Finite Group Databasebeta

← #10826083

Subgroups — #10826083

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79257C211yesmaximal, characteristic
2#79045C211yesmaximal, characteristic, complemented
3C3×#2102C211yesmaximal, characteristic, complemented
4#70726C211yesmaximal, characteristic, complemented
5#70531C211yesmaximal, characteristic
6#69043C212yesmaximal, complemented
7#69014C212yesmaximal, complemented
8#68779C212yesmaximal, complemented
9#68740C212yesmaximal, complemented
10#68698C211yesmaximal, characteristic
11#68697C211yesmaximal, characteristic, complemented
12#23834—31nomaximal, Sylow
13#5748C2211yescharacteristic
14C2×#907C2212yes—
15#5470C2211yescharacteristic
16C3×#430C2211yescharacteristic
17C3×#423C2211yescharacteristic
18#5139C2212yes—
19#5132C2212yes—
20#5122C2211yescharacteristic
21#5122C2211yescharacteristic
22#5119C2211yescharacteristic
23#5119C2211yescharacteristic
24#5006C2212yes—
25#5002C2212yes—
26#4997C2212yes—
27C2×#805C2212yes—
28C2×#803C2211yescharacteristic
29#4895—22no—
30#4894—22no—
31#4828—22no—
32#4828—22no—
33C3×#344C2212yes—
34C3×#339C2212yes—
35C3×H(3,ℤ/4)C2211yescharacteristic
36C3⋊#338C2211yescharacteristic
37#4558C2212yes—
38#4556C2212yes—
39#4531C2212yes—
40#2961—31no—
41#2955—31no—
42#2103—31no—
43#2102D311yescharacteristic, complemented
44#2080—32no—
45#2053—31no—
46#2050—31no—
47#2031—32no—
48#1765—32no—
49#1757—32no—
50#1747—31no—
51C2×#235C2312yes—
52C2×C3×M16C2311yescharacteristic
53C2×C3×M16C2312yes—
54C2×C3×M16C2311yescharacteristic
55C3×#117C2312yes—
56C2×C3×C42C2311yescharacteristic
57#926C2311yescharacteristic
58#926C2311yescharacteristic
59#924—22no—
60C2×#208C2312yes—
61C22×C3⋊C8C2311yescharacteristic
62C22×C3⋊C8C2311yescharacteristic
63#909—22no—
64#908—22no—
65#907—22no—
66#907—22no—
67#907—22no—
68#907—22no—
69#904—22no—
70C2×ΓL(1,𝔽25)—22no—
71C2×D3×C8—22no—
72#870—21no—
73#869—21no—
74C4×Dic6—22no—
75C3×#108—22no—
76C3×C4⋊C8—22no—
77#825—24no—
78#824—22no—
79#823—22no—
80#815—21no—
81#815D412yes—
82Dic3⋊C8—22no—
83Dic3⋊C8—22no—
84C8×Dic3—21no—
85C8×Dic3D412yes—
86#810—22no—
87#809—22no—
88#806—24no—
89#805—22no—
90#805—22no—
91#803—21no—
92#803—21no—
93#803—22no—
94C4×C3⋊C8—22no—
95#575—31no—
96C2×#131—32no—
97#533—31no—
98#446—62no—
99#446—62no—
100#444—62no—
101#443—62no—
102#430C2×D311yescharacteristic
103#424—31no—
104#424—31no—
105#423C2×D311yescharacteristic
106#421—32no—
107C2×#104—32no—
108#416—32no—
109#413—32no—
110#410—32no—
111#405—31no—
112#405—31no—
113#405—32no—
114C2×#97—31no—
115#344—32no—
116#344C2×D312yes—
117#341—32no—
118#339C2×D312yes—
119#338—31no—
120H(3,ℤ/4)—32no—
121H(3,ℤ/4)C2×D311yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4C2411yescharacteristic
124C2×#67—22no—
125#235—22no—
126#235—22no—
127#235—22no—
128#235—22no—
129C2×D12#4112yes—
130C2×C4×D3—22no—
131C2×Dic6—22no—
132C2×Dic6#4112yes—
133C3×M16—22no—
134C3×M16—22no—
135C3×M16—22no—
136C3×M16—22no—
137C2×C3×C8—21no—
138C2×C3×C8—22no—
139C2×C3×C8C2×D412yes—
140C3×C4⋊C4—22no—
141C3×C22⋊C4—22no—
142C3×C42—21no—
143C3×C42—22no—
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146#217—21no—
147#217#4112yes—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—24no—
152C4×Dic3—21no—
153C4×Dic3C2×D412yes—
154#208—22no—
155#208—22no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8C2×D411yescharacteristic
160C2×C3⋊C8C2×D411yescharacteristic
161C2×C3⋊C8#4112yes—
162C2×C3⋊C8#4112yes—
163Dic12—22no—
164D24—22no—
165#204—24no—
166ΓL(1,𝔽25)—42no—
167ΓL(1,𝔽25)—42no—
168D3×C8—42no—
169D3×C8—42no—
170C2×#41—32no—
171#137—62no—
172AGL(1,ℤ/8)—62no—
173#135—62no—
174#131—62no—
175#131—62no—
176#131—62no—
177#131—62no—
178C2×M16—32no—
179C2×M16—62no—
180C2×M16C22×D312yes—
181C2×M16C22×D311yescharacteristic
182C2×M16C22×D311yescharacteristic
183C22×C8—31no—
184C22×C8—31no—
185C22×C8—62no—
186C4⋊Q8—61no—
187#125—61no—
188C22⋊Q8—62no—
189C4×Q8—62no—
190#117—31no—
191#117—31no—
192#117C22×D312yes—
193C2×C42C22×D311yescharacteristic
194#108—22no—
195C4⋊C8—22no—
196C4⋊C8—62no—
197C4⋊C8—62no—
198#104—64no—
199#104—62no—
200#104—62no—
201#103—62no—
202#102—62no—
203#101—62no—
204#100—62no—
205#99—64no—
206#97—32no—
207#97—61no—
208#97—61no—
209#97—62no—
210#97—61no—
211C4×C8—32no—
212C4×C8—62no—
213C4×C8—61no—
214C23×C3C2×#4111yescharacteristic
215C22×D3—22no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4C22×D411yescharacteristic
221C2×C3×C4C2×#4111yescharacteristic
222C2×C3×C4C2×#4112yes—
223C2×C3×C4C2×#4112yes—
224#67—42no—
225#67—42no—
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—22no—
229D12—22no—
230D12—22no—
231C4×D3—42no—
232C4×D3—42no—
233Dic6—22no—
234Dic6—22no—
235Dic6—82no—
236C3×C8—22no—
237C3×C8—41no—
238C3×C8—42no—
239C3⋊C8—41no—
240C3⋊C8—42no—
241C3⋊C8#12411yescharacteristic
242C3⋊C8#12412yes—
243C3⋊C8#12411yescharacteristic
244#41—62no—
245#41—62no—
246#41—62no—
247#41—62no—
248C2×Q8—32no—
249C2×Q8—62no—
250C2×D4—32no—
251C2×D4—62no—
252C22×C4—21no—
253C22×C4—62no—
254C22×C4C23×D311yescharacteristic
255Q16—62no—
256SD16—64no—
257D8—62no—
258M16—22no—
259M16—22no—
260M16—22no—
261M16—22no—
262M16—62no—
263M16—62no—
264M16—122no—
265M16—122no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—31no—
269C2×C8—31no—
270C2×C8—32no—
271C2×C8—32no—
272C2×C8—61no—
273C2×C8—62no—
274C2×C8—62no—
275C2×C8—122no—
276C2×C8—122no—
277C2×C8D3×D412yes—
278C4⋊C4—22no—
279C4⋊C4—64no—
280C4⋊C4—62no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C22⋊C4—22no—
284C22⋊C4—32no—
285C22⋊C4—61no—
286C42—21no—
287C42—22no—
288C42—32no—
289C42—61no—
290C42#23911yescharacteristic
291C42#23911yescharacteristic
292C22×C3—41no—
293C22×C3#52511yescharacteristic
294C22×C3#52912yes—
295C2×D3—42no—
296C2×D3—82no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—82no—
300C3×C4#52511yescharacteristic
301C3×C4C2×#12411yescharacteristic
302C3×C4#52912yes—
303Dic3—42no—
304Dic3—42no—
305Dic3—42no—
306C23—62no—
307C23D3×#4111yescharacteristic
308Q8—62no—
309Q8—62no—
310Q8—242no—
311D4—62no—
312D4—62no—
313D4—122no—
314D4—122no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—62no—
319C2×C4—62no—
320C2×C4—62no—
321C2×C4—81no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4C2×D3×D411yescharacteristic
325C2×C4C2×#23911yescharacteristic
326C2×C4D3×#4112yes—
327C2×C4D3×#4112yes—
328C8—22no—
329C8—31no—
330C8—32no—
331C8—31no—
332C8—41no—
333C8—42no—
334C8—121no—
335C8—122no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3#239811yescharacteristic
339D3—82no—
340C22—41no—
341C22—122no—
342C22—242no—
343C22#562711yescharacteristic
344C22#570612yes—
345C4—41no—
346C4—41no—
347C4—82no—
348C4—122no—
349C4—122no—
350C4—122no—
351C4#565611yescharacteristic
352C4D3×#12411yescharacteristic
353C4#569112yes—
354C3#2383411yescharacteristic, complemented, Sylow
355C2—21no—
356C2—22no—
357C2—242no—
358C2#7813111yescharacteristic
subgroupG/H
centerC4D3×#124
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2102C2
socleC2×C3#2398
prime radical#5748C22
hypercenter#2102D3
hypocenterC3#23834
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2102

Complements

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

NH ≅ G/Ntype
2. #79045C2semidirect
3. C3×#2102C2semidirect
4. #70726C2semidirect
6. #69043C2semidirect
7. #69014C2semidirect
8. #68779C2semidirect
9. #68740C2semidirect
11. #68697C2semidirect
43. #2102D3semidirect
354. C3#23834semidirect

Maximal subgroups

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

subgroupindex
#792572
#790452
C3×#21022
#707262
#705312
#690432
#690142
#687792
#687402
#686982
#686972
#238343