Finite Group Databasebeta

← #10691715

Subgroups — #10691715

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#83215C211yesmaximal, characteristic
2#79004C211yesmaximal, characteristic
3C2×C3×#332C211yesmaximal, characteristic, complemented
4#71789C211yesmaximal, characteristic, complemented
5#70432C211yesmaximal, characteristic
6#67707C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#9271—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#901—21no—
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#836—21no—
15C2×#836—21no—
16#5110—21no—
17#5088—21no—
18#5087—22no—
19#5078—21no—
20#4967C414yes—
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4838—21no—
25#4836—22no—
26#4834—21no—
27C3×#332—21no—
28C3×#332—21no—
29#4486—21no—
30#4485—21no—
31#3226—31no—
32#2935—31no—
33#1966—31no—
34#1547—31no—
35C2×#332D311yescharacteristic, complemented
36#1301—31no—
37#1283—31no—
38C22×C3×Q8—21no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C412yes—
48C2×C3×C42C2311yescharacteristic
49#930—21no—
50#930C2×C412yes—
51C2×#208—21no—
52C2×#208D411yescharacteristic
53C2×#208D411yescharacteristic
54#901—42no—
55#901—42no—
56C2×ΓL(1,𝔽25)—22no—
57C2×D3×C8—22no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12D412yes—
62#869D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D411yescharacteristic
67C3×C4⋊C8D411yescharacteristic
68#836—42no—
69#836—42no—
70#830C2×C412yes—
71(C2×D3)⋊C8—22no—
72(C2×D3)⋊C8—22no—
73#809—24no—
74#807—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78#803—21no—
79C4×C3⋊C8—21no—
80C4×C3⋊C8—21no—
81C2×#131—61no—
82#532—31no—
83C2×C4⋊Q8C2×D311yescharacteristic
84#500—61no—
85#486—61no—
86#480—34no—
87#474—62no—
88#436—61no—
89#435—62no—
90C8×D4—61no—
91#431—31no—
92#423—31no—
93C2×C4⋊C8C2×D311yescharacteristic
94C2×C4⋊C8C2×D311yescharacteristic
95C2×#101—61no—
96C2×#101—61no—
97#408—61no—
98C4×M16—31no—
99#332—21no—
100#332—21no—
101#332—61no—
102#331—61no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C22×C3×C4—22no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×#67—22no—
114#235—41no—
115#235—41no—
116#235—41no—
117#235—41no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12C22⋊C412yes—
121C2×C4×D3—22no—
122C2×Dic6—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C2×C3×C8—81no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3C22⋊C412yes—
141#208—21no—
142#208—21no—
143#208—21no—
144#208—41no—
145#208—41no—
146#208D812yes—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—41no—
151ΓL(1,𝔽25)—44no—
152D3×C8—44no—
153C2×#41—61no—
154C22×Q8—21no—
155#131—122no—
156#131—122no—
157C2×M16—31no—
158C2×M16—31no—
159C2×M16—62no—
160C2×M16—61no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—62no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—31no—
169#127—31no—
170#124—61no—
171#121—32no—
172#121—61no—
173C4×D4—32no—
174C4×D4—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C4×D312yes—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C4⋊C8#6711yescharacteristic
186C4⋊C8#6711yescharacteristic
187#102—64no—
188#101—122no—
189#101—122no—
190C22⋊C8—32no—
191C22⋊C8—62no—
192C22⋊C8—62no—
193#97—61no—
194C4×C8—61no—
195C4×C8—61no—
196C23×C3C4×D411yescharacteristic
197C22×D3—22no—
198C3×Q8—81no—
199C3×Q8—81no—
200C3×Q8—81no—
201C3×Q8—81no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4C2×D811yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218#67—42no—
219#67—42no—
220C2×Dic3—22no—
221D12—42no—
222D12—41no—
223D12—41no—
224D12—41no—
225D12—41no—
226C4×D3—42no—
227C4×D3—42no—
228Dic6—41no—
229Dic6—41no—
230C3×C8—41no—
231C3×C8—41no—
232C3×C8—81no—
233C3⋊C8—22no—
234C3⋊C8—42no—
235C3⋊C8—41no—
236C3⋊C8—81no—
237#41—121no—
238#41—121no—
239#41—121no—
240#41—121no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—61no—
248C2×D4—32no—
249C2×D4—62no—
250C2×D4—61no—
251C2×D4—61no—
252C22×C4—22no—
253C22×C4—62no—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4#23711yescharacteristic
257M16—32no—
258M16—61no—
259M16—61no—
260M16—61no—
261M16—124no—
262M16—121no—
263M16—121no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—61no—
270C2×C8—62no—
271C2×C8—61no—
272C2×C8—81no—
273C2×C8—124no—
274C2×C8—121no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—32no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—61no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C42#23511yescharacteristic
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C42C2×#6711yescharacteristic
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—82no—
299C3×C4#44711yescharacteristic
300C3×C4#44911yescharacteristic
301C3×C4#45911yescharacteristic
302C3×C4#46311yescharacteristic
303C3×C4#47111yescharacteristic
304C3×C4#47311yescharacteristic
305Dic3—42no—
306C23—62no—
307C23#67⋊C411yescharacteristic
308Q8—81no—
309Q8—81no—
310Q8—81no—
311Q8—81no—
312Q8—121no—
313Q8—121no—
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—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—62no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4D3×C22⋊C411yescharacteristic
331C2×C4#88311yescharacteristic
332C2×C4#88411yescharacteristic
333C2×C4D3×D811yescharacteristic
334C2×C4#91111yescharacteristic
335C2×C4#91411yescharacteristic
336C2×C4#91611yescharacteristic
337C2×C4C4×#6711yescharacteristic
338C2×C4#93711yescharacteristic
339C2×C4#93811yescharacteristic
340C8—41no—
341C8—41no—
342C8—62no—
343C8—81no—
344C8—122no—
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#475611yescharacteristic
359C22#478511yescharacteristic
360C22#495611yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—82no—
365C4—122no—
366C4#478611yescharacteristic
367C4#478711yescharacteristic
368C4#515611yescharacteristic
369C4#517211yescharacteristic
370C4#517711yescharacteristic
371C4#519211yescharacteristic
372C3#927111yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—242no—
376C2#7012411yescharacteristic
377C2#7020811yescharacteristic
378C2#7028611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9271
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. C2×C3×#332C2semidirect
4. #71789C2semidirect
6. #67707C2semidirect
7. #67706C2semidirect
35. C2×#332D3semidirect
372. C3#9271semidirect

Maximal subgroups

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

subgroupindex
#832152
#790042
C2×C3×#3322
#717892
#704322
#677072
#677062
#92713