Finite Group Databasebeta

← #10877353

Subgroups — #10877353

Order 768.

640 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83734C211yesmaximal, characteristic
2#79252C211yesmaximal, characteristic, complemented
3C3×#1810C211yesmaximal, characteristic, complemented
4#71977C211yesmaximal, characteristic, complemented
5#70718C211yesmaximal, characteristic, complemented
6#70707C211yesmaximal, characteristic, complemented
7#70706C211yesmaximal, characteristic, complemented
8#69992C212yesmaximal
9#69987C212yesmaximal, complemented
10#68945C212yesmaximal, complemented
11#68936C212yesmaximal, complemented
12#27691—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14#5791C2211yescharacteristic
15#5745C2212yes—
16C2×#901C2211yescharacteristic
17C2×C3×#108C2211yescharacteristic
18C2×C3×#108C2211yescharacteristic
19C3×#407C2212yes—
20#5207C2211yescharacteristic
21#5207C2212yes—
22#5132C2212yes—
23#5125C2212yes—
24#5124C2212yes—
25#5124C2212yes—
26#5111—22no—
27#5110C2211yescharacteristic
28C8×#67—22no—
29C2×#819C2211yescharacteristic
30C2×#819C2211yescharacteristic
31#5102C2211yescharacteristic
32#5101C2211yescharacteristic
33#4893—22no—
34D3×#108—22no—
35C3×#341C2212yes—
36C3⋊#338C2212yes—
37#4559C2212yes—
38#4556C2212yes—
39#4552C2212yes—
40#2929—31no—
41#2916—31no—
42#2103—32no—
43#2091—32no—
44#1953—31no—
45#1953—31no—
46#1891—32no—
47#1858—32no—
48#1813—31no—
49#1810—31no—
50#1810D311yescharacteristic, complemented
51C23×C3×C4C2311yescharacteristic
52C2×#235C2311yescharacteristic
53C2×C3×M16C2312yes—
54C2×C3×M16C2312yes—
55C22×C3×C8—21no—
56C22×C3×C8—21no—
57C22×C3×C8—22no—
58C22×C3×C8—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61#953Q812yes—
62#930—21no—
63#929Q812yes—
64C4×#67—22no—
65#926C2311yescharacteristic
66#924—21no—
67C2×#208C2311yescharacteristic
68C2×#208C2312yes—
69C2×#208C2312yes—
70C2×#208C2311yescharacteristic
71#907—42no—
72D3×M16—24no—
73#901—22no—
74#901—22no—
75#901—42no—
76C2×ΓL(1,𝔽25)D412yes—
77C2×D3×C8D412yes—
78C3×#108—21no—
79C3×#108—21no—
80C3×#108—22no—
81C3×C22⋊C8—42no—
82#830—22no—
83#830—42no—
84#824—22no—
85#823—22no—
86#822—24no—
87#822—42no—
88(C2×D3)⋊C8—22no—
89#819—21no—
90#819—21no—
91#819—22no—
92#819—42no—
93#815—21no—
94Dic3⋊C8—22no—
95C8×Dic3—21no—
96#568—32no—
97C2×#131—31no—
98C23×C8C2×D311yescharacteristic
99#525—31no—
100#435—62no—
101C8×D4—62no—
102#430—32no—
103#430—32no—
104#430—32no—
105#430—62no—
106C2×#108—31no—
107C2×#108—31no—
108C2×#108—62no—
109C2×#108C2×D311yescharacteristic
110C2×#108C2×D311yescharacteristic
111#424—31no—
112#413—32no—
113#408—31no—
114#407—31no—
115#407—32no—
116#407C2×D312yes—
117#405—31no—
118#344—32no—
119#344—32no—
120#341—32no—
121#341C2×D312yes—
122#338—32no—
123C24×C3C2×Q811yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4C2×Q811yescharacteristic
131C22×C3×C4C2411yescharacteristic
132C2×#67—22no—
133C2×#67C2×Q812yes—
134#235—21no—
135#235—21no—
136#235—21no—
137#235—21no—
138C2×D12#4111yescharacteristic
139C2×C4×D3C2×D412yes—
140C2×Dic6#4111yescharacteristic
141C3×M16—22no—
142C3×M16—22no—
143C3×M16—22no—
144C3×M16—22no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—42no—
151C2×C3×C8—42no—
152C2×C3×C8C2×D411yescharacteristic
153C2×C3×C8C2×D411yescharacteristic
154C2×C3×C8#4112yes—
155C2×C3×C8#4112yes—
156#217—22no—
157#217C2×Q812yes—
158(C2×D3)⋊C4—22no—
159C4⋊Dic3—21no—
160Dic3⋊C4—22no—
161C4×Dic3—21no—
162#208—21no—
163#208—21no—
164#208—22no—
165#208—22no—
166#208—22no—
167#208—21no—
168#208—22no—
169#208—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8C2×D412yes—
174ΓL(1,𝔽25)—24no—
175ΓL(1,𝔽25)—44no—
176D3×C8—24no—
177D3×C8—44no—
178C2×#41—31no—
179C23×C4C22×D311yescharacteristic
180#131—62no—
181#131—62no—
182#131—122no—
183#131—122no—
184C2×M16—31no—
185C2×M16—32no—
186C2×M16—32no—
187C2×M16—32no—
188C2×M16—31no—
189C2×M16—64no—
190C2×M16C22×D312yes—
191C2×M16C22×D312yes—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—32no—
197C22×C8C22×D311yescharacteristic
198C22×C8C22×D311yescharacteristic
199#123—32no—
200C22⋊Q8—61no—
201#121—61no—
202#120—32no—
203C4×D4—62no—
204#117—31no—
205#108—21no—
206#108—21no—
207#108—22no—
208#108—64no—
209#108—62no—
210#108—61no—
211#108—61no—
212#108—122no—
213#108—122no—
214C4⋊C8—62no—
215#101—62no—
216#100—62no—
217C22⋊C8—42no—
218C22⋊C8—62no—
219C22⋊C8—62no—
220C22⋊C8—122no—
221#97—61no—
222C4×C8—61no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—42no—
227C23×C3C22×Q811yescharacteristic
228C22×D3C22⋊Q812yes—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4C22×D411yescharacteristic
237C2×C3×C4C2×#4111yescharacteristic
238C2×C3×C4C2×#4111yescharacteristic
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4H(5,𝔽2)11yescharacteristic
241C2×C3×C4H(5,𝔽2)11yescharacteristic
242#67—22no—
243#67—22no—
244#67—44no—
245C2×Dic3—22no—
246C2×Dic3C22⋊Q812yes—
247D12—21no—
248D12—21no—
249C4×D3—22no—
250C4×D3—22no—
251Dic6—21no—
252Dic6—21no—
253C3×C8—41no—
254C3×C8—42no—
255C3×C8—42no—
256C3×C8#12312yes—
257C3×C8#12312yes—
258C3⋊C8—22no—
259C3⋊C8—42no—
260C3⋊C8—41no—
261C3⋊C8—42no—
262C24D3×Q811yescharacteristic
263#41—61no—
264#41—61no—
265#41—61no—
266#41—61no—
267C2×Q8—31no—
268C2×D4—32no—
269C2×D4—31no—
270C2×D4—62no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—32no—
278C22×C4D3×Q811yescharacteristic
279C22×C4C23×D311yescharacteristic
280M16—22no—
281M16—22no—
282M16—22no—
283M16—22no—
284M16—61no—
285M16—61no—
286M16—64no—
287M16—62no—
288M16—62no—
289M16—61no—
290M16—62no—
291M16—62no—
292M16—61no—
293M16—124no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—32no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—64no—
303C2×C8—61no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—124no—
307C2×C8#23512yes—
308C2×C8D3×D411yescharacteristic
309C2×C8D3×D411yescharacteristic
310C2×C8#23712yes—
311C4⋊C4—62no—
312C4⋊C4—61no—
313C22⋊C4—32no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C42—61no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3C2×C22⋊Q811yescharacteristic
326C22×C3#55411yescharacteristic
327C22×C3#55411yescharacteristic
328C2×D3—22no—
329C2×D3—42no—
330C3×C4—41no—
331C3×C4C2×#12311yescharacteristic
332C3×C4#53511yescharacteristic
333C3×C4#55311yescharacteristic
334C3×C4#55311yescharacteristic
335Dic3—22no—
336Dic3—42no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—32no—
341C23—42no—
342C23C2×D3×Q811yescharacteristic
343Q8—61no—
344Q8—61no—
345D4—62no—
346D4—62no—
347D4—61no—
348D4—61no—
349D4—124no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—32no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—62no—
361C2×C4C2×#23511yescharacteristic
362C2×C4C2×D3×D411yescharacteristic
363C2×C4C2×#23711yescharacteristic
364C2×C4#100411yescharacteristic
365C2×C4D3×#4111yescharacteristic
366C2×C4#100911yescharacteristic
367C8—41no—
368C8—42no—
369C8—42no—
370C8—62no—
371C8—122no—
372C8—122no—
373C8—121no—
374C8#88312yes—
375C8#88312yes—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—42no—
380C2×C3#248711yescharacteristic
381D3—42no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—62no—
391C22—122no—
392C22#551111yescharacteristic
393C22D3×C22⋊Q811yescharacteristic
394C22#563511yescharacteristic
395C4—41no—
396C4—62no—
397C4—122no—
398C4C2×#88311yescharacteristic
399C4#553011yescharacteristic
400C4#560711yescharacteristic
401C4#561411yescharacteristic
402C3#2769111yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—21no—
406C2—42no—
407C2—122no—
408C2#7391211yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1810C2
socleC2×C3#2487
prime radical#5791C22
hypercenter#1810D3
hypocenterC3#27691
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. #79252C2semidirect
3. C3×#1810C2semidirect
4. #71977C2semidirect
5. #70718C2semidirect
6. #70707C2semidirect
7. #70706C2semidirect
9. #69987C2semidirect
10. #68945C2semidirect
11. #68936C2semidirect
50. #1810D3semidirect
402. C3#27691semidirect

Maximal subgroups

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

subgroupindex
#837342
#792522
C3×#18102
#719772
#707182
#707072
#707062
#699922
#699872
#689452
#689362
#276913