Finite Group Databasebeta

← #10694083

Subgroups — #10694083

Order 768.

586 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#213C211yesmaximal, characteristic
2C3×#1954C211yesmaximal, characteristic, complemented
3C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic, complemented
5C2×#4479C211yesmaximal, characteristic, complemented
6#69787C211yesmaximal, characteristic, complemented
7C2×#4455C211yesmaximal, characteristic, complemented
8#9663—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#213C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8—22no—
14C2×C3×C4×C8C2211yescharacteristic
15C2×#832C411yescharacteristic
16C2×(C2×D3)⋊C8—22no—
17C4×#213—22no—
18C4×#213—22no—
19C2×#809C411yescharacteristic
20C2×#807C411yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24D4⋊(C3⋊C8)—22no—
25#4483—22no—
26#4479—22no—
27#4455—22no—
28C2×C4×D8—31no—
29#1954—31no—
30#1954D311yescharacteristic, complemented
31C2×#335—31no—
32C2×D4⋊C8—31no—
33C2×D4⋊C8—31no—
34C2×#322—31no—
35C22×C3×D4C2×C411yescharacteristic
36C23×C3×C4—21no—
37C22×D12C2×C411yescharacteristic
38C22×C4×D3—21no—
39C22×C3×C8—21no—
40C22×C3×C8D412yes—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#213—22no—
49C2×#213—22no—
50C2×#213—22no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C22×C3⋊C8—21no—
54C22×C3⋊C8—21no—
55C22×C3⋊C8C2×C411yescharacteristic
56C22×C3⋊C8C2×C411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—44no—
64C3×C4×C8—21no—
65C3×C4×C8—21no—
66#832—22no—
67(C2×D3)⋊C8—44no—
68#809—22no—
69#807—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8D412yes—
74C4×C3⋊C8D412yes—
75C4×C3⋊C8D412yes—
76C22×D8—31no—
77C2×C4×D4—31no—
78C2×C4×D4C2×D311yescharacteristic
79C4×D8—62no—
80C4×D8—62no—
81C2×#107—31no—
82C2×C4⋊C8—31no—
83C2×C4⋊C8C2×D311yescharacteristic
84C2×#102—31no—
85C2×#102—31no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—62no—
88C2×C4×C8—31no—
89C2×C4×C8—31no—
90C2×C4×C8C2×D311yescharacteristic
91#335—62no—
92D4⋊C8—62no—
93D4⋊C8—62no—
94#322—62no—
95C24×C3—21no—
96C23×D3—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4M1612yes—
103C22×C3×C4—21no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4—42no—
107C22×C3×C4—42no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12M1612yes—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—42no—
121C2×C4×D3—42no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C2×C3×C8—42no—
126C2×C3×C8D814yescomplemented
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#213—28no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—42no—
148C2×C3⋊C8C22⋊C412yes—
149C2×C3⋊C8C22⋊C412yes—
150C22×D4—31no—
151C22×D4C4×D311yescharacteristic
152C23×C4—21no—
153C23×C4—61no—
154C2×D8—62no—
155C2×D8—62no—
156C2×D8—62no—
157C22×C8—21no—
158C22×C8—31no—
159C22×C8—31no—
160C22×C8—61no—
161C22×C8—61no—
162C22×C8#6712yes—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—61no—
168C4×D4—61no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4C4×D311yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—61no—
175C2×C42C22×D311yescharacteristic
176#107—62no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181#102—62no—
182#102—62no—
183C22⋊C8—44no—
184C22⋊C8—124no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—32no—
188C4×C8—32no—
189C4×C8—32no—
190C4×C8—61no—
191C23×C3—22no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3C4×D411yescharacteristic
197C22×D3—22no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—42no—
201C22×D3—42no—
202C3×D4—22no—
203C3×D4—22no—
204C3×D4—41no—
205C3×D4—41no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—82no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4C2×C22⋊C411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12411yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×C3×C4C2×M1611yescharacteristic
219C2×C3×C4C2×M1611yescharacteristic
220C2×C3×C4#13111yescharacteristic
221C2×C3×C4#13111yescharacteristic
222C2×C3×C4C2×D812yes—
223C2×C3×C4#13512yes—
224C2×Dic3—41no—
225C2×Dic3—42no—
226D12—22no—
227D12—22no—
228D12—41no—
229D12—41no—
230C4×D3—82no—
231C4×D3—81no—
232C4×D3—81no—
233C3×C8—24no—
234C3×C8—81no—
235C3⋊C8—22no—
236C3⋊C8—22no—
237C3⋊C8—42no—
238C3⋊C8—81no—
239C24—21no—
240C24—61no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—32no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4ΓL(1,𝔽25)12yes—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—61no—
259C22×C4—122no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4—122no—
263C22×C4C2×C4×D311yescharacteristic
264C22×C4#23511yescharacteristic
265C22×C4C2×#6711yescharacteristic
266D8—68no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—32no—
270C2×C8—32no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—62no—
278C2×C8—122no—
279C2×C8—122no—
280C2×C8—121no—
281C2×C8#21314yescomplemented
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C42D3×D411yescharacteristic
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C42#23711yescharacteristic
294C22×C3—44no—
295C22×C3—42no—
296C22×C3—82no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3#39011yescharacteristic
300C22×C3#43511yescharacteristic
301C22×C3#43511yescharacteristic
302C22×C3C4×D812yes—
303C22×C3#44412yes—
304C2×D3—44no—
305C2×D3—42no—
306C2×D3—82no—
307C2×D3—81no—
308C2×D3—81no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—81no—
312C3×C4#40711yescharacteristic
313C3×C4#40811yescharacteristic
314C3×C4#43511yescharacteristic
315C3×C4#43511yescharacteristic
316C3×C4#46612yes—
317C3×C4#49311yescharacteristic
318C3×C4#49511yescharacteristic
319Dic3—81no—
320C23—22no—
321C23—41no—
322C23—42no—
323C23—41no—
324C23—42no—
325C23—62no—
326C23—122no—
327C23—121no—
328C23—122no—
329C23—121no—
330C23C4×#6711yescharacteristic
331D4—22no—
332D4—22no—
333D4—41no—
334D4—41no—
335D4—62no—
336D4—62no—
337D4—121no—
338D4—121no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—121no—
346C2×C4—122no—
347C2×C4—242no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4D3×C22⋊C411yescharacteristic
351C2×C4#67⋊C411yescharacteristic
352C2×C4#88411yescharacteristic
353C2×C4#88511yescharacteristic
354C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
355C2×C4#90111yescharacteristic
356C2×C4D3×M1611yescharacteristic
357C2×C4#90711yescharacteristic
358C2×C4C2×#21312yes—
359C2×C4#93911yescharacteristic
360C2×C4#94011yescharacteristic
361C2×C4#95012yes—
362C8—24no—
363C8—62no—
364C8—62no—
365C8—81no—
366C8—122no—
367C8—241no—
368C2×C3—82no—
369C2×C3#158112yes—
370C2×C3#158112yes—
371C2×C3#195511yescharacteristic
372C2×C3#197011yescharacteristic
373C2×C3#197511yescharacteristic
374D3—82no—
375C22—42no—
376C22—44no—
377C22—81no—
378C22—81no—
379C22—82no—
380C22—122no—
381C22—124no—
382C22—241no—
383C22—241no—
384C22—242no—
385C22#495711yescharacteristic
386C22C4×#21312yes—
387C22#511111yescharacteristic
388C22#512711yescharacteristic
389C22#514512yes—
390C4—21no—
391C4—21no—
392C4—81no—
393C4—241no—
394C4#472611yescharacteristic
395C4#472711yescharacteristic
396C4#472811yescharacteristic
397C4#473011yescharacteristic
398C4#503812yes—
399C4#505911yescharacteristic
400C4#507311yescharacteristic
401C3#966311yescharacteristic, complemented, Sylow
402C2—82no—
403C2—242no—
404C2#6964711yescharacteristic
405C2#6991711yescharacteristic
406C2#7000011yescharacteristic
407C2#7084112yes—
408C2#7086212yes—
subgroupG/H
centerC22×C4C2×#67
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1954C2
socleC23×C3C4×D4
prime radicalC22×#213C4
cosocleC2×C3×C42C23
hypercenter#1954D3
hypocenterC3#9663
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. C3×#1954C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. C2×#4483C2semidirect
5. C2×#4479C2semidirect
6. #69787C2semidirect
7. C2×#4455C2semidirect
30. #1954D3semidirect
126. C2×C3×C8D8semidirect
281. C2×C8#213semidirect
401. C3#9663semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2132
C3×#19542
C2×D4⋊(C3⋊C8)2
C2×#44832
C2×#44792
#697872
C2×#44552
#96633