristic
36C23×C3×C4—21no—
37C22×#67—22no—
38C22×D12C2×C411yescharacteristic
39C22×C4×D3—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#217—22no—
50#940—22no—
51#940—22no—
52C4×#67—44no—
53C4×#67—44no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×Dic3⋊C4—22no—
58C2×C4×Dic3C2×C411yescharacteristic
59C2×C4×Dic3C2×C411yescharacteristic
60C2×C4×Dic3D412yes—
61C2×C4×Dic3D412yes—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4×C8—21no—
71C3×C4×C8D412yescomplemented
72#815—28no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yescomplemented
77C2×#127—31no—
78C2×C4×D4—31no—
79C2×C4×D4—62no—
80C2×C4×D4C2×D311yescharacteristic
81C2×C4⋊C8—31no—
82C2×C4⋊C8C2×D311yescharacteristic
83C2×#97—62no—
84C2×C4×C8—31no—
85C2×C4×C8C2×D311yescharacteristic
86#390—31no—
87#390—31no—
88#390—62no—
89#384—31no—
90C43—31no—
91#336—62no—
92D4⋊C8—22no—
93D4⋊C8—62no—
94D4⋊C8—62no—
95D4⋊C8—62no—
96C24×C3—21no—
97C23×D3—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4M1612yes—
104C22×C3×C4—21no—
105C22×C3×C4—41no—
106C22×C3×C4—41no—
107C22×C3×C4—42no—
108C22×C3×C4—42no—
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—28no—
113C2×#67—44no—
114C2×#67—44no—
115C22×Dic3—21no—
116C22×Dic3C22⋊C412yes—
117C22×Dic3C22⋊C412yes—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12M1612yes—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C2×C4×D3—42no—
127C2×C4×D3—42no—
128C2×C3×C8—24no—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C2×C3×C8—42no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140#217—44no—
141(C2×D3)⋊C4—44no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146Dic3⋊C4—44no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—22no—
152C4×Dic3—22no—
153C4×Dic3—24no—
154C2×C3⋊C8—24no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—42no—
158C22×D4—31no—
159C22×D4—62no—
160C22×D4C4×D311yescharacteristic
161C23×C4—21no—
162C23×C4—61no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—61no—
166C22×C8—61no—
167#127—62no—
168#127—62no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C4×D4—124no—
178C4×D4—124no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4C4×D311yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—62no—
185C2×C22⋊C4—61no—
186C2×C42—31no—
187C2×C42—31no—
188C2×C42—32no—
189C2×C42—32no—
190C2×C42C22×D311yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195#97—68no—
196C4×C8—21no—
197C4×C8—32no—
198C4×C8—61no—
199C4×C8#6712yescomplemented
200C23×C3—22no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—42no—
204C23×C3—42no—
205C23×C3C2×C22⋊C411yescharacteristic
206C22×D3—22no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—42no—
210C22×D3—42no—
211C3×D4—22no—
212C3×D4—22no—
213C3×D4—41no—
214C3×D4—41no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—82no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12411yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×C3×C4C2×M1611yescharacteristic
228C2×C3×C4C2×M1611yescharacteristic
229C2×C3×C4#13111yescharacteristic
230C2×C3×C4#13111yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)12yes—
232C2×C3×C4AGL(1,ℤ/8)12yes—
233#67—416no—
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3#10414yes—
242D12—22no—
243D12—22no—
244D12—41no—
245D12—41no—
246C4×D3—82no—
247C4×D3—81no—
248C4×D3—81no—
249C3×C8—24no—
250C3×C8—81no—
251C3⋊C8—24no—
252C3⋊C8—81no—
253C24—21no—
254C24—61no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—32no—
261C2×D4—68no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—124no—
268C2×D4—124no—
269C2×D4ΓL(1,𝔽25)12yes—
270C22×C4—21no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—61no—
278C22×C4—121no—
279C22×C4—122no—
280C22×C4—122no—
281C22×C4—121no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—64no—
290C2×C8—62no—
291C2×C8—122no—
292C2×C8—121no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—124no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—124no—
301C22⋊C4—124no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C42—64no—
305C42—61no—
306C42—61no—
307C42—61no—
308C42—61no—
309C42—62no—
310C42—62no—
311C42#23511yescharacteristic
312C42D3×D411yescharacteristic
313C42#23711yescharacteristic
314C42C2×#6711yescharacteristic
315C22×C3—42no—
316C22×C3—44no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—82no—
320C22×C3#39011yescharacteristic
321C22×C3#40711yescharacteristic
322C22×C3#40811yescharacteristic
323C22×C3#41812yes—
324C22×C3C2×#10412yes—
325C2×D3—42no—
326C2×D3—44no—
327C2×D3—81no—
328C2×D3—81no—
329C2×D3—82no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—81no—
333C3×C4#43511yescharacteristic
334C3×C4#43511yescharacteristic
335C3×C4#43511yescharacteristic
336C3×C4#43511yescharacteristic
337C3×C4#49012yes—
338C3×C4#49611yescharacteristic
339C3×C4#49611yescharacteristic
340Dic3—24no—
341Dic3—42no—
342Dic3—81no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—62no—
349C23—121no—
350C23—121no—
351C23—122no—
352C23—122no—
353C23D3×C22⋊C411yescharacteristic
354D4—22no—
355D4—22no—
356D4—41no—
357D4—41no—
358D4—62no—
359D4—62no—
360D4—1216no—
361D4—121no—
362D4—121no—
363C2×C4—22no—
364C2×C4—34no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—62no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4—242no—
380C2×C4#67⋊C411yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4#88511yescharacteristic
383C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
384C2×C4#90111yescharacteristic
385C2×C4D3×M1611yescharacteristic
386C2×C4#90711yescharacteristic
387C2×C4#91112yes—
388C2×C4#91412yes—
389C2×C4C4×#6711yescharacteristic
390C2×C4#93911yescharacteristic
391C2×C4#94011yescharacteristic
392C8—24no—
393C8—64no—
394C8—81no—
395C8—241no—
396C2×C3—82no—
397C2×C3#148412yes—
398C2×C3#149012yes—
399C2×C3#195511yescharacteristic
400C2×C3#196111yescharacteristic
401C2×C3#196211yescharacteristic
402D3—82no—
403C22—42no—
404C22—44no—
405C22—81no—
406C22—82no—
407C22—81no—
408C22—122no—
409C22—124no—
410C22—241no—
411C22—242no—
412C22—241no—
413C22#472611yescharacteristic
414C22#472711yescharacteristic
415C22#477012yes—
416C22D3×#10412yes—
417C22#495711yescharacteristic
418C4—21no—
419C4—21no—
420C4—64no—
421C4—81no—
422C4—122no—
423C4—241no—
424C4#472811yescharacteristic
425C4#473011yescharacteristic
426C4#476412yes—
427C4#511111yescharacteristic
428C4#512711yescharacteristic
429C4#515411yescharacteristic
430C4#517611yescharacteristic
431C3#964811yescharacteristic, complemented, Sylow
432C2—82no—
433C2—242no—
434C2#6817112yes—
435C2#6817512yes—
436C2#6991711yescharacteristic
437C2#7016011yescharacteristic
438C2#7035611yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#940C4
cosocleC2×C3×C42C23
hypercenterC2×D4⋊C8D3
hypocenterC3#9648
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×D4⋊C8C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. #70443C2semidirect
5. C2×#4483C2semidirect
6. C4×#815C2semidirect
7. C2×#4459C2semidirect
34. C2×D4⋊C8D3semidirect
71. C3×C4×C8D4semidirect
76. C4×C3⋊C8D4semidirect
199. C4×C8#67semidirect
431. C3#9648semidirect

Maximal subgroups

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

subgroupindex
C4×#9402
C2×C3×D4⋊C82
C2×D4⋊(C3⋊C8)2
#704432
C2×#44832
C4×#8152
C2×#44592
#96483