Finite Group Databasebeta

← #10896449

Subgroups — #10896449

Order 768.

612 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79249C211yesmaximal, characteristic
2#78445C211yesmaximal, characteristic, complemented
3C3×#1945C211yesmaximal, characteristic, complemented
4#70706C211yesmaximal, characteristic, complemented
5#70658C211yesmaximal, characteristic
6#70657C211yesmaximal, characteristic, complemented
7#70549C212yesmaximal, complemented
8#70546C212yesmaximal, complemented
9#69795C211yesmaximal, characteristic
10#68779C212yesmaximal, complemented
11#68773C212yesmaximal, complemented
12#29270—31nomaximal, Sylow
13#5745C2212yes—
14C2×#904C2211yescharacteristic
15#5470C2211yescharacteristic
16C2×C3×#108C2211yescharacteristic
17C3×#427C2211yescharacteristic
18C3×#424C2212yes—
19C2×C3×C4×C8C2211yescharacteristic
20#5139C2212yes—
21#5124C2212yes—
22#5110C2212yes—
23C2×#819C2211yescharacteristic
24#5102C2211yescharacteristic
25#5102C2211yescharacteristic
26#5101C2211yescharacteristic
27#5101C2211yescharacteristic
28#5006C2212yes—
29#5005C2212yes—
30C2×#805C2212yes—
31#4924C2211yescharacteristic
32#4895—22no—
33#4894—22no—
34#4679—22no—
35C8×Dic6—22no—
36C3×#339C2212yes—
37#4556C2212yes—
38#4533C2212yes—
39#4531C2212yes—
40#2960—31no—
41#2957—31no—
42#2098—32no—
43#2031—32no—
44#1976—31no—
45#1974—31no—
46#1953—31no—
47#1945D311yescharacteristic, complemented
48#1868—32no—
49#1861—32no—
50#1833—31no—
51C2×#235C2312yes—
52C2×C3×M16C2312yes—
53C22×C3×C8C2311yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C3×#117C2312yes—
56C2×C3×C42C2311yescharacteristic
57#926C2311yescharacteristic
58#926C2311yescharacteristic
59#924—22no—
60C2×#208C2311yescharacteristic
61C2×#208C2312yes—
62C2×#208C2311yescharacteristic
63#907—24no—
64D3×M16—42no—
65C2×Dic12Q811yescharacteristic, complemented
66#904—21no—
67#904—21no—
68#904—21no—
69#904—21no—
70C2×D24Q811yescharacteristic, complemented
71C2×#204Q812yescomplemented
72#901—42no—
73#870—21no—
74#869—21no—
75C4×Dic6—22no—
76C3×#108—21no—
77C3×#108—21no—
78C3×#107—21no—
79C3×#106—21no—
80C3×C4⋊C8—42no—
81C3×C4×C8—21no—
82C3×C4×C8—21no—
83#825—24no—
84#822—42no—
85(C2×D3)⋊C8—42no—
86#819—21no—
87#819—21no—
88#815—21no—
89#815D412yes—
90Dic3⋊C8—22no—
91Dic3⋊C8—22no—
92C8×Dic3—21no—
93C8×Dic3D412yes—
94#806—24no—
95#805—22no—
96#805—22no—
97#805—42no—
98C4⋊(C3⋊C8)—22no—
99C2×#135—31no—
100#568—32no—
101#533—31no—
102#446—62no—
103C8×Q8—62no—
104#444—62no—
105#443—62no—
106#430—32no—
107C2×#108—31no—
108C2×#108C2×D311yescharacteristic
109#427C2×D311yescharacteristic
110#424—31no—
111#424—31no—
112#424C2×D312yes—
113#423—31no—
114#421—32no—
115#421—32no—
116C2×#104—32no—
117#410—32no—
118#408—32no—
119#405—31no—
120#405—31no—
121C2×C4×C8C2×D311yescharacteristic
122#344—32no—
123#344—32no—
124#339C2×D312yes—
125H(3,ℤ/4)—32no—
126C22×C3×C4—21no—
127C22×C3×C4C2411yescharacteristic
128C2×#67—22no—
129#235—22no—
130#235—22no—
131#235—22no—
132#235—22no—
133C2×D12C2×Q812yes—
134C2×C4×D3—22no—
135C2×Dic6—22no—
136C2×Dic6C2×Q812yes—
137C3×M16—22no—
138C3×M16—22no—
139C2×C3×C8—21no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8—22no—
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8C2×Q811yescharacteristic
146C2×C3×C8C2×Q811yescharacteristic
147C2×C3×C8#4111yescharacteristic
148C2×C3×C8#4111yescharacteristic
149C3×C4⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C42—21no—
152C3×C42—22no—
153C3×C42#4111yescharacteristic
154C3×C42#4111yescharacteristic
155#217—21no—
156#217#4112yes—
157C4⋊Dic3—22no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—24no—
161C4×Dic3—21no—
162C4×Dic3C2×D412yes—
163#208—21no—
164#208—21no—
165#208—22no—
166#208—22no—
167#208—21no—
168#208—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8C2×D412yes—
172Dic12—21no—
173Dic12—21no—
174D24—21no—
175D24—21no—
176#204—22no—
177#204—22no—
178ΓL(1,𝔽25)—44no—
179D3×C8—44no—
180C2×#41—32no—
181#135—61no—
182#135—61no—
183#135—61no—
184#135—61no—
185C2×Q16—31no—
186C2×SD16—32no—
187C2×D8—31no—
188#131—64no—
189#131—122no—
190C2×M16—31no—
191C2×M16—32no—
192C2×M16—31no—
193C2×M16—122no—
194C2×M16C22×D312yes—
195C22×C8C22×D311yescharacteristic
196C22×C8C22×D311yescharacteristic
197C4⋊Q8—61no—
198#125—61no—
199C22⋊Q8—62no—
200C4×Q8—62no—
201#117—31no—
202#117—31no—
203#117C22×D312yes—
204C2×C42C22×D311yescharacteristic
205#108—21no—
206#108—21no—
207#108—61no—
208#108—61no—
209#108—122no—
210#107—21no—
211#106—21no—
212C4⋊C8—42no—
213C4⋊C8—62no—
214C4⋊C8—62no—
215C4⋊C8—62no—
216#104—64no—
217#104—62no—
218#104—62no—
219#104—122no—
220#99—64no—
221C22⋊C8—122no—
222#97—32no—
223#97—61no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—32no—
227C4×C8—61no—
228C23×C3C2×#4111yescharacteristic
229C22×D3—22no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4C22×D411yescharacteristic
235C2×C3×C4C22×Q811yescharacteristic
236C2×C3×C4C2×#4111yescharacteristic
237C2×C3×C4C2×#4111yescharacteristic
238C2×C3×C4H(5,𝔽2)11yescharacteristic
239C2×C3×C4H(5,𝔽2)11yescharacteristic
240#67—42no—
241#67—42no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245D12—22no—
246D12—22no—
247C4×D3—42no—
248C4×D3—42no—
249Dic6—22no—
250Dic6—22no—
251Dic6—44no—
252C3×C8—42no—
253C3×C8—41no—
254C3×C8C22⋊Q811yescharacteristic
255C3×C8C22⋊Q811yescharacteristic
256C3×C8C22⋊Q811yescharacteristic
257C3×C8C22⋊Q811yescharacteristic
258C3⋊C8—22no—
259C3⋊C8—41no—
260C3⋊C8—42no—
261#41—62no—
262#41—62no—
263#41—62no—
264#41—62no—
265C2×Q8—32no—
266C2×Q8—62no—
267C2×D4—32no—
268C2×D4—62no—
269C22×C4—21no—
270C22×C4—62no—
271C22×C4C23×D311yescharacteristic
272Q16—61no—
273Q16—61no—
274SD16—62no—
275SD16—62no—
276D8—61no—
277D8—61no—
278M16—22no—
279M16—22no—
280M16—61no—
281M16—61no—
282M16—62no—
283M16—61no—
284M16—62no—
285M16—61no—
286M16—124no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—32no—
292C2×C8—61no—
293C2×C8—62no—
294C2×C8—124no—
295C2×C8#23511yescharacteristic
296C2×C8#23511yescharacteristic
297C2×C8D3×D411yescharacteristic
298C2×C8D3×D411yescharacteristic
299C2×C8D3×Q811yescharacteristic
300C2×C8D3×Q811yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—64no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C22⋊C4—22no—
307C22⋊C4—32no—
308C22⋊C4—61no—
309C42—21no—
310C42—22no—
311C42—32no—
312C42—61no—
313C42#23911yescharacteristic
314C42#23911yescharacteristic
315C22×C3—41no—
316C22×C3#52511yescharacteristic
317C22×C3#55411yescharacteristic
318C22×C3#55911yescharacteristic
319C2×D3—42no—
320C2×D3—82no—
321C3×C4—22no—
322C3×C4—41no—
323C3×C4—82no—
324C3×C4C2×C22⋊Q811yescharacteristic
325C3×C4#53511yescharacteristic
326C3×C4#55411yescharacteristic
327C3×C4#55911yescharacteristic
328Dic3—42no—
329Dic3—42no—
330Dic3—42no—
331C23—62no—
332C23D3×#4111yescharacteristic
333Q8—62no—
334Q8—62no—
335Q8—124no—
336D4—62no—
337D4—62no—
338D4—122no—
339D4—122no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4C2×#23511yescharacteristic
350C2×C4C2×D3×D411yescharacteristic
351C2×C4#100411yescharacteristic
352C2×C4C2×D3×Q811yescharacteristic
353C2×C4C2×#23911yescharacteristic
354C2×C4#100911yescharacteristic
355C8—41no—
356C8—42no—
357C8—62no—
358C8—122no—
359C8—121no—
360C8#89611yescharacteristic
361C8#89611yescharacteristic
362C8#89611yescharacteristic
363C8#89611yescharacteristic
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3#258011yescharacteristic
368D3—82no—
369C22—41no—
370C22—122no—
371C22—242no—
372C22#555911yescharacteristic
373C22#562711yescharacteristic
374C22#569011yescharacteristic
375C4—22no—
376C4—41no—
377C4—82no—
378C4—122no—
379C4—122no—
380C4—122no—
381C4C2×#89611yescharacteristic
382C4#558511yescharacteristic
383C4#560711yescharacteristic
384C4#573011yescharacteristic
385C3#2927011yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—21no—
389C2—242no—
390C2#7607511yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1945C2
socleC2×C3#2580
prime radicalC2×#904C22
hypercenter#1945D3
hypocenterC3#29270
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1945

Complements

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

NH ≅ G/Ntype
2. #78445C2semidirect
3. C3×#1945C2semidirect
4. #70706C2semidirect
6. #70657C2semidirect
7. #70549C2semidirect
8. #70546C2semidirect
10. #68779C2semidirect
11. #68773C2semidirect
47. #1945D3semidirect
65. C2×Dic12Q8semidirect
70. C2×D24Q8semidirect
71. C2×#204Q8semidirect
385. C3#29270semidirect

Maximal subgroups

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

subgroupindex
#792492
#784452
C3×#19452
#707062
#706582
#706572
#705492
#705462
#697952
#687792
#687732
#292703