Finite Group Databasebeta

← #10691719

Subgroups — #10691719

Order 768.

496 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79268C211yesmaximal, characteristic
2#79004C211yesmaximal, characteristic
3#70522C211yesmaximal, characteristic, complemented
4#70503C211yesmaximal, characteristic
5C2×#4482C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#9271—31nomaximal, Sylow
9C2×#901—21no—
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#824—21no—
16C2×#824—21no—
17#5110—21no—
18C2×C4⋊(C3⋊C8)C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4846—22no—
21#4838—21no—
22#4836—22no—
23#4834—21no—
24#4736C414yes—
25#4694—21no—
26#4682—21no—
27#4485—21no—
28#4482—21no—
29#4482—21no—
30#4482—21no—
31#3226—31no—
32#2935—31no—
33#1966—31no—
34#1547—31no—
35C2×#332—31no—
36#1301D311yescharacteristic, complemented
37#1283—31no—
38C2×#235—21no—
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C22×C3×C8—21no—
42C2×C3×C42C2311yescharacteristic
43#930—21no—
44#930C2×C412yes—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3C2×C412yes—
47C2×#208D411yescharacteristic
48C2×#208D411yescharacteristic
49C22×C3⋊C8—21no—
50#901—42no—
51#901—42no—
52C2×ΓL(1,𝔽25)—22no—
53C2×D3×C8—22no—
54#875—21no—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12D412yes—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D411yescharacteristic
68C3×C4⋊C8D411yescharacteristic
69C3×C22⋊C8C2×C412yes—
70C3×C4×C8—21no—
71#824—42no—
72#824—42no—
73#821—24no—
74(C2×D3)⋊C8—22no—
75(C2×D3)⋊C8—22no—
76#818—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79#803—21no—
80C4×C3⋊C8—21no—
81C2×#131—61no—
82#532—31no—
83C2×C4⋊Q8—31no—
84#500—61no—
85#486—61no—
86#480—34no—
87#474—62no—
88#436—61no—
89#435—62no—
90C8×D4—61no—
91#431C2×D311yescharacteristic
92#423C2×D311yescharacteristic
93C2×C4⋊C8—31no—
94C2×C4⋊C8C2×D311yescharacteristic
95C2×#101—61no—
96C2×#101—61no—
97#408—61no—
98C4×M16—31no—
99#332—61no—
100#332—61no—
101#332—61no—
102#331—61no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×#67—22no—
107C22×Dic3—22no—
108#235—41no—
109#235—41no—
110#235—41no—
111#235—41no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12C22⋊C412yes—
115C2×C4×D3—22no—
116C2×Dic6—21no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×Dic6—41no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—22no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3C22⋊C412yes—
143#208—21no—
144#208—21no—
145#208—21no—
146#208D812yes—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—81no—
151ΓL(1,𝔽25)—44no—
152D3×C8—44no—
153C2×#41—61no—
154C22×Q8—61no—
155#131—122no—
156#131—122no—
157C2×M16—21no—
158C2×M16—31no—
159C2×M16—31no—
160C2×M16—62no—
161C22×C8—21no—
162C22×C8—62no—
163C22×C8—61no—
164C4⋊Q8—31no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169#127—31no—
170#124—61no—
171#121—32no—
172#121—61no—
173C4×D4—32no—
174C4×D4—61no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C4⋊C8D1211yescharacteristic
186C4⋊C8D1211yescharacteristic
187#102—64no—
188#101—122no—
189#101—122no—
190C22⋊C8—62no—
191C22⋊C8—62no—
192C22⋊C8C4×D312yes—
193#97—61no—
194C4×C8—21no—
195C4×C8—61no—
196C23×C3C4×D411yescharacteristic
197C22×D3—22no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4C2×C22⋊C411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12311yescharacteristic
207C2×C3×C4C2×D811yescharacteristic
208C2×C3×C4#13511yescharacteristic
209C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
210C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
211#67—42no—
212#67—42no—
213C2×Dic3—22no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217D12—42no—
218D12—41no—
219D12—41no—
220D12—41no—
221D12—41no—
222C4×D3—42no—
223C4×D3—42no—
224Dic6—41no—
225Dic6—41no—
226Dic6—81no—
227Dic6—81no—
228Dic6—81no—
229Dic6—81no—
230C3×C8—42no—
231C3×C8—41no—
232C3×C8—41no—
233C3×C8—81no—
234C3⋊C8—22no—
235C3⋊C8—41no—
236C3⋊C8—81no—
237#41—121no—
238#41—121no—
239#41—121no—
240#41—121no—
241C2×Q8—61no—
242C2×Q8—121no—
243C2×Q8—121no—
244C2×Q8—121no—
245C2×Q8—121no—
246C2×Q8—121no—
247C2×Q8—121no—
248C2×D4—32no—
249C2×D4—62no—
250C2×D4—61no—
251C2×D4—61no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4#23911yescharacteristic
257M16—32no—
258M16—41no—
259M16—41no—
260M16—61no—
261M16—61no—
262M16—61no—
263M16—124no—
264C2×C8—22no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—124no—
273C2×C8—121no—
274C2×C8—241no—
275C4⋊C4—32no—
276C4⋊C4—61no—
277C4⋊C4—62no—
278C4⋊C4—62no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C42C2×D1211yescharacteristic
284C42#23511yescharacteristic
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#38611yescharacteristic
290C22×C3C4×D811yescharacteristic
291C22×C3#44211yescharacteristic
292C2×D3—42no—
293C2×D3—42no—
294C2×D3—42no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4#44711yescharacteristic
299C3×C4#44911yescharacteristic
300C3×C4#45911yescharacteristic
301C3×C4#46311yescharacteristic
302C3×C4#47111yescharacteristic
303C3×C4#47311yescharacteristic
304Dic3—42no—
305Dic3—82no—
306C23—62no—
307C23#89311yescharacteristic
308Q8—121no—
309Q8—121no—
310Q8—241no—
311Q8—241no—
312Q8—241no—
313Q8—241no—
314D4—122no—
315D4—122no—
316D4—122no—
317D4—121no—
318D4—121no—
319D4—121no—
320D4—121no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—62no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4C4×D1211yescharacteristic
331C2×C4D3×C22⋊C411yescharacteristic
332C2×C4#88211yescharacteristic
333C2×C4#88411yescharacteristic
334C2×C4#89411yescharacteristic
335C2×C4#89511yescharacteristic
336C2×C4D3×D811yescharacteristic
337C2×C4#91411yescharacteristic
338C2×C4#91411yescharacteristic
339C2×C4#91911yescharacteristic
340C8—42no—
341C8—41no—
342C8—41no—
343C8—62no—
344C8—81no—
345C8—121no—
346C8—241no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3#187911yescharacteristic
350C2×C3#190811yescharacteristic
351C2×C3#191511yescharacteristic
352D3—82no—
353C22—21no—
354C22—21no—
355C22—122no—
356C22—122no—
357C22—122no—
358C22#466511yescharacteristic
359C22#486911yescharacteristic
360C22#487211yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—122no—
365C4—242no—
366C4#477311yescharacteristic
367C4#478611yescharacteristic
368C4#481011yescharacteristic
369C4#481911yescharacteristic
370C4#489611yescharacteristic
371C4#489811yescharacteristic
372C3#927111yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—242no—
376C2#6889711yescharacteristic
377C2#6894911yescharacteristic
378C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9271
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #70522C2semidirect
5. C2×#4482C2semidirect
6. C3×#1301C2semidirect
7. #67744C2semidirect
36. #1301D3semidirect
372. C3#9271semidirect

Maximal subgroups

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

subgroupindex
#792682
#790042
#705222
#705032
C2×#44822
C3×#13012
#677442
#92713