Finite Group Databasebeta

← #10693991

Subgroups — #10693991

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 400 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
6#69750C211yesmaximal, characteristic, complemented
7C2×#4459C211yesmaximal, characteristic
8#9645—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#947C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5229C411yescharacteristic
15#4990—22no—
16#4988C411yescharacteristic
17#4977C411yescharacteristic
18C4×(C2×D3)⋊C4—22no—
19C42×Dic3C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22C3×Q8⋊C8—22no—
23#4550—22no—
24#4550—22no—
25#4535—22no—
26#4483—22no—
27#4459—22no—
28C4×#124—31no—
29#1831—31no—
30#1744—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#947—22no—
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—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8D412yes—
74C2×#124—31no—
75C2×C4×Q8C2×D311yescharacteristic
76C2×C4×D4—31no—
77C2×C4⋊C8—31no—
78C2×C4⋊C8C2×D311yescharacteristic
79C2×C4×C8—31no—
80C2×C4×C8C2×D311yescharacteristic
81#391—31no—
82#390—31no—
83#388—62no—
84#382—31no—
85C4×C22⋊C4—62no—
86C43—31no—
87#336—62no—
88#336—62no—
89Q8⋊C8—22no—
90Q8⋊C8—62no—
91D4⋊C8—62no—
92D4⋊C8—62no—
93C23×D3—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8M1612yes—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3C22⋊C412yes—
107C22×Dic3C22⋊C412yes—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12M1612yes—
114C2×C4×D3—41no—
115C2×C4×D3—41no—
116C2×C4×D3—42no—
117C2×C4×D3—42no—
118C2×C3×C8—22no—
119C2×C3×C8—42no—
120C2×C3×C8—41no—
121C2×C3×C8—42no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42—41no—
127C3×C42—41no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C3×C42#4111yescharacteristic
132(C2×D3)⋊C4—28no—
133(C2×D3)⋊C4—44no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3—21no—
141C4×Dic3—21no—
142C4×Dic3—22no—
143C4×Dic3—22no—
144C4×Dic3—42no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—42no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—42no—
149C22×Q8C4×D311yescharacteristic
150C22×D4—31no—
151C23×C4—61no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—61no—
155C22×C8—61no—
156#124—62no—
157#124—62no—
158C4×Q8—21no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×D4—61no—
163C4×D4—61no—
164C4×D4—61no—
165C4×D4—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4C4×D311yescharacteristic
169C2×C22⋊C4—62no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—61no—
172C2×C42—21no—
173C2×C42—31no—
174C2×C42—31no—
175C2×C42—32no—
176C2×C42—32no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4×C8—21no—
183C4×C8—32no—
184C4×C8—61no—
185C4×C8#6712yes—
186#95—62no—
187#95—62no—
188C23×C3C2×C22⋊C411yescharacteristic
189C22×D3—22no—
190C22×D3—41no—
191C22×D3—41no—
192C22×D3—42no—
193C22×D3—42no—
194C3×Q8—22no—
195C3×Q8—22no—
196C3×Q8—41no—
197C3×Q8—41no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12411yescharacteristic
209C2×C3×C4#12711yescharacteristic
210C2×C3×C4C2×M1611yescharacteristic
211C2×C3×C4C2×M1611yescharacteristic
212C2×C3×C4#13111yescharacteristic
213C2×C3×C4#13111yescharacteristic
214C2×C3×C4#13512yes—
215C2×C3×C4#13512yes—
216C2×Dic3—22no—
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224D12—22no—
225D12—22no—
226D12—41no—
227D12—41no—
228C4×D3—82no—
229C4×D3—81no—
230C4×D3—81no—
231C3×C8—42no—
232C3×C8—81no—
233C3⋊C8—42no—
234C3⋊C8—81no—
235C24—61no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8ΓL(1,𝔽25)12yes—
242C2×D4—32no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—32no—
251C22×C4—32no—
252C22×C4—61no—
253C22×C4—121no—
254C22×C4—122no—
255C22×C4—122no—
256C22×C4—121no—
257C22×C4C2×C4×D311yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260C2×C8—22no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—62no—
265C2×C8—122no—
266C2×C8—122no—
267C2×C8—121no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C22⋊C4—68no—
275C22⋊C4—124no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C42—41no—
279C42—41no—
280C42—61no—
281C42—61no—
282C42—61no—
283C42—61no—
284C42—62no—
285C42—62no—
286C42—122no—
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42#23711yescharacteristic
290C42C2×#6711yescharacteristic
291C22×C3#39011yescharacteristic
292C22×C3#40711yescharacteristic
293C22×C3#40811yescharacteristic
294C22×C3#41612yes—
295C22×C3#42112yes—
296C2×D3—42no—
297C2×D3—44no—
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—82no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—42no—
304C3×C4—81no—
305C3×C4#43511yescharacteristic
306C3×C4#43511yescharacteristic
307C3×C4#43511yescharacteristic
308C3×C4#43511yescharacteristic
309C3×C4#48812yes—
310C3×C4#49511yescharacteristic
311C3×C4#49511yescharacteristic
312Dic3—42no—
313Dic3—42no—
314Dic3—81no—
315C23—62no—
316C23—121no—
317C23—121no—
318C23—122no—
319C23—122no—
320C23D3×C22⋊C411yescharacteristic
321Q8—22no—
322Q8—22no—
323Q8—41no—
324Q8—41no—
325D4—62no—
326D4—62no—
327D4—121no—
328D4—121no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—121no—
342C2×C4—122no—
343C2×C4—241no—
344C2×C4—241no—
345C2×C4—242no—
346C2×C4#67⋊C411yescharacteristic
347C2×C4#88411yescharacteristic
348C2×C4#88511yescharacteristic
349C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
350C2×C4#90111yescharacteristic
351C2×C4D3×M1611yescharacteristic
352C2×C4#90711yescharacteristic
353C2×C4#91612yes—
354C2×C4#91912yes—
355C2×C4C4×#6711yescharacteristic
356C2×C4#93911yescharacteristic
357C2×C4#94011yescharacteristic
358C8—42no—
359C8—81no—
360C8—122no—
361C8—241no—
362C2×C3#148212yes—
363C2×C3#148812yes—
364C2×C3#195511yescharacteristic
365C2×C3#195811yescharacteristic
366C2×C3#196311yescharacteristic
367D3—82no—
368C22—122no—
369C22—124no—
370C22—241no—
371C22—242no—
372C22—241no—
373C22#472611yescharacteristic
374C22#472711yescharacteristic
375C22#480412yes—
376C22#482112yes—
377C22#495711yescharacteristic
378C4—21no—
379C4—21no—
380C4—42no—
381C4—81no—
382C4—122no—
383C4—122no—
384C4—241no—
385C4#472811yescharacteristic
386C4#473011yescharacteristic
387C4#480012yes—
388C4#511111yescharacteristic
389C4#512711yescharacteristic
390C4#516811yescharacteristic
391C4#519111yescharacteristic
392C3#964511yescharacteristic, complemented, Sylow
393C2—242no—
394C2#6823312yes—
395C2#6823712yes—
396C2#6991711yescharacteristic
397C2#7022611yescharacteristic
398C2#7035711yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenterC2×Q8⋊C8D3
hypocenterC3#9645
LCSG ▷ C22×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. #69750C2semidirect
32. C2×Q8⋊C8D3semidirect
392. C3#9645semidirect

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
#697502
C2×#44592
#96453