Finite Group Databasebeta

← #82798

Subgroups — #82798

Order 384.

562 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#894C211yesmaximal, characteristic, complemented
3#5231C211yesmaximal, characteristic, complemented
4#5231C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4992C211yesmaximal, characteristic, complemented
8#4988C211yesmaximal, characteristic, complemented
9#4987C211yesmaximal, characteristic, complemented
10#4985C211yesmaximal, characteristic, complemented
11#4980C211yesmaximal, characteristic, complemented
12#4961C211yesmaximal, characteristic, complemented
13#4960C211yesmaximal, characteristic
14#4956C211yesmaximal, characteristic, complemented
15#4945C211yesmaximal, characteristic, complemented
16#2696—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C3×C22⋊Q8—24no—
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×C4⋊Dic3C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43#894—24no—
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#123—31no—
55C2×C22⋊Q8D311yescharacteristic, complemented
56#400—31no—
57#397—31no—
58#394—31no—
59#393—31no—
60#393—31no—
61#392—31no—
62#390—31no—
63#388—31no—
64#387—31no—
65#386—31no—
66#386—31no—
67#386—31no—
68#380—31no—
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C23×D3C2311yescharacteristic
72C2×C3×Q8—22no—
73C2×C3×Q8—41no—
74C2×C3×Q8—41no—
75C22×C3×C4—22no—
76C22×C3×C4—41no—
77C22×C3×C4—41no—
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—22no—
85C2×#67—22no—
86C2×#67—22no—
87C2×#67—41no—
88C2×#67—41no—
89C2×#67—41no—
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—41no—
99C2×C4×D3—22no—
100C2×C4×D3—22no—
101C2×C4×D3—41no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4D414yescomplemented
105C3×C22⋊C4—22no—
106C3×C22⋊C4—22no—
107#217—22no—
108#217—41no—
109(C2×D3)⋊C4—22no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4—41no—
117(C2×D3)⋊C4—41no—
118C4⋊Dic3—41no—
119Dic3⋊C4—22no—
120C4×Dic3—41no—
121C22×Q8C2×D311yescharacteristic
122C22×D4—31no—
123C22×D4—31no—
124C22×D4—31no—
125C23×C4—31no—
126C23×C4C2×D311yescharacteristic
127#123—64no—
128C22⋊Q8—24no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4C2×D311yescharacteristic
132C2×C4⋊C4C2×D311yescharacteristic
133C2×C4⋊C4C2×D311yescharacteristic
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4C2×D311yescharacteristic
146C2×C22⋊C4C2×D311yescharacteristic
147C2×C42—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95—31no—
156#95—31no—
157#95—31no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3#4112yes—
165C23×C3C2411yescharacteristic
166C22×D3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3#4112yes—
180C3×Q8—44no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—44no—
197C2×C3×C4C2×D412yes—
198C2×C3×C4C2×D412yes—
199C2×C3×C4C2×D412yes—
200#67—44no—
201#67—44no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220D12—44no—
221C4×D3—44no—
222C24—31no—
223C24—31no—
224C24C22×D311yescharacteristic
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—62no—
231C2×D4—62no—
232C2×D4—62no—
233C2×D4—121no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4—121no—
237C22×C4—22no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—121no—
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D311yescharacteristic
251C22×C4C22×D311yescharacteristic
252C22×C4C22×D311yescharacteristic
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—62no—
258C4⋊C4—121no—
259C4⋊C4#6714yescomplemented
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—62no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C42—121no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3C22×D411yescharacteristic
282C22×C3C2×#4111yescharacteristic
283C22×C3C2×#4111yescharacteristic
284C22×C3H(5,𝔽2)11yescharacteristic
285C22×C3H(5,𝔽2)11yescharacteristic
286C22×C3H(5,𝔽2)11yescharacteristic
287C22×C32−1+411yescharacteristic
288C2×D3—22no—
289C2×D3—22no—
290C2×D3—22no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—41no—
308Dic3—41no—
309Dic3—41no—
310Dic3—41no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—32no—
321C23—61no—
322C23—61no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23—61no—
328C23—61no—
329C23—61no—
330C23—61no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23#23912yes—
335C23C23×D311yescharacteristic
336Q8—44no—
337D4—124no—
338D4—124no—
339D4—124no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—44no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—124no—
375C2×C4C2×#6712yes—
376C2×C4C2×#6712yes—
377C2×C4C2×#6712yes—
378C2×C3—42no—
379C2×C3#53511yescharacteristic
380C2×C3#53811yescharacteristic
381C2×C3#53911yescharacteristic
382C2×C3#54611yescharacteristic
383C2×C3#54811yescharacteristic
384C2×C3#55111yescharacteristic
385C2×C3#56211yescharacteristic
386D3—42no—
387D3—81no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—62no—
396C22—62no—
397C22—62no—
398C22—121no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22#100411yescharacteristic
410C22#100411yescharacteristic
411C22C2×#23911yescharacteristic
412C22#100711yescharacteristic
413C22D3×#4111yescharacteristic
414C22#100911yescharacteristic
415C22C22×#6711yescharacteristic
416C4—22no—
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—121no—
423C4—121no—
424C4—121no—
425C4—121no—
426C4—121no—
427C4—121no—
428C3#269611yescharacteristic, complemented, Sylow
429C2—42no—
430C2—122no—
431C2—241no—
432C2#563711yescharacteristic
433C2#563811yescharacteristic
434C2#564411yescharacteristic
435C2#564711yescharacteristic
436C2#580511yescharacteristic
437C2#581711yescharacteristic
438C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radical#4960C2
hypercenterC2×C22⋊Q8D3
hypocenterC3#2696
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
2. C2×#894C2semidirect
3. #5231C2semidirect
4. #5231C2semidirect
5. #5213C2semidirect
6. #5213C2semidirect
7. #4992C2semidirect
8. #4988C2semidirect
9. #4987C2semidirect
10. #4985C2semidirect
11. #4980C2semidirect
12. #4961C2semidirect
14. #4956C2semidirect
15. #4945C2semidirect
55. C2×C22⋊Q8D3semidirect
104. C3×C4⋊C4D4semidirect
259. C4⋊C4#67semidirect
428. C3#2696semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×#8942
#52312
#52132
#49922
#49882
#49872
#49852
#49802
#49612
#49602
#49562
#49452
#26963