Finite Group Databasebeta

← #10694110

Subgroups — #10694110

Order 768.

510 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#213C211yesmaximal, characteristic
2C3×#1460C211yesmaximal, characteristic, complemented
3#71597C211yesmaximal, characteristic, complemented
4#70783C211yesmaximal, characteristic, complemented
5#70757C211yesmaximal, characteristic, complemented
6#70745C211yesmaximal, characteristic, complemented
7#70740C211yesmaximal, characteristic, complemented
8#9668—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#213C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×#832C411yescharacteristic
15C4×#213—21no—
16C4×#213—21no—
17C4×#213—21no—
18C4×#213—21no—
19C2×#809C411yescharacteristic
20C2×#807C411yescharacteristic
21#4925C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#323—22no—
25D4⋊(C3⋊C8)—22no—
26#4480—21no—
27#4479—21no—
28#4468—22no—
29#4459—22no—
30#4454—21no—
31C8×C3⋊C8—21no—
32C2×C4×D8—31no—
33#1564—31no—
34#1478—31no—
35#1478—31no—
36#1460—31no—
37#1460D311yescharacteristic, complemented
38#1450—31no—
39C22×C3×D4C2×C411yescharacteristic
40C23×C3×C4—21no—
41C22×D12C2×C411yescharacteristic
42C22×C4×D3—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#213—21no—
51C2×#213—21no—
52C2×#213—22no—
53C2×#213C814yescomplemented
54C2×(C2×D3)⋊C4—21no—
55C2×C4⋊Dic3C2×C411yescharacteristic
56C22×C3⋊C8C2×C411yescharacteristic
57C22×C3⋊C8C2×C411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×C22⋊C8D412yes—
66C3×C4×C8—21no—
67C3×C4×C8—21no—
68#832—21no—
69#832—21no—
70#830—21no—
71#830—21no—
72#809—21no—
73#809—21no—
74#807—21no—
75#807—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82C4×C3⋊C8D411yescharacteristic
83C4×C3⋊C8D411yescharacteristic
84C22×D8—31no—
85C2×C4×D4—31no—
86C2×C4×D4C2×D311yescharacteristic
87C4×D8—61no—
88C4×D8—61no—
89C4×D8—61no—
90C4×D8—61no—
91#431—31no—
92#431—31no—
93#431C2×D311yescharacteristic
94#431C2×D311yescharacteristic
95C2×#107—31no—
96C2×#102—31no—
97C2×#102—31no—
98C2×C4×C8—31no—
99#335—61no—
100#334—61no—
101D4⋊C8—62no—
102D4⋊C8—62no—
103#323—22no—
104#323—62no—
105#322—61no—
106C82—61no—
107C24×C3—21no—
108C23×D3—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4C2×C812yes—
115C22×C3×C4—21no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—82no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12C2×C812yes—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×C4×D3—82no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#213—24no—
143#213—24no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8C22⋊C411yescharacteristic
153C2×C3⋊C8C22⋊C411yescharacteristic
154C2×C3⋊C8C22⋊C411yescharacteristic
155C2×C3⋊C8C22⋊C411yescharacteristic
156C2×C3⋊C8C2×C812yes—
157C2×C3⋊C8M1612yes—
158C22×D4—31no—
159C22×D4C4×D311yescharacteristic
160C23×C4—21no—
161C23×C4—61no—
162C2×D8—34no—
163C2×D8—62no—
164C2×D8—61no—
165C2×D8—61no—
166C22×C8—31no—
167C22×C8—31no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4C4×D311yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—61no—
180C2×C42C22×D311yescharacteristic
181#107—61no—
182#107—61no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—32no—
186C4⋊C8—61no—
187#102—61no—
188#102—61no—
189#102—61no—
190#102—61no—
191C22⋊C8—21no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194C22⋊C8#6712yes—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—31no—
198C4×C8—31no—
199C4×C8—31no—
200C4×C8—31no—
201C4×C8—61no—
202C4×C8—61no—
203C23×C3—22no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—81no—
207C23×C3—81no—
208C23×C3C4×D411yescharacteristic
209C22×D3—22no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—81no—
213C22×D3—81no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—41no—
217C3×D4—41no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—82no—
222C2×C3×C4—82no—
223C2×C3×C4—81no—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12411yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×C3×C4C22×C811yescharacteristic
231C2×C3×C4C2×M1611yescharacteristic
232C2×C3×C4#13111yescharacteristic
233C2×C3×C4#13111yescharacteristic
234C2×Dic3—41no—
235C2×Dic3—81no—
236D12—22no—
237D12—22no—
238D12—41no—
239D12—41no—
240C4×D3—82no—
241C4×D3—82no—
242C3×C8—42no—
243C3×C8—81no—
244C3⋊C8—21no—
245C3⋊C8—21no—
246C3⋊C8—42no—
247C3⋊C8—81no—
248C3⋊C8C22⋊C812yes—
249C3⋊C8C22⋊C812yes—
250C24—21no—
251C24—61no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—32no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4D3×C812yes—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—61no—
268C22×C4—82no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—242no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4#23511yescharacteristic
274C22×C4C2×#6711yescharacteristic
275D8—64no—
276D8—64no—
277C2×C8—22no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8—32no—
281C2×C8—32no—
282C2×C8—31no—
283C2×C8—31no—
284C2×C8—41no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—62no—
288C2×C8—121no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C42D3×D411yescharacteristic
298C42D3×D411yescharacteristic
299C42D3×D411yescharacteristic
300C42#23711yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—42no—
304C22×C3—82no—
305C22×C3—82no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3#39011yescharacteristic
309C22×C3C8×D411yescharacteristic
310C22×C3#43611yescharacteristic
311C2×D3—42no—
312C2×D3—82no—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—82no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—81no—
320C3×C4C2×C22⋊C811yescharacteristic
321C3×C4#40811yescharacteristic
322C3×C4C8×D411yescharacteristic
323C3×C4#43611yescharacteristic
324C3×C4#47212yes—
325Dic3—81no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—62no—
330C23—81no—
331C23—81no—
332C23—121no—
333C23—121no—
334C23—241no—
335C23—241no—
336C23C4×#6711yescharacteristic
337D4—22no—
338D4—22no—
339D4—41no—
340D4—41no—
341D4—62no—
342D4—62no—
343D4—121no—
344D4—121no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—121no—
352C2×C4—242no—
353C2×C4—242no—
354C2×C4—241no—
355C2×C4D3×C22⋊C411yescharacteristic
356C2×C4#67⋊C411yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4#88511yescharacteristic
359C2×C4C2×D3×C811yescharacteristic
360C2×C4#90111yescharacteristic
361C2×C4D3×M1611yescharacteristic
362C2×C4#90711yescharacteristic
363C2×C4#93911yescharacteristic
364C2×C4#94011yescharacteristic
365C8—32no—
366C8—32no—
367C8—42no—
368C8—61no—
369C8—61no—
370C8—81no—
371C8—122no—
372C8—241no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—82no—
376C2×C3#195411yescharacteristic
377C2×C3#197611yescharacteristic
378C2×C3#197811yescharacteristic
379D3—82no—
380C22—21no—
381C22—21no—
382C22—42no—
383C22—82no—
384C22—82no—
385C22—81no—
386C22—81no—
387C22—122no—
388C22—242no—
389C22—242no—
390C22—241no—
391C22—241no—
392C22#495711yescharacteristic
393C22C8×#6711yescharacteristic
394C22#512811yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—81no—
399C4—241no—
400C4D3×C22⋊C811yescharacteristic
401C4#67⋊C811yescharacteristic
402C4#472711yescharacteristic
403C4#472911yescharacteristic
404C4#474912yes—
405C3#966811yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—82no—
409C2—242no—
410C2#6991611yescharacteristic
411C2#7004211yescharacteristic
412C2#7007411yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1460C2
socleC22×C3#390
prime radicalC22×#213C4
cosocleC2×C3×C42C23
hypercenter#1460D3
hypocenterC3#9668
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. C3×#1460C2semidirect
3. #71597C2semidirect
4. #70783C2semidirect
5. #70757C2semidirect
6. #70745C2semidirect
7. #70740C2semidirect
37. #1460D3semidirect
53. C2×#213C8semidirect
405. C3#9668semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2132
C3×#14602
#715972
#707832
#707572
#707452
#707402
#96683