Finite Group Databasebeta

← #10693965

Subgroups — #10693965

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#947C211yesmaximal, characteristic
2C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
3C2×#4535C211yesmaximal, characteristic, complemented
4#70443C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic
6C4×C8×Dic3C211yesmaximal, characteristic, complemented
7C2×#4459C211yesmaximal, characteristic
8#9639—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#947C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5229C411yescharacteristic
15C2×C8×Dic3—22no—
16#4990—22no—
17#4988C411yescharacteristic
18#4977C411yescharacteristic
19C4×(C2×D3)⋊C4—22no—
20C42×Dic3C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×Q8⋊C8—22no—
24#4550—22no—
25#4535—22no—
26#4483—22no—
27#4459—22no—
28C4×#124—31no—
29#1831—31no—
30C42×C8—31no—
31C2×Q8⋊C8—31no—
32C2×Q8⋊C8D311yescharacteristic, complemented
33C2×D4⋊C8—31no—
34C2×D4⋊C8—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×D12C2×C411yescharacteristic
37C22×C4×D3—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48#947—22no—
49#947C814yescomplemented
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55C2×C4×Dic3C2×C411yescharacteristic
56C2×C4×Dic3D412yes—
57C2×C4×Dic3D412yes—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4×C8—21no—
67C3×C4×C8D412yes—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C8×Dic3—28no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4×C3⋊C8—21no—
74C4×C3⋊C8D412yes—
75C2×#124—31no—
76C2×C4×Q8C2×D311yescharacteristic
77C2×C4×D4—31no—
78C2×C4⋊C8—31no—
79C2×C4⋊C8C2×D311yescharacteristic
80C2×C4×C8—31no—
81C2×C4×C8—62no—
82C2×C4×C8C2×D311yescharacteristic
83#391—31no—
84#390—31no—
85#388—62no—
86#382—31no—
87C4×C22⋊C4—62no—
88C43—31no—
89#336—62no—
90Q8⋊C8—22no—
91Q8⋊C8—62no—
92D4⋊C8—62no—
93D4⋊C8—62no—
94C23×D3—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8C2×C812yes—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3C22⋊C412yes—
108C22×Dic3C22⋊C412yes—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12C2×C812yes—
115C2×C4×D3—41no—
116C2×C4×D3—41no—
117C2×C4×D3—42no—
118C2×C4×D3—42no—
119C2×C3×C8—24no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C2×C3×C8—42no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42—41no—
128C3×C42—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133(C2×D3)⋊C4—28no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—44no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3—21no—
141C4×Dic3—22no—
142C4×Dic3—22no—
143C4×Dic3—24no—
144C4×Dic3C2×C812yes—
145C4×Dic3M1612yes—
146C2×C3⋊C8—24no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—42no—
150C22×Q8C4×D311yescharacteristic
151C22×D4—31no—
152C23×C4—61no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—61no—
156C22×C8—61no—
157#124—34no—
158#124—62no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×D4—61no—
164C4×D4—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4C4×D311yescharacteristic
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—61no—
173C2×C42—21no—
174C2×C42—31no—
175C2×C42—31no—
176C2×C42—32no—
177C2×C42—32no—
178C2×C42C22×D311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4×C8—21no—
184C4×C8—32no—
185C4×C8—68no—
186C4×C8—61no—
187C4×C8#6712yes—
188#95—62no—
189#95—62no—
190C23×C3C2×C22⋊C411yescharacteristic
191C22×D3—22no—
192C22×D3—41no—
193C22×D3—41no—
194C22×D3—42no—
195C22×D3—42no—
196C3×Q8—22no—
197C3×Q8—22no—
198C3×Q8—41no—
199C3×Q8—41no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12411yescharacteristic
211C2×C3×C4#12711yescharacteristic
212C2×C3×C4C22×C811yescharacteristic
213C2×C3×C4C2×M1611yescharacteristic
214C2×C3×C4#13111yescharacteristic
215C2×C3×C4#13111yescharacteristic
216C2×C3×C4#13512yes—
217C2×C3×C4#13512yes—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3C22⋊C814yes—
225C2×Dic3#10414yes—
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—24no—
236C3⋊C8—81no—
237C24—61no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8D3×C812yes—
244C2×D4—32no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—32no—
253C22×C4—32no—
254C22×C4—61no—
255C22×C4—121no—
256C22×C4—122no—
257C22×C4—122no—
258C22×C4—121no—
259C22×C4C2×C4×D311yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4D3×D411yescharacteristic
262C2×C8—24no—
263C2×C8—22no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—64no—
267C2×C8—62no—
268C2×C8—122no—
269C2×C8—121no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C22⋊C4—68no—
277C22⋊C4—124no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C42—32no—
281C42—32no—
282C42—41no—
283C42—41no—
284C42—64no—
285C42—61no—
286C42—61no—
287C42—62no—
288C42—62no—
289C42#23511yescharacteristic
290C42D3×D411yescharacteristic
291C42#23711yescharacteristic
292C42C2×#6711yescharacteristic
293C22×C3#39011yescharacteristic
294C22×C3C2×C22⋊C811yescharacteristic
295C22×C3#40811yescharacteristic
296C22×C3#41612yes—
297C22×C3C2×#10412yes—
298C2×D3—42no—
299C2×D3—44no—
300C2×D3—81no—
301C2×D3—81no—
302C2×D3—82no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—42no—
306C3×C4—81no—
307C3×C4C8×D411yescharacteristic
308C3×C4C8×D411yescharacteristic
309C3×C4#43611yescharacteristic
310C3×C4#43611yescharacteristic
311C3×C4#48812yes—
312C3×C4#49511yescharacteristic
313C3×C4#49511yescharacteristic
314Dic3—24no—
315Dic3—24no—
316Dic3—81no—
317C23—62no—
318C23—121no—
319C23—121no—
320C23—122no—
321C23—122no—
322C23D3×C22⋊C411yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—41no—
326Q8—41no—
327D4—62no—
328D4—62no—
329D4—121no—
330D4—121no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—34no—
334C2×C4—34no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—241no—
346C2×C4—241no—
347C2×C4—242no—
348C2×C4#67⋊C411yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4#88511yescharacteristic
351C2×C4C2×D3×C811yescharacteristic
352C2×C4#90111yescharacteristic
353C2×C4D3×M1611yescharacteristic
354C2×C4#90711yescharacteristic
355C2×C4#91612yes—
356C2×C4#91912yes—
357C2×C4C4×#6711yescharacteristic
358C2×C4#93911yescharacteristic
359C2×C4#94011yescharacteristic
360C8—24no—
361C8—64no—
362C8—81no—
363C8—241no—
364C2×C3#147612yes—
365C2×C3#149412yes—
366C2×C3#195411yescharacteristic
367C2×C3#195811yescharacteristic
368C2×C3#196211yescharacteristic
369D3—82no—
370C22—122no—
371C22—124no—
372C22—241no—
373C22—242no—
374C22—241no—
375C22D3×C22⋊C811yescharacteristic
376C22#472711yescharacteristic
377C22#480412yes—
378C22D3×#10412yes—
379C22#495711yescharacteristic
380C4—21no—
381C4—21no—
382C4—42no—
383C4—64no—
384C4—64no—
385C4—81no—
386C4—241no—
387C4#67⋊C811yescharacteristic
388C4#472911yescharacteristic
389C4#480012yes—
390C4C8×#6711yescharacteristic
391C4#512811yescharacteristic
392C4#516811yescharacteristic
393C4#519111yescharacteristic
394C3#963911yescharacteristic, complemented, Sylow
395C2—242no—
396C2#6822912yes—
397C2#6824112yes—
398C2#6991611yescharacteristic
399C2#7022611yescharacteristic
400C2#7035611yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenterC2×Q8⋊C8D3
hypocenterC3#9639
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8⋊C8C2semidirect
3. C2×#4535C2semidirect
4. #70443C2semidirect
6. C4×C8×Dic3C2semidirect
32. C2×Q8⋊C8D3semidirect
49. #947C8semidirect
394. C3#9639semidirect

Maximal subgroups

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

subgroupindex
C4×#9472
C2×C3×Q8⋊C82
C2×#45352
#704432
C2×#44832
C4×C8×Dic32
C2×#44592
#96393