Finite Group Databasebeta

← #11777635

Subgroups — #11777635

Order 1152.

431 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#3753C211yesmaximal, characteristic
2C3×D4×C3⋊C8C211yesmaximal, characteristic, complemented
3#10590338C211yesmaximal, characteristic, complemented
4C3×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5#10589098C211yesmaximal, characteristic, complemented
6#10589082C211yesmaximal, characteristic, complemented
7#10589078C211yesmaximal, characteristic, complemented
8#70844—31nomaximal
9#68157—31nomaximal
10C32×C4×D4C2211yescharacteristic
11C2×#3753C411yescharacteristic
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13C22×C3×C3⋊C8—21no—
14C2×#3712—21no—
15#63659C411yescharacteristic
16#63636C411yescharacteristic
17#63634C411yescharacteristic
18C4×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×#830—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4×C3⋊C8C2211yescharacteristic
23C3×C4×C3⋊C8C2211yescharacteristic
24#63557—21no—
25C3×C8×D4—31no—
26C4×#213—31no—
27C4×#213—31no—
28C4×#213—61no—
29D4×C3⋊C8D311yescharacteristic, complemented
30#4838—31no—
31#4688—31no—
32C3×D4⋊C8—31no—
33D4⋊(C3⋊C8)D311yescharacteristic, complemented
34#4483—31no—
35#4480—31no—
36#4480—31no—
37#4459—31no—
38#4455—31no—
39#4454—31no—
40C2×C32×D4C2×C411yescharacteristic
41C22×C32×C4—21no—
42C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
43C2×C4×C3⋊D3—21no—
44C32×C4⋊C4C2×C411yescharacteristic
45C32×C22⋊C4—21no—
46C32×C42C2311yescharacteristic
47#3753—22no—
48#3752—21no—
49C4⋊(C3⋊Dic3)C2×C411yescharacteristic
50C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
51C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
52C2×C3×C3⋊C8—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—42no—
56C2×C3×C3⋊C8D411yescharacteristic
57C2×C3×C3⋊C8D411yescharacteristic
58#3712—42no—
59#1586—91noSylow
60C22×C3×C8—61no—
61C3×C4×D4—21no—
62C3×C4×D4C2×D311yescharacteristic
63C3×C4×D4C2×D311yescharacteristic
64C2×#213—31no—
65C2×#213—31no—
66C2×#213—61no—
67C22×C3⋊C8—21no—
68C2×ΓL(1,𝔽25)—61no—
69C2×ΓL(1,𝔽25)—61no—
70C4×D12—31no—
71C4×D12—31no—
72C4×D12—61no—
73C3×C4⋊C8—31no—
74C3×C4⋊C8—31no—
75C3×C22⋊C8—61no—
76C3×C4×C8—31no—
77C3×C4×C8—31no—
78#832—31no—
79#832—31no—
80#832—61no—
81#830—21no—
82(C2×D3)⋊C8—61no—
83(C2×D3)⋊C8—61no—
84#809—31no—
85#809—31no—
86#809—61no—
87#807—31no—
88#807—31no—
89#807—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8—31no—
93C4×C3⋊C8—31no—
94C4×C3⋊C8—61no—
95C4×C3⋊C8C2×D311yescharacteristic
96C4×C3⋊C8C2×D311yescharacteristic
97C23×C32—21no—
98C22×C3⋊D3—21no—
99C32×D4—21no—
100C32×D4M1612yescomplemented
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4C22×C411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C32×C4#4111yescharacteristic
107C2×C3⋊Dic3—21no—
108C4⋊(C3⋊D3)—21no—
109C4⋊(C3⋊D3)—21no—
110C4×C3⋊D3—41no—
111C4×C3⋊D3—41no—
112C3⋊(C3⋊C8)—21no—
113C3⋊(C3⋊C8)—21no—
114C3×C3⋊C8—22no—
115C3×C3⋊C8—21no—
116C3×C3⋊C8—21no—
117C3×C3⋊C8—41no—
118C3×C3⋊C8—41no—
119C4×D8—91no—
120#436—91no—
121C8×D4—31no—
122#334—91no—
123D4⋊C8—31no—
124D4⋊C8—91no—
125#322—91no—
126C2×C3×D4—21no—
127C2×C3×D4C4×D311yescharacteristic
128C2×C3×D4C4×D311yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—41no—
132C2×D12—31no—
133C2×D12—31no—
134C2×D12—61no—
135C2×C4×D3—61no—
136C2×C4×D3—61no—
137C2×C4×D3—121no—
138C2×C3×C8—31no—
139C2×C3×C8—31no—
140C2×C3×C8—61no—
141C2×C3×C8—61no—
142C2×C3×C8—61no—
143C2×C3×C8—122no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4C4×D311yescharacteristic
146C3×C4⋊C4C4×D311yescharacteristic
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—41no—
150C3×C42—21no—
151C3×C42C22×D311yescharacteristic
152C3×C42C22×D311yescharacteristic
153#213—64no—
154#213—62no—
155#213—62no—
156(C2×D3)⋊C4—61no—
157(C2×D3)⋊C4—61no—
158(C2×D3)⋊C4—121no—
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161C4⋊Dic3—61no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8—42no—
170C2×C3⋊C8—61no—
171C2×C3⋊C8—61no—
172C2×C3⋊C8#6711yescharacteristic
173C2×C3⋊C8#6711yescharacteristic
174ΓL(1,𝔽25)—122no—
175ΓL(1,𝔽25)—122no—
176C22×C32—22no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32C4×D411yescharacteristic
180C2×C3⋊D3—41no—
181C2×C3⋊D3—41no—
182C2×C3⋊D3—41no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4C2×M1611yescharacteristic
186C32×C4#13111yescharacteristic
187C32×C4AGL(1,ℤ/8)11yescharacteristic
188C32×C4AGL(1,ℤ/8)11yescharacteristic
189C3⋊Dic3—41no—
190C2×D8—91no—
191C2×M16—181no—
192C22×C8—61no—
193C4×D4—91no—
194C4×D4D3211yescharacteristic
195#107—91no—
196C4⋊C8—31no—
197C4⋊C8—31no—
198#102—91no—
199#102—91no—
200C22⋊C8—61no—
201C22⋊C8—181no—
202C4×C8—31no—
203C4×C8—31no—
204C4×C8—91no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—41no—
208C22×D3—61no—
209C22×D3—61no—
210C22×D3—121no—
211C3×D4—21no—
212C3×D4—21no—
213C3×D4—22no—
214C3×D4—41no—
215C3×D4ΓL(1,𝔽25)12yescomplemented
216C3×D4ΓL(1,𝔽25)12yescomplemented
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4C2×C4×D311yescharacteristic
230C2×C3×C4C2×C4×D311yescharacteristic
231C2×C3×C4#23511yescharacteristic
232C2×C3×C4D3×D411yescharacteristic
233C2×C3×C4#23711yescharacteristic
234C2×C3×C4C2×#6711yescharacteristic
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—121no—
238D12—62no—
239D12—61no—
240D12—61no—
241D12—61no—
242D12—61no—
243D12—121no—
244C4×D3—121no—
245C4×D3—121no—
246C4×D3—121no—
247C4×D3—121no—
248C4×D3—241no—
249C4×D3—241no—
250C3×C8—62no—
251C3×C8—61no—
252C3×C8—61no—
253C3×C8—121no—
254C3×C8—121no—
255C3⋊C8—22no—
256C3⋊C8—21no—
257C3⋊C8—21no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C3⋊C8—62no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C3⋊C8—121no—
266C2×C32—42no—
267C2×C32#43511yescharacteristic
268C2×C32#44211yescharacteristic
269C2×C32#44311yescharacteristic
270C3⋊D3—81no—
271C2×D4—91no—
272C2×D4#62411yescharacteristic
273C22×C4—21no—
274C22×C4—181no—
275D8—182no—
276M16—362no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—91no—
283C2×C8—91no—
284C2×C8—122no—
285C4⋊C4—91no—
286C4⋊C4#62411yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—181no—
289C42C2×D3211yescharacteristic
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3#67⋊C411yescharacteristic
301C22×C3C4×#6711yescharacteristic
302C2×D3—122no—
303C2×D3—121no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—121no—
307C2×D3—121no—
308C2×D3—121no—
309C2×D3—241no—
310C2×D3—241no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—41no—
320C3×C4—81no—
321C3×C4C2×ΓL(1,𝔽25)11yescharacteristic
322C3×C4C2×ΓL(1,𝔽25)11yescharacteristic
323C3×C4#90111yescharacteristic
324C3×C4#90711yescharacteristic
325C3×C4#91111yescharacteristic
326C3×C4#91411yescharacteristic
327C3×C4#93211yescharacteristic
328C3×C4#94911yescharacteristic
329Dic3—121no—
330Dic3—121no—
331Dic3—241no—
332C32#158611yescharacteristic, complemented, Sylow
333C23—21no—
334C23—181no—
335D4—21no—
336D4—181no—
337D4—181no—
338D4#371212yescomplemented
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—181no—
343C2×C4—361no—
344C2×C4—361no—
345C2×C4#380411yescharacteristic
346C2×C4C2×#62411yescharacteristic
347C2×C4D3×#6711yescharacteristic
348C8—62no—
349C8—61no—
350C8—61no—
351C8—121no—
352C8—121no—
353C8—181no—
354C8—181no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—42no—
359C2×C3—42no—
360C2×C3—82no—
361C2×C3#473011yescharacteristic
362C2×C3#478511yescharacteristic
363C2×C3#482411yescharacteristic
364C2×C3#501411yescharacteristic
365C2×C3#511111yescharacteristic
366C2×C3#514411yescharacteristic
367D3—242no—
368D3—241no—
369D3—241no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—361no—
374C22—361no—
375C22—361no—
376C22#6395211yescharacteristic
377C4—21no—
378C4—41no—
379C4—361no—
380C4#6381711yescharacteristic
381C4C2×#371211yescharacteristic
382C4#6392511yescharacteristic
383C4#6393411yescharacteristic
384C3—21no—
385C3#6815711yescharacteristic, complemented
386C3#7084411yescharacteristic, complemented
387C2—42no—
388C2—721no—
389C2#1059115711yescharacteristic
390C2#1059140411yescharacteristic
391C2#1059166211yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4×D4C22
socleC22×C32C4×D4
prime radicalC2×#3753C4
cosocleC32×C42C23
hypercenterC4×D4D32
hypocenterC32#1586
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×D4×C3⋊C8C2semidirect
3. #10590338C2semidirect
4. C3×D4⋊(C3⋊C8)C2semidirect
5. #10589098C2semidirect
6. #10589082C2semidirect
7. #10589078C2semidirect
29. D4×C3⋊C8D3semidirect
33. D4⋊(C3⋊C8)D3semidirect
100. C32×D4M16semidirect
215. C3×D4ΓL(1,𝔽25)semidirect
216. C3×D4ΓL(1,𝔽25)semidirect
332. C32#1586semidirect
338. D4#3712semidirect
385. C3#68157semidirect
386. C3#70844semidirect

Maximal subgroups

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

subgroupindex
C4×#37532
C3×D4×C3⋊C82
#105903382
C3×D4⋊(C3⋊C8)2
#105890982
#105890822
#105890782
#708443
#681573