Finite Group Databasebeta

← #10876274

Subgroups — #10876274

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76482C211yesmaximal, characteristic
2#74224C211yesmaximal, characteristic, complemented
3C3×#2074C211yesmaximal, characteristic, complemented
4#70267C212yesmaximal
5#70223C212yesmaximal, complemented
6#69685C211yesmaximal, characteristic, complemented
7#69557C211yesmaximal, characteristic, complemented
8#68875C212yesmaximal, complemented
9#68792C212yesmaximal, complemented
10#68682C211yesmaximal, characteristic, complemented
11#68665C211yesmaximal, characteristic, complemented
12#27597—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C2×D3×C4⋊C4C2211yescharacteristic
15C2×#872C2211yescharacteristic
16C2×C3×#103C2212yes—
17C3×#391C2211yescharacteristic
18C3×#382C2211yescharacteristic
19#5229C2211yescharacteristic
20C2×#835C2212yes—
21C2×#821C2212yes—
22C2×(C2×D3)⋊C8C2212yes—
23C2×#817C2212yes—
24#4988C2211yescharacteristic
25#4987C2212yes—
26Dic3×C4⋊C4C2212yes—
27C2×#809C2212yes—
28C2×#808C2212yes—
29#4939C2211yescharacteristic
30#4935C2211yescharacteristic
31#4670—22no—
32#4667—22no—
33#4665—22no—
34#4655—22no—
35C3×#340C2212yes—
36C3×#336C2211yescharacteristic
37#4550C2212yes—
38#4529C2212yes—
39#4524C2211yescharacteristic
40#2314—31no—
41#2311—31no—
42#2096—31no—
43#2094—31no—
44#2074D311yescharacteristic, complemented
45#2073—31no—
46#1927—32no—
47#1926—32no—
48#1853—32no—
49#1818—31no—
50#1788—32no—
51C22×C3×Q8C2311yescharacteristic
52C2×#239—22no—
53C2×#239—22no—
54C2×#239—24no—
55C2×#239—42no—
56C22×D12—21no—
57C22×D12C2311yescharacteristic
58C22×C4×D3—21no—
59C22×C4×D3—21no—
60C22×C4×D3C2311yescharacteristic
61C22×C3×C8C2312yes—
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C42C2311yescharacteristic
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4D412yes—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2311yescharacteristic
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4D412yes—
74C2×C4×Dic3—22no—
75C2×C4×Dic3C2311yescharacteristic
76C22×C3⋊C8C2312yes—
77D3×C4⋊C4—42no—
78#872—24no—
79C3×#103—24no—
80C3×#95—22no—
81C3×#95—22no—
82#835—42no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86#821—42no—
87(C2×D3)⋊C8—24no—
88#817—42no—
89#809—24no—
90#808—42no—
91C22×#41—31no—
92C2×#117—31no—
93C22×C4⋊C4—31no—
94C2×#106—32no—
95C2×#106—32no—
96C2×#103—32no—
97C2×#103C2×D312yes—
98C2×#102—32no—
99C2×#102—32no—
100C2×C22⋊C8—32no—
101#391—31no—
102#391C2×D311yescharacteristic
103#390—31no—
104#390—31no—
105#388—62no—
106#387—62no—
107#386—62no—
108#384—31no—
109#382C2×D311yescharacteristic
110#380—32no—
111#380—62no—
112C4×C4⋊C4—32no—
113#340—32no—
114#340C2×D312yes—
115#336—32no—
116#336—31no—
117#336C2×D311yescharacteristic
118C23×D3—21no—
119C23×D3C2×D411yescharacteristic
120C2×C3×Q8—21no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8#4112yes—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2411yescharacteristic
129C22×Dic3—22no—
130C22×Dic3—21no—
131C22×Dic3C2×D412yes—
132C22×Dic3C2×D411yescharacteristic
133#239—48no—
134#239—48no—
135C2×D12—22no—
136C2×D12—22no—
137C2×D12—24no—
138C2×D12—21no—
139C2×D12—42no—
140C2×D12—42no—
141C2×D12#4112yes—
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—22no—
145C2×C4×D3—24no—
146C2×C4×D3—42no—
147C2×C4×D3—42no—
148C2×C4×D3—42no—
149C2×C4×D3—82no—
150C2×C4×D3#4114yes—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C42—22no—
158(C2×D3)⋊C4—24no—
159(C2×D3)⋊C4—82no—
160(C2×D3)⋊C4—82no—
161(C2×D3)⋊C4—82no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—82no—
166Dic3⋊C4—24no—
167Dic3⋊C4—42no—
168C4×Dic3—22no—
169C4×Dic3—82no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×#41—64no—
174C2×#41—62no—
175C2×#41—62no—
176C2×#41—122no—
177C22×Q8C22×D311yescharacteristic
178C22×D4—31no—
179C22×D4—61no—
180C23×C4—31no—
181C23×C4—61no—
182C23×C4—61no—
183C22×C8—32no—
184C22×C8C22×D312yes—
185#117—64no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—122no—
192C2×C4⋊C4C22×D311yescharacteristic
193C2×C4⋊C4C22×D312yes—
194C2×C22⋊C4—32no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C42—31no—
199C2×C42—62no—
200C2×C42C22×D311yescharacteristic
201#106—122no—
202#106—122no—
203#103—24no—
204#103—122no—
205#102—64no—
206#102—122no—
207C22⋊C8—64no—
208#95—22no—
209#95—22no—
210#95—62no—
211#95—62no—
212#95—62no—
213C23×C3C22×D411yescharacteristic
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—22no—
217C22×D3—22no—
218C22×D3—22no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—42no—
222C22×D3—42no—
223C22×D3#12312yes—
224C3×Q8—24no—
225C3×Q8—42no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4C2×#4111yescharacteristic
238C2×C3×C4C2×#4112yes—
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4H(5,𝔽2)12yes—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3#12312yes—
252D12—24no—
253D12—48no—
254D12—42no—
255C4×D3—24no—
256C4×D3—24no—
257C4×D3—48no—
258C4×D3—84no—
259C3×C8—42no—
260C3⋊C8—42no—
261C24—31no—
262C24—61no—
263#41—128no—
264#41—128no—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8#23712yes—
269C2×D4—32no—
270C2×D4—64no—
271C2×D4—61no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—122no—
275C2×D4—122no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—31no—
280C22×C4—34no—
281C22×C4—32no—
282C22×C4—64no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4—122no—
291C22×C4—242no—
292C22×C4D3×D412yes—
293C22×C4C23×D311yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—62no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—64no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—122no—
308C4⋊C4—242no—
309C22⋊C4—64no—
310C22⋊C4—242no—
311C22⋊C4—242no—
312C22⋊C4—242no—
313C42—22no—
314C42—62no—
315C42—242no—
316C22×C3C2×#12311yescharacteristic
317C22×C3#52511yescharacteristic
318C22×C3#53511yescharacteristic
319C22×C3#57712yes—
320C22×C3#57712yes—
321C2×D3—24no—
322C2×D3—22no—
323C2×D3—42no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—44no—
327C2×D3—42no—
328C2×D3—82no—
329C2×D3—81no—
330C2×D3—81no—
331C3×C4—42no—
332C3×C4—42no—
333C3×C4—82no—
334C3×C4—81no—
335C3×C4#55112yes—
336C3×C4#55112yes—
337Dic3—22no—
338Dic3—42no—
339Dic3—82no—
340Dic3—81no—
341C23—32no—
342C23—61no—
343C23—62no—
344C23—62no—
345C23—61no—
346C23—62no—
347C23—121no—
348C23—121no—
349C23—122no—
350C23—122no—
351C23C2×D3×D411yescharacteristic
352Q8—24no—
353Q8—42no—
354D4—64no—
355D4—128no—
356D4—122no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—32no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—64no—
372C2×C4—64no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—128no—
377C2×C4—121no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—244no—
383C2×C4C2×#23711yescharacteristic
384C2×C4D3×#4112yes—
385C2×C4D3×#4111yescharacteristic
386C2×C4#100912yes—
387C8—42no—
388C8—122no—
389C2×C3#248211yescharacteristic
390C2×C3#309912yes—
391C2×C3#312512yes—
392C2×C3#322111yescharacteristic
393C2×C3#322211yescharacteristic
394D3—42no—
395D3—82no—
396C22—62no—
397C22—64no—
398C22—121no—
399C22—122no—
400C22—121no—
401C22—122no—
402C22—124no—
403C22—241no—
404C22—242no—
405C22—241no—
406C22#560611yescharacteristic
407C22D3×#12311yescharacteristic
408C22#567411yescharacteristic
409C22#577012yes—
410C22#577812yes—
411C4—42no—
412C4—42no—
413C4—62no—
414C4—82no—
415C4—81no—
416C4—122no—
417C4—242no—
418C4—241no—
419C4#565812yes—
420C4#570512yes—
421C3#2759711yescharacteristic, complemented, Sylow
422C2—122no—
423C2—242no—
424C2#7767611yescharacteristic
425C2#8057512yes—
426C2#8152112yes—
427C2#8164611yescharacteristic
428C2#8174311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2074C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#2074D3
hypocenterC3#27597
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2074

Complements

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

NH ≅ G/Ntype
2. #74224C2semidirect
3. C3×#2074C2semidirect
5. #70223C2semidirect
6. #69685C2semidirect
7. #69557C2semidirect
8. #68875C2semidirect
9. #68792C2semidirect
10. #68682C2semidirect
11. #68665C2semidirect
44. #2074D3semidirect
421. C3#27597semidirect

Maximal subgroups

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

subgroupindex
#764822
#742242
C3×#20742
#702672
#702232
#696852
#695572
#688752
#687922
#686822
#686652
#275973