Finite Group Databasebeta

← #10691411

Subgroups — #10691411

Order 768.

562 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4846C211yesmaximal, characteristic
2C2×#4838C211yesmaximal, characteristic
3C2×#4482C211yesmaximal, characteristic, complemented
4#69584C211yesmaximal, characteristic
5#69579C211yesmaximal, characteristic, complemented
6C3×#1296C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#9227—31nomaximal, Sylow
9C22×ΓL(1,𝔽25)—21no—
10C2×C4×D12C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×#384C2211yescharacteristic
15C2×#821—21no—
16C2×(C2×D3)⋊C8—21no—
17C2×#818—21no—
18C2×#810—21no—
19C2×#810—21no—
20C2×#809—21no—
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4846—22no—
24#4846C414yescomplemented
25#4838—22no—
26#4838—22no—
27#4529—21no—
28#4482—22no—
29C2×#474—31no—
30C2×#436—31no—
31#1956—31no—
32#1802—31no—
33C2×#332—31no—
34#1296D311yescharacteristic, complemented
35#1277—31no—
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C22×Dic6—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C22×C3⋊C8—21no—
50C22×C3⋊C8D411yescharacteristic
51C22×C3⋊C8D411yescharacteristic
52C2×ΓL(1,𝔽25)—24no—
53C2×ΓL(1,𝔽25)—42no—
54C2×ΓL(1,𝔽25)—42no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12C2×C412yes—
58C4×D12D412yes—
59#869—21no—
60#869—21no—
61#869—21no—
62#869C2×C412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8C2×C412yes—
66C3×C4⋊C8D412yes—
67#821—24no—
68(C2×D3)⋊C8—24no—
69#818—24no—
70#810—42no—
71#810—42no—
72#809—42no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C22×M16—61no—
78C2×C4⋊Q8—31no—
79C2×C4×D4—31no—
80#474—34no—
81#474—62no—
82#436—62no—
83#436—62no—
84C2×#107—61no—
85C2×C4⋊C8—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87C2×C4⋊C8C2×D311yescharacteristic
88C2×#103—61no—
89C2×#103—61no—
90C2×#102—61no—
91C2×#102—61no—
92C2×C22⋊C8—61no—
93C2×C4×C8—31no—
94#384C2×D311yescharacteristic
95#340—61no—
96#332—62no—
97C23×D3—21no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×Dic3—21no—
103C22×Dic3—41no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12C22⋊C412yes—
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3—41no—
113C2×C4×D3—41no—
114C2×C4×D3—41no—
115C2×C4×D3—41no—
116C2×Dic6—22no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—81no—
133C3×C42C22×C411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3C22⋊C412yes—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8D812yes—
154ΓL(1,𝔽25)—48no—
155C22×Q8—61no—
156C22×D4—31no—
157C23×C4—61no—
158C2×M16—64no—
159C2×M16—122no—
160C2×M16—122no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—31no—
164C22×C8—31no—
165C22×C8—61no—
166C4⋊Q8—32no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4×D4—32no—
171C4×D4—32no—
172C4×D4—61no—
173C4×D4—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C42C22×D311yescharacteristic
182#107—64no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187C4⋊C8C4×D312yes—
188C4⋊C8D1212yes—
189#103—122no—
190#103—122no—
191#102—64no—
192#102—122no—
193C22⋊C8—64no—
194C4×C8—61no—
195C4×C8—61no—
196C23×C3#12111yescharacteristic
197C22×D3—22no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—41no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4C2×C22⋊C411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12311yescharacteristic
215C2×C3×C4C2×D811yescharacteristic
216C2×C3×C4#13511yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
219C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
220C2×C3×C4#13711yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×C3×C4#13711yescharacteristic
223C2×Dic3—22no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229D12—22no—
230D12—22no—
231D12—41no—
232D12—41no—
233C4×D3—42no—
234C4×D3—42no—
235C4×D3—42no—
236C4×D3—42no—
237Dic6—42no—
238Dic6—42no—
239Dic6—81no—
240Dic6—81no—
241C3×C8—42no—
242C3×C8—42no—
243C3⋊C8—22no—
244C3⋊C8—41no—
245C3⋊C8—81no—
246C24—61no—
247C2×Q8—62no—
248C2×Q8—121no—
249C2×Q8—121no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×D4—32no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C22×C4—41no—
260C22×C4—61no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4#23511yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271M16—128no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—32no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—121no—
285C2×C8—121no—
286C2×C8—121no—
287C4⋊C4—32no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—62no—
294C4⋊C4—62no—
295C4⋊C4—81no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C42C2×C4×D311yescharacteristic
301C42C2×D1211yescharacteristic
302C42D3×D411yescharacteristic
303C42#23911yescharacteristic
304C22×C3#38611yescharacteristic
305C22×C3#45911yescharacteristic
306C22×C3#46311yescharacteristic
307C22×C3#46811yescharacteristic
308C22×C3#47011yescharacteristic
309C22×C3#47111yescharacteristic
310C22×C3#47311yescharacteristic
311C2×D3—42no—
312C2×D3—42no—
313C2×D3—42no—
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C2×D3—81no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—161no—
321C3×C4#41711yescharacteristic
322C3×C4#41911yescharacteristic
323C3×C4C4×D811yescharacteristic
324C3×C4#44211yescharacteristic
325C3×C4#44711yescharacteristic
326C3×C4#44911yescharacteristic
327C3×C4#46811yescharacteristic
328C3×C4#47011yescharacteristic
329Dic3—42no—
330Dic3—82no—
331C23—62no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23—121no—
338C23#88411yescharacteristic
339Q8—122no—
340Q8—122no—
341Q8—241no—
342Q8—241no—
343D4—62no—
344D4—62no—
345D4—121no—
346D4—121no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—62no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—241no—
361C2×C4—241no—
362C2×C4C4×D1211yescharacteristic
363C2×C4D3×C22⋊C411yescharacteristic
364C2×C4#88211yescharacteristic
365C2×C4#89311yescharacteristic
366C2×C4#89411yescharacteristic
367C2×C4#89511yescharacteristic
368C2×C4D3×D811yescharacteristic
369C2×C4#91111yescharacteristic
370C2×C4#91411yescharacteristic
371C2×C4#91411yescharacteristic
372C2×C4#91511yescharacteristic
373C2×C4#91511yescharacteristic
374C2×C4#91811yescharacteristic
375C2×C4#91911yescharacteristic
376C8—42no—
377C8—42no—
378C8—62no—
379C8—121no—
380C8—241no—
381C2×C3#168511yescharacteristic
382C2×C3#168911yescharacteristic
383C2×C3#169911yescharacteristic
384C2×C3#170311yescharacteristic
385C2×C3#187911yescharacteristic
386C2×C3#189411yescharacteristic
387C2×C3#190811yescharacteristic
388D3—82no—
389C22—122no—
390C22—122no—
391C22—122no—
392C22—241no—
393C22—241no—
394C22—241no—
395C22—241no—
396C22#466511yescharacteristic
397C22#474811yescharacteristic
398C22#475511yescharacteristic
399C22#478011yescharacteristic
400C22#478611yescharacteristic
401C22#481611yescharacteristic
402C22#481911yescharacteristic
403C4—21no—
404C4—21no—
405C4—122no—
406C4—161no—
407C4—242no—
408C4#477111yescharacteristic
409C4#477311yescharacteristic
410C4#480211yescharacteristic
411C4#481011yescharacteristic
412C4#486711yescharacteristic
413C4#486911yescharacteristic
414C4#487211yescharacteristic
415C4#488611yescharacteristic
416C3#922711yescharacteristic, complemented, Sylow
417C2—242no—
418C2#6839811yescharacteristic
419C2#6844911yescharacteristic
420C2#6845311yescharacteristic
421C2#6850311yescharacteristic
422C2#6888411yescharacteristic
423C2#6889711yescharacteristic
424C2#6900711yescharacteristic
subgroupG/H
centerC23#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#121
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1296D3
hypocenterC3#9227
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. C2×#4482C2semidirect
5. #69579C2semidirect
6. C3×#1296C2semidirect
7. #67703C2semidirect
24. #4846C4semidirect
34. #1296D3semidirect
416. C3#9227semidirect

Maximal subgroups

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

subgroupindex
C2×#48462
C2×#48382
C2×#44822
#695842
#695792
C3×#12962
#677032
#92273