Finite Group Databasebeta

← #11785520

Subgroups — #11785520

Order 1152.

547 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63814C211yesmaximal, characteristic
2C3×#5132C211yesmaximal, characteristic
3C2×C3×#822C211yesmaximal, characteristic, complemented
4C2×#63577C211yesmaximal, characteristic, complemented
5#10591076C211yesmaximal, characteristic
6#10589426C211yesmaximal, characteristic, complemented
7#10589159C211yesmaximal, characteristic, complemented
8#70718—31nomaximal
9#70134—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×#3748C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×C3×#208C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3711C412yes—
18#63814—22no—
19#63814—22no—
20C2×D3×C3⋊C8C412yes—
21C3×#824—21no—
22C3×#823—21no—
23C3×#822—22no—
24#63577—22no—
25#63560—21no—
26#63558—21no—
27C2×#948—31no—
28C2×#901—31no—
29C2×C3×#108—31no—
30C3×#413—31no—
31#5237—31no—
32#5132—31no—
33#5132D311yescharacteristic
34C2×#822—31no—
35C2×#822D311yescharacteristic, complemented
36C2×#819—31no—
37C3⋊#338—31no—
38C3⋊#338—31no—
39C3⋊#338—31no—
40C3⋊#338—31no—
41C3⋊#338—61no—
42C22×C32×C4C2311yescharacteristic
43C2×C3×#67Q812yes—
44C3×#235—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235—21no—
48C2×C3×D12D411yescharacteristic
49C2×C3×C4×D3C2×C412yes—
50C2×C3×Dic6D411yescharacteristic
51C32×M16—21no—
52C32×M16—21no—
53C2×C32×C8—21no—
54#3748—21no—
55#3748—21no—
56#3748—21no—
57#3748—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C2×C3⋊(C3⋊C8)C2×C412yes—
60C3×#208—21no—
61C3×#208—21no—
62C3×#208—21no—
63C3×#208—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8C2×C411yescharacteristic
67C2×C3×C3⋊C8C2×C411yescharacteristic
68C2×C3×C3⋊C8D412yes—
69C2×C3×C3⋊C8D412yes—
70#3711—24no—
71D3×C3⋊C8—24no—
72#1813—91noSylow
73C2×C3×#41—31no—
74C2×#235C2×D311yescharacteristic
75C2×C3×M16—21no—
76C2×C3×M16—31no—
77C2×C3×M16—31no—
78C2×C3×M16C2×D311yescharacteristic
79C2×C3×M16C2×D311yescharacteristic
80C22×C3×C8—31no—
81#948—62no—
82#948—62no—
83C2×#208—31no—
84C2×#208—31no—
85C2×#208—32no—
86C2×#208—31no—
87C2×#208—31no—
88C2×#208—61no—
89C2×#208—61no—
90C2×#208C2×D311yescharacteristic
91C2×#208C2×D311yescharacteristic
92C22×C3⋊C8—32no—
93C22×C3⋊C8C2×D311yescharacteristic
94#901—62no—
95#901—62no—
96C2×ΓL(1,𝔽25)—32no—
97C2×D3×C8—32no—
98C3×#108—62no—
99C3×#101—61no—
100C3×#100—61no—
101#836—61no—
102#833—61no—
103#824—21no—
104#824—61no—
105#823—21no—
106#823—61no—
107#822—22no—
108#822—62no—
109#819—62no—
110C23×C32C2×Q811yescharacteristic
111C22×C3×D3C4⋊C412yes—
112C2×C32×C4C22×C411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4#4111yescharacteristic
117C2×C32×C4#4111yescharacteristic
118C3×#67—22no—
119C3×#67—22no—
120C2×C3×Dic3C4⋊C412yes—
121C3×D12—21no—
122C3×D12—21no—
123C3×C4×D3—22no—
124C3×C4×D3—22no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C32×C8—41no—
128C3⋊(C3⋊C8)—22no—
129C3⋊(C3⋊C8)—41no—
130C3×C3⋊C8—41no—
131C3×C3⋊C8—41no—
132C3×C3⋊C8C22⋊C412yes—
133C3×C3⋊C8C22⋊C412yes—
134C2×#131—91no—
135C2×#108—31no—
136C2×#108—91no—
137#413—31no—
138#413—91no—
139#338—91no—
140#338—91no—
141C3×#41—61no—
142C3×#41—61no—
143C3×#41—61no—
144C3×#41—61no—
145C2×C3×Q8—31no—
146C2×C3×D4—32no—
147C2×C3×D4—31no—
148C22×C3×C4—21no—
149C22×C3×C4—32no—
150C22×C3×C4C22×D311yescharacteristic
151C22×C3×C4C22×D311yescharacteristic
152C2×#67Dic612yes—
153#235—21no—
154#235—21no—
155#235—21no—
156#235—21no—
157C2×D12#6711yescharacteristic
158C2×C4×D3C4×D312yes—
159C2×Dic6#6711yescharacteristic
160C3×M16—21no—
161C3×M16—21no—
162C3×M16—21no—
163C3×M16—21no—
164C3×M16—41no—
165C3×M16—41no—
166C3×M16—61no—
167C3×M16—61no—
168C3×M16—61no—
169C3×M16—61no—
170C2×C3×C8—22no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—31no—
174C2×C3×C8—31no—
175C2×C3×C8—32no—
176C2×C3×C8—32no—
177C2×C3×C8—61no—
178C2×C3×C8—61no—
179#208—21no—
180#208—21no—
181#208—21no—
182#208—21no—
183#208—61no—
184#208—61no—
185#208—61no—
186#208—61no—
187#208—64no—
188#208—61no—
189#208—61no—
190#208—61no—
191#208—61no—
192#208—121no—
193#208—121no—
194#208—121no—
195#208—121no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—32no—
199C2×C3⋊C8—32no—
200C2×C3⋊C8—62no—
201C2×C3⋊C8—64no—
202C2×C3⋊C8—62no—
203C2×C3⋊C8—61no—
204C2×C3⋊C8—61no—
205C2×C3⋊C8C4×D311yescharacteristic
206C2×C3⋊C8C4×D311yescharacteristic
207C2×C3⋊C8D1212yes—
208C2×C3⋊C8#6712yes—
209ΓL(1,𝔽25)—64no—
210D3×C8—64no—
211C22×C32—41no—
212C22×C32C2×C4⋊C411yescharacteristic
213C22×C32C22⋊Q811yescharacteristic
214C22×C32C22⋊Q811yescharacteristic
215C2×C3×D3—22no—
216C2×C3×D3—42no—
217C32×C4C2×C22⋊C411yescharacteristic
218C32×C4#11711yescharacteristic
219C32×C4#12111yescharacteristic
220C32×C4#12111yescharacteristic
221C3×Dic3—22no—
222C2×#41—31no—
223#131—182no—
224#131—182no—
225C2×M16—31no—
226C2×M16—31no—
227C2×M16—92no—
228C2×M16—91no—
229C2×M16—91no—
230C2×M16D3211yescharacteristic
231C22×C8—31no—
232C22×C8—92no—
233#108—62no—
234#108—182no—
235#101—61no—
236#101—181no—
237#100—61no—
238#100—181no—
239C23×C3—21no—
240C23×C3—32no—
241C23×C3C2×Dic611yescharacteristic
242C23×C3D3×Q811yescharacteristic
243C22×D3Dic3⋊C412yes—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×D4—62no—
247C3×D4—62no—
248C3×D4—61no—
249C3×D4—61no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—32no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4C2×C4×D311yescharacteristic
260C2×C3×C4C2×C4×D311yescharacteristic
261C2×C3×C4C2×D1211yescharacteristic
262C2×C3×C4D3×D411yescharacteristic
263C2×C3×C4D3×D411yescharacteristic
264C2×C3×C4D3×D411yescharacteristic
265C2×C3×C4#23711yescharacteristic
266C2×C3×C4#23711yescharacteristic
267C2×C3×C4#23711yescharacteristic
268C2×C3×C4#23911yescharacteristic
269C2×C3×C4C2×#6711yescharacteristic
270C2×C3×C4C2×#6711yescharacteristic
271#67—22no—
272#67—22no—
273C2×Dic3Dic3⋊C412yes—
274D12—21no—
275D12—21no—
276C4×D3—22no—
277C4×D3—22no—
278Dic6—21no—
279Dic6—21no—
280C3×C8—32no—
281C3×C8—32no—
282C3×C8—42no—
283C3×C8—41no—
284C3×C8—41no—
285C3×C8—121no—
286C3×C8—121no—
287C3⋊C8—41no—
288C3⋊C8—41no—
289C3⋊C8—62no—
290C3⋊C8—62no—
291C3⋊C8—122no—
292C3⋊C8—122no—
293C3⋊C8—121no—
294C3⋊C8—121no—
295C3⋊C8(C2×D3)⋊C412yes—
296C3⋊C8(C2×D3)⋊C412yes—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32#38011yescharacteristic
301C3×D3—42no—
302#41—61no—
303#41—61no—
304#41—61no—
305#41—61no—
306C2×Q8—31no—
307C2×D4—32no—
308C2×D4—31no—
309C22×C4—32no—
310C22×C4C2×D3211yescharacteristic
311M16—21no—
312M16—21no—
313M16—61no—
314M16—61no—
315M16—61no—
316M16—61no—
317M16—184no—
318M16—181no—
319M16—181no—
320M16—181no—
321M16—181no—
322C2×C8—21no—
323C2×C8—31no—
324C2×C8—31no—
325C2×C8—32no—
326C2×C8—32no—
327C2×C8—61no—
328C2×C8—61no—
329C2×C8—92no—
330C2×C8—184no—
331C2×C8—181no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—62no—
338C22×C3—81no—
339C22×C3—122no—
340C22×C3#87811yescharacteristic
341C22×C3#87811yescharacteristic
342C22×C3D3×C4⋊C411yescharacteristic
343C22×C3#89711yescharacteristic
344C22×C3C2×Dic3⋊C411yescharacteristic
345C22×C3#94611yescharacteristic
346C2×D3—22no—
347C2×D3—42no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—62no—
353C3×C4#87711yescharacteristic
354C3×C4D3×C22⋊C411yescharacteristic
355C3×C4#89211yescharacteristic
356C3×C4#89511yescharacteristic
357C3×C4C2×(C2×D3)⋊C411yescharacteristic
358C3×C4#93811yescharacteristic
359C3×C4#93911yescharacteristic
360C3×C4#93911yescharacteristic
361Dic3—22no—
362C32#181311yescharacteristic, complemented, Sylow
363C23—32no—
364C23D3×Dic611yescharacteristic
365Q8—61no—
366Q8—61no—
367D4—62no—
368D4—62no—
369D4—61no—
370D4—61no—
371C2×C4—32no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4#379611yescharacteristic
375C2×C4C4×D3211yescharacteristic
376C2×C4D3×D1211yescharacteristic
377C2×C4#380511yescharacteristic
378C2×C4D3×#6711yescharacteristic
379C2×C4D3×#6711yescharacteristic
380C8—32no—
381C8—32no—
382C8—41no—
383C8—121no—
384C8—121no—
385C8—182no—
386C8—361no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—122no—
398C2×C3#494211yescharacteristic
399C2×C3#498711yescharacteristic
400D3—42no—
401C22—41no—
402C22—62no—
403C22—122no—
404C22D3×Dic3⋊C411yescharacteristic
405C22#6388711yescharacteristic
406C22#6389311yescharacteristic
407C4—62no—
408C4#6387811yescharacteristic
409C4#6388611yescharacteristic
410C4D3×(C2×D3)⋊C411yescharacteristic
411C4#6397011yescharacteristic
412C3—21no—
413C3#7013411yescharacteristic, complemented
414C3#7071811yescharacteristic, complemented
415C2—21no—
416C2—21no—
417C2—21no—
418C2—122no—
419C2#1059136611yescharacteristic
subgroupG/H
centerC4#63878
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#380
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1813
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C2×C3×#822C2semidirect
4. C2×#63577C2semidirect
6. #10589426C2semidirect
7. #10589159C2semidirect
35. C2×#822D3semidirect
362. C32#1813semidirect
413. C3#70134semidirect
414. C3#70718semidirect

Maximal subgroups

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

subgroupindex
C2×#638142
C3×#51322
C2×C3×#8222
C2×#635772
#105910762
#105894262
#105891592
#707183
#701343