Finite Group Databasebeta

← #10691730

Subgroups — #10691730

Order 768.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83302C211yesmaximal, characteristic
2#83277C211yesmaximal, characteristic, complemented
3C3×#1547C211yesmaximal, characteristic, complemented
4#70521C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic, complemented
6#67714C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#9273—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#948—21no—
11C2×#874C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5234—21no—
15C2×#823—21no—
16C2×#823—21no—
17#5088—21no—
18#5073—21no—
19#5036C414yes—
20#5015—22no—
21#5012—21no—
22#5011—22no—
23D4×C3⋊C8—21no—
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C4×#208C2211yescharacteristic
27C3×#332—21no—
28C3×#331—21no—
29#4485—21no—
30#4485—21no—
31#3205—31no—
32#2935—31no—
33#1965—31no—
34#1547D311yescharacteristic, complemented
35C2×#331—31no—
36#1301—31no—
37#1283—31no—
38C2×C3×#41—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12C2×C412yes—
42C2×C3×M16—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121C2×C412yes—
49C3×C4×D4—21no—
50C3×C4×D4D412yes—
51C2×C3×C42C2311yescharacteristic
52#948—42no—
53#948—42no—
54C2×#213—24no—
55C2×#208—22no—
56C2×#208D411yescharacteristic
57C2×#208D411yescharacteristic
58C22×C3⋊C8—22no—
59C22×C3⋊C8—21no—
60#874—21no—
61#874—21no—
62#874—21no—
63#874—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68#830—22no—
69#830—22no—
70#830C2×C412yes—
71#823—42no—
72#823—42no—
73#809—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D411yescharacteristic
77C4⋊(C3⋊C8)D411yescharacteristic
78#803—21no—
79C4×C3⋊C8—21no—
80C4×C3⋊C8—21no—
81C2×#131—61no—
82#532C2×D311yescharacteristic
83C2×#127—31no—
84#493—61no—
85#486—61no—
86#459—62no—
87#447—34no—
88#436—61no—
89#435—62no—
90C8×D4—61no—
91#431—31no—
92#423C2×D311yescharacteristic
93C2×C4⋊C8—31no—
94C2×C4⋊C8C2×D311yescharacteristic
95C2×#100—61no—
96C2×#100—61no—
97#408—61no—
98C4×M16—31no—
99#332—21no—
100#331—21no—
101#331—61no—
102#331—61no—
103C23×D3—22no—
104C3×#41—41no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C2×C3×Q8—21no—
109C2×C3×D4—22no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4C22⋊C412yes—
113C22×C3×C4—22no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×D12—22no—
118C2×D12—22no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—82no—
128C2×D12—81no—
129C2×D12—81no—
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—81no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4C22⋊C412yes—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#213—84no—
144#208—21no—
145#208—21no—
146#208—21no—
147#208—44no—
148#208D812yescomplemented
149C2×C3⋊C8—21no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—44no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×#41—21no—
158C22×D4—32no—
159C22×D4—61no—
160C22×D4—61no—
161C2×D8—64no—
162#131—122no—
163#131—122no—
164C2×M16—21no—
165C2×M16—31no—
166C2×M16—31no—
167C2×M16—62no—
168C22×C8—21no—
169C22×C8—62no—
170C22×C8—61no—
171C4⋊Q8#6711yescharacteristic
172#127—61no—
173#127—61no—
174#127—61no—
175#127—61no—
176#127#6711yescharacteristic
177#124—21no—
178#121—21no—
179#121C4×D312yes—
180C4×D4—21no—
181C4×D4D1212yes—
182C2×C42C22×D311yescharacteristic
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—31no—
188C4⋊C8—31no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#102—64no—
192#100—122no—
193#100—122no—
194C22⋊C8—32no—
195C22⋊C8—62no—
196C22⋊C8—62no—
197#97—61no—
198C4×C8—61no—
199C4×C8—61no—
200C23×C3—22no—
201C23×C3C4×D411yescharacteristic
202C22×D3—42no—
203C22×D3—42no—
204C22×D3—42no—
205C22×D3—82no—
206C22×D3—82no—
207C3×Q8—41no—
208C3×Q8—41no—
209C3×D4—42no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—41no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—41no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232D12—44no—
233D12—82no—
234D12—82no—
235D12—81no—
236D12—81no—
237D12—81no—
238D12—81no—
239D12—81no—
240D12—81no—
241C3×C8—81no—
242C3×C8—81no—
243C3⋊C8—22no—
244C3⋊C8—42no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247C3⋊C8—41no—
248C24—62no—
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—21no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—241no—
268C2×D4—241no—
269C2×D4—242no—
270C2×D4(C2×D3)⋊C412yes—
271C22×C4—22no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4#23711yescharacteristic
275D8—244no—
276M16—32no—
277M16—41no—
278M16—41no—
279M16—61no—
280M16—61no—
281M16—61no—
282M16—124no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—62no—
290C2×C8—81no—
291C2×C8—124no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4(C2×D3)⋊C412yes—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C42C2×D1211yescharacteristic
299C42D3×D411yescharacteristic
300C42#23711yescharacteristic
301C42C2×#6711yescharacteristic
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3#38611yescharacteristic
308C22×C3C4×D811yescharacteristic
309C22×C3#44211yescharacteristic
310C2×D3—82no—
311C2×D3—82no—
312C2×D3—82no—
313C2×D3—162no—
314C2×D3—162no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4#44711yescharacteristic
320C3×C4#44911yescharacteristic
321C3×C4#45911yescharacteristic
322C3×C4#46311yescharacteristic
323C3×C4#47111yescharacteristic
324C3×C4#47211yescharacteristic
325C23—22no—
326C23—122no—
327C23—122no—
328C23—122no—
329C23—242no—
330C23—242no—
331C23#67⋊C411yescharacteristic
332Q8—41no—
333Q8—41no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—124no—
342D4—242no—
343D4—242no—
344D4—241no—
345D4—241no—
346D4—241no—
347D4—241no—
348D4—241no—
349D4—241no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4#67⋊C411yescharacteristic
357C2×C4#88211yescharacteristic
358C2×C4#88611yescharacteristic
359C2×C4D3×D811yescharacteristic
360C2×C4#91111yescharacteristic
361C2×C4#91411yescharacteristic
362C2×C4#91611yescharacteristic
363C2×C4C2×(C2×D3)⋊C411yescharacteristic
364C2×C4#93911yescharacteristic
365C2×C4#93911yescharacteristic
366C8—62no—
367C8—81no—
368C8—81no—
369C8—122no—
370C8—121no—
371C8—121no—
372C8—121no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—82no—
376C2×C3#187911yescharacteristic
377C2×C3#190711yescharacteristic
378C2×C3#191411yescharacteristic
379D3—162no—
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—242no—
386C22—242no—
387C22—242no—
388C22—482no—
389C22—482no—
390C22#475611yescharacteristic
391C22#478511yescharacteristic
392C22#495811yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4#477311yescharacteristic
398C4#478111yescharacteristic
399C4#515011yescharacteristic
400C4#516511yescharacteristic
401C4#519911yescharacteristic
402C4#520311yescharacteristic
403C3#927311yescharacteristic, complemented, Sylow
404C2—21no—
405C2—21no—
406C2—82no—
407C2—482no—
408C2#7013711yescharacteristic
409C2#7017211yescharacteristic
410C2#7030311yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1547C2
socleC22×C3#386
prime radical#83302C2
cosocleC2×C3×C42C23
hypercenter#1547D3
hypocenterC3#9273
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #83277C2semidirect
3. C3×#1547C2semidirect
4. #70521C2semidirect
5. C2×#4485C2semidirect
6. #67714C2semidirect
7. #67706C2semidirect
34. #1547D3semidirect
148. #208D8semidirect
403. C3#9273semidirect

Maximal subgroups

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

subgroupindex
#833022
#832772
C3×#15472
#705212
C2×#44852
#677142
#677062
#92733