Finite Group Databasebeta

← #10691393

Subgroups — #10691393

Order 768.

578 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5083C211yesmaximal, characteristic
2C2×#4838C211yesmaximal, characteristic
3C2×C3×#332C211yesmaximal, characteristic, complemented
4#71787C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic
6#67703C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9225—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C22×ΓL(1,𝔽25)—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#835—21no—
16C2×#821—21no—
17C2×(C2×D3)⋊C8—21no—
18#5083—22no—
19#5083C414yescomplemented
20#4935C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4838—22no—
24#4838—22no—
25C3×#332—22no—
26#4551—21no—
27#4529—21no—
28#4529—21no—
29C2×#461—31no—
30C2×#436—31no—
31#1990—31no—
32#1802—31no—
33C2×#332D311yescharacteristic, complemented
34#1296—31no—
35#1277—31no—
36C22×C3×Q8—21no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8C2×C412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#215—24no—
49C2×#215—42no—
50C2×#215—42no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C22×C3⋊C8—21no—
57C22×C3⋊C8D411yescharacteristic
58C22×C3⋊C8D411yescharacteristic
59C2×ΓL(1,𝔽25)—24no—
60C2×ΓL(1,𝔽25)—42no—
61C2×ΓL(1,𝔽25)—42no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12C2×C412yes—
65C4×D12D412yes—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70#835—24no—
71#821—42no—
72(C2×D3)⋊C8—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)C2×C412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C22×SD16—61no—
78C22×M16—61no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C4×D4—31no—
81#461—34no—
82#461—62no—
83#436—62no—
84#436—62no—
85C2×C4⋊C8—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87C2×C4⋊C8C2×D311yescharacteristic
88C2×#103—61no—
89C2×#102—61no—
90C2×C22⋊C8—61no—
91C2×C4×C8—31no—
92#384—31no—
93#340—61no—
94#340—61no—
95#340—61no—
96#332—22no—
97C23×D3—21no—
98C2×C3×Q8—22no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—41no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12C22⋊C412yes—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#215—48no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—81no—
145C4⋊Dic3C22⋊C412yes—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8SD1612yescomplemented
155ΓL(1,𝔽25)—48no—
156C22×Q8—21no—
157C22×D4—31no—
158C23×C4—61no—
159C2×SD16—64no—
160C2×SD16—122no—
161C2×SD16—122no—
162C2×M16—64no—
163C2×M16—122no—
164C2×M16—122no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—31no—
168C22×C8—31no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C4×D312yes—
174C4×D4—32no—
175C4×D4—32no—
176C4×D4—61no—
177C4×D4—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—32no—
190C4⋊C8—61no—
191C4⋊C8#6712yes—
192#103—64no—
193#102—122no—
194C22⋊C8—64no—
195C4×C8—61no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C22×D3—22no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3—41no—
205C3×Q8—42no—
206C3×Q8—42no—
207C3×Q8—81no—
208C3×Q8—81no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—42no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4C2×C22⋊C411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12311yescharacteristic
220C2×C3×C4C2×SD1611yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C4#13711yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234D12—22no—
235D12—22no—
236D12—41no—
237D12—41no—
238C4×D3—42no—
239C4×D3—42no—
240C4×D3—42no—
241C4×D3—42no—
242C3×C8—42no—
243C3×C8—81no—
244C3⋊C8—22no—
245C3⋊C8—42no—
246C3⋊C8—41no—
247C24—61no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—32no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C22×C4—41no—
261C22×C4—61no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4#23511yescharacteristic
270C22×C4D3×D411yescharacteristic
271C22×C4D3×D411yescharacteristic
272SD16—128no—
273M16—128no—
274C2×C8—22no—
275C2×C8—32no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—62no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—121no—
288C2×C8—121no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—32no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4—241no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C42C2×C4×D311yescharacteristic
303C42D3×D411yescharacteristic
304C42#23711yescharacteristic
305C42C2×#6711yescharacteristic
306C22×C3#38611yescharacteristic
307C22×C3#46111yescharacteristic
308C22×C3#46311yescharacteristic
309C22×C3#46811yescharacteristic
310C22×C3#46911yescharacteristic
311C22×C3#47111yescharacteristic
312C22×C3#47311yescharacteristic
313C2×D3—42no—
314C2×D3—42no—
315C2×D3—42no—
316C2×D3—81no—
317C2×D3—81no—
318C2×D3—81no—
319C2×D3—81no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—82no—
323C3×C4#41711yescharacteristic
324C3×C4#41811yescharacteristic
325C3×C4C4×SD1611yescharacteristic
326C3×C4#44011yescharacteristic
327C3×C4#45011yescharacteristic
328C3×C4#45211yescharacteristic
329C3×C4#46811yescharacteristic
330C3×C4#46911yescharacteristic
331Dic3—42no—
332Dic3—161no—
333C23—62no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23—121no—
338C23—121no—
339C23—121no—
340C23#88411yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345D4—62no—
346D4—62no—
347D4—121no—
348D4—121no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—62no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—241no—
362C2×C4—241no—
363C2×C4—241no—
364C2×C4D3×C22⋊C411yescharacteristic
365C2×C4#67⋊C411yescharacteristic
366C2×C4#88311yescharacteristic
367C2×C4#91111yescharacteristic
368C2×C4#91111yescharacteristic
369C2×C4D3×SD1611yescharacteristic
370C2×C4#91411yescharacteristic
371C2×C4#91411yescharacteristic
372C2×C4#91511yescharacteristic
373C2×C4#91811yescharacteristic
374C2×C4#91911yescharacteristic
375C2×C4C4×#6711yescharacteristic
376C2×C4#93711yescharacteristic
377C2×C4#93811yescharacteristic
378C8—42no—
379C8—62no—
380C8—81no—
381C8—121no—
382C8—122no—
383C2×C3#169111yescharacteristic
384C2×C3#169311yescharacteristic
385C2×C3#169911yescharacteristic
386C2×C3#170111yescharacteristic
387C2×C3#187711yescharacteristic
388C2×C3#189511yescharacteristic
389C2×C3#190811yescharacteristic
390D3—82no—
391C22—122no—
392C22—122no—
393C22—122no—
394C22—241no—
395C22—241no—
396C22—241no—
397C22—241no—
398C22#475411yescharacteristic
399C22#475511yescharacteristic
400C22#478011yescharacteristic
401C22#478211yescharacteristic
402C22#481811yescharacteristic
403C22#481911yescharacteristic
404C22#495611yescharacteristic
405C4—21no—
406C4—21no—
407C4—82no—
408C4—122no—
409C4—481no—
410C4#477011yescharacteristic
411C4#477111yescharacteristic
412C4#478811yescharacteristic
413C4#481711yescharacteristic
414C4#515911yescharacteristic
415C4#516911yescharacteristic
416C4#517411yescharacteristic
417C4#518711yescharacteristic
418C3#922511yescharacteristic, complemented, Sylow
419C2—242no—
420C2#6845011yescharacteristic
421C2#6845111yescharacteristic
422C2#6845311yescharacteristic
423C2#6845411yescharacteristic
424C2#7012311yescharacteristic
425C2#7020311yescharacteristic
426C2#7027611yescharacteristic
subgroupG/H
centerC23#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#121
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9225
LCSG ▷ 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
3. C2×C3×#332C2semidirect
4. #71787C2semidirect
6. #67703C2semidirect
7. #67702C2semidirect
19. #5083C4semidirect
33. C2×#332D3semidirect
154. C2×C3⋊C8SD16semidirect
418. C3#9225semidirect

Maximal subgroups

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

subgroupindex
C2×#50832
C2×#48382
C2×C3×#3322
#717872
#704792
#677032
#677022
#92253