Finite Group Databasebeta

← #86747

Subgroups — #86747

Order 384.

1060 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×Q8C211yesmaximal, characteristic, complemented
2Q8×#67C212yesmaximal, complemented
3C2×#944C212yesmaximal, complemented
4#5729C212yesmaximal, complemented
5D3×C4⋊Q8C211yesmaximal, characteristic, complemented
6#5722C212yesmaximal, complemented
7#5721C211yesmaximal, characteristic, complemented
8#5636C214yesmaximal, complemented
9#5632C212yesmaximal, complemented
10#5623C214yesmaximal, complemented
11#5622C212yesmaximal, complemented
12#5578C211yesmaximal, characteristic, complemented
13#5567C211yesmaximal, characteristic, complemented
14#5546C212yesmaximal
15#5539C212yesmaximal, complemented
16D4×Dic6C211yesmaximal, characteristic, complemented
17C4×#237C211yesmaximal, characteristic
18#3503—31nomaximal, Sylow
19C22×C3×Q8C2212yes—
20C2×D3×Q8C2212yes—
21C2×#237C2211yescharacteristic
22C22×Dic6C2212yes—
23C3×C4⋊Q8C2211yescharacteristic
24C3×C4⋊Q8C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×C22⋊Q8C2214yes—
27C3×C4×Q8C2211yescharacteristic
28C3×C4×D4C2211yescharacteristic
29C3×C4×D4C2212yes—
30#946C2214yes—
31#946C2211yescharacteristic
32Q8×Dic3C2212yes—
33#944C2212yes—
34#944C2214yes—
35#944C2214yes—
36#944C2212yes—
37#944C2214yes—
38#944C2211yescharacteristic
39#944C2212yes—
40D4×Dic3C2211yescharacteristic
41D4×Dic3C2212yes—
42C4×#67C2212yes—
43C4×#67C2214yes—
44#924C2212yes—
45#924C2214yes—
46C2×Dic3⋊C4C2214yes—
47C2×Dic3⋊C4C2214yes—
48C2×C4×Dic3C2212yes—
49#897C2211yescharacteristic
50#896C2214yes—
51#896C2212yes—
52#892C2211yescharacteristic
53D3×C4⋊C4C2212yes—
54D3×C4⋊C4C2212yes—
55#890C2212yes—
56#889C2214yes—
57#889C2214yes—
58#889C2214yes—
59#889C2212yes—
60#888C2214yes—
61#888C2212yes—
62#888C2212yes—
63#888C2212yes—
64#888C2211yescharacteristic
65#887C2212yes—
66#887C2214yes—
67#887C2212yes—
68#887C2211yescharacteristic
69#67⋊C4C2214yes—
70#67⋊C4C2212yes—
71#878C2214yes—
72#878C2214yes—
73#878C2214yes—
74#877C2212yes—
75#877C2214yes—
76#872C2212yes—
77C42×D3C2211yescharacteristic
78#870C2212yes—
79#869C2211yescharacteristic
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
84Q82—32no—
85#557—31no—
86#557—31no—
87#554—34no—
88#554—32no—
89D4×Q8—32no—
90D4×Q8—32no—
91D4×Q8—31no—
92D4×Q8D311yescharacteristic, complemented
93#544—32no—
94#544—34no—
95#533—32no—
96#533—31no—
97C2×C4⋊Q8—32no—
98C2×C4⋊Q8—31no—
99#520—32no—
100C4×#41—31no—
101C2×C3×Q8—44no—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2312yes—
104C2×C3×Q8C2314yes—
105C2×C3×D4C2311yescharacteristic
106C22×C3×C4C2312yes—
107C22×C3×C4C2314yes—
108C2×#67C2312yes—
109C22×Dic3C2312yes—
110C22×Dic3C2314yes—
111C22×Dic3C2312yes—
112D3×Q8—44no—
113#237Q818yescomplemented
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2311yescharacteristic
116C2×C4×D3C2314yes—
117C2×C4×D3C2311yescharacteristic
118C2×Dic6—44no—
119C2×Dic6C2312yes—
120C2×Dic6C2311yescharacteristic
121C2×Dic6C2312yes—
122C2×Dic6C2314yes—
123C2×Dic6C2314yes—
124C2×Dic6C2311yescharacteristic
125C3×C4⋊C4C2311yescharacteristic
126C3×C4⋊C4C2311yescharacteristic
127C3×C4⋊C4C2314yes—
128C3×C4⋊C4C2312yes—
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2312yes—
131C3×C22⋊C4C2314yes—
132C3×C22⋊C4C2312yes—
133C3×C42C2311yescharacteristic
134C3×C42C2312yes—
135#217C2314yes—
136#217C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2312yes—
140C4⋊Dic3C2312yes—
141C4⋊Dic3C2311yescharacteristic
142C4⋊Dic3C2312yes—
143C4⋊Dic3C2311yescharacteristic
144C4⋊Dic3C2312yes—
145C4⋊Dic3C2314yes—
146Dic3⋊C4C2312yes—
147Dic3⋊C4C2312yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2314yes—
152Dic3⋊C4C2314yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161C4×Dic3C2311yescharacteristic
162C4×Dic3C2312yes—
163C4×Dic3C2314yes—
164C4×Dic3C2312yes—
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2314yes—
168C4×Dic3C2312yes—
169C4×Dic3C2312yes—
170C2×#41—31no—
171C22×Q8—32no—
172C22×Q8—32no—
173C22×Q8C2×D312yes—
174C4⋊Q8—32no—
175C4⋊Q8—31no—
176C4⋊Q8—34no—
177C4⋊Q8—32no—
178C4⋊Q8—32no—
179C4⋊Q8—34no—
180C4⋊Q8—34no—
181C4⋊Q8—32no—
182C4⋊Q8—32no—
183C4⋊Q8—31no—
184C4⋊Q8—34no—
185C4⋊Q8—32no—
186C4⋊Q8—31no—
187C4⋊Q8C2×D311yescharacteristic
188C4⋊Q8C2×D312yes—
189#125—34no—
190#125—32no—
191#125—34no—
192#125—34no—
193#125—32no—
194#125—32no—
195C22⋊Q8—31no—
196C22⋊Q8—34no—
197C22⋊Q8—32no—
198C22⋊Q8—34no—
199C22⋊Q8—31no—
200C22⋊Q8—32no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8C2×D312yes—
206C22⋊Q8C2×D314yes—
207C4×Q8—32no—
208C4×Q8—31no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—34no—
212C4×Q8—31no—
213C4×Q8—32no—
214C4×Q8—31no—
215C4×Q8—32no—
216C4×Q8C2×D311yescharacteristic
217C4×D4—31no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—34no—
221C4×D4—32no—
222C4×D4—34no—
223C4×D4C2×D311yescharacteristic
224C4×D4C2×D312yes—
225#117—31no—
226#117—32no—
227#117—34no—
228#117—32no—
229C2×C4⋊C4—34no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C42—32no—
234C2×C42—31no—
235C23×C3C2412yes—
236C22×D3C2411yescharacteristic
237C3×Q8—24no—
238C3×Q8—22no—
239C3×Q8—44no—
240C3×D4C2×Q814yescomplemented
241C2×C3×C4—24no—
242C2×C3×C4—44no—
243C2×C3×C4C2412yes—
244C2×C3×C4C2412yes—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2414yes—
247C2×C3×C4C2412yes—
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250#67C2×Q818yescomplemented
251C2×Dic3—24no—
252C2×Dic3—44no—
253C2×Dic3C2×Q818yescomplemented
254C2×Dic3C2412yes—
255C2×Dic3C2412yes—
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2412yes—
258C2×Dic3C2414yes—
259C2×Dic3C2412yes—
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2411yescharacteristic
262C4×D3—22no—
263C4×D3—22no—
264C4×D3—44no—
265C4×D3C2×Q814yescomplemented
266Dic6—22no—
267Dic6—24no—
268Dic6—24no—
269Dic6—44no—
270Dic6—44no—
271Dic6C2×Q814yescomplemented
272#41—38no—
273C2×Q8—31no—
274C2×Q8—32no—
275C2×Q8—32no—
276C2×Q8—31no—
277C2×Q8—34no—
278C2×Q8—34no—
279C2×Q8—44no—
280C2×Q8—124no—
281C2×Q8—124no—
282C2×Q8C22×D312yes—
283C2×Q8C22×D311yescharacteristic
284C2×Q8C22×D314yes—
285C2×D4—32no—
286C2×D4C22×D311yescharacteristic
287C22×C4—32no—
288C22×C4—31no—
289C22×C4—32no—
290C22×C4—34no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—34no—
294C22×C4C22×D312yes—
295C22×C4C22×D314yes—
296C4⋊C4—34no—
297C4⋊C4—32no—
298C4⋊C4—34no—
299C4⋊C4—32no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—32no—
303C4⋊C4—34no—
304C4⋊C4—34no—
305C4⋊C4—32no—
306C4⋊C4—32no—
307C4⋊C4—32no—
308C4⋊C4—31no—
309C4⋊C4—32no—
310C4⋊C4—31no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4—34no—
314C4⋊C4—32no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4C22×D312yes—
318C4⋊C4C22×D312yes—
319C4⋊C4C22×D311yescharacteristic
320C4⋊C4C22×D311yescharacteristic
321C4⋊C4C22×D314yes—
322C4⋊C4C22×D312yes—
323C22⋊C4—34no—
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4—32no—
327C22⋊C4—32no—
328C22⋊C4C22×D314yes—
329C22⋊C4C22×D312yes—
330C42—31no—
331C42—32no—
332C42—31no—
333C42—32no—
334C42—32no—
335C42—31no—
336C42—34no—
337C42—34no—
338C42—32no—
339C42C22×D311yescharacteristic
340C42C22×D312yes—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3C22×Q814yes—
344C22×C3C2511yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3C22×Q812yes—
348C3×C4—22no—
349C3×C4—24no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—21no—
353C3×C4C22×Q812yes—
354C3×C4H(5,𝔽2)12yes—
355Dic3—24no—
356Dic3—22no—
357Dic3—21no—
358Dic3—22no—
359Dic3—22no—
360Dic3C22×Q814yes—
361Dic3C22×Q812yes—
362Dic32−1+412yes—
363C23—31no—
364C23C23×D312yes—
365Q8—22no—
366Q8—24no—
367Q8—34no—
368Q8—44no—
369Q8—64no—
370Q8—62no—
371Q8—64no—
372Q8—124no—
373Q8—124no—
374D4—38no—
375D4D3×Q814yescomplemented
376C2×C4—24no—
377C2×C4—38no—
378C2×C4—32no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—34no—
382C2×C4—32no—
383C2×C4—32no—
384C2×C4—31no—
385C2×C4—34no—
386C2×C4—32no—
387C2×C4—44no—
388C2×C4—62no—
389C2×C4—64no—
390C2×C4—62no—
391C2×C4—124no—
392C2×C4—124no—
393C2×C4C23×D312yes—
394C2×C4C23×D312yes—
395C2×C4C23×D311yescharacteristic
396C2×C4C23×D314yes—
397C2×C4C23×D312yes—
398C2×C4C23×D311yescharacteristic
399C2×C4C23×D311yescharacteristic
400C2×C3—24no—
401C2×C3C23×Q811yescharacteristic
402C2×C3C2×H(5,𝔽2)11yescharacteristic
403C2×C3C2×2−1+411yescharacteristic
404D3—22no—
405C22—22no—
406C22—22no—
407C22—32no—
408C22—61no—
409C22—61no—
410C22C2×D3×Q814yes—
411C22C24×D311yescharacteristic
412C4—22no—
413C4—22no—
414C4—24no—
415C4—21no—
416C4—22no—
417C4—32no—
418C4—34no—
419C4—32no—
420C4—61no—
421C4—64no—
422C4—62no—
423C4—62no—
424C4—62no—
425C4#100412yes—
426C4C2×D3×Q812yes—
427C3#350311yescharacteristic, complemented, Sylow
428C2—24no—
429C2—62no—
430C2C2×#100411yescharacteristic
431C2C22×D3×Q811yescharacteristic
432C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×D4×Q8C2
prime radicalC2×#237C22
hypercenterD4×Q8D3
hypocenterC3#3503
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×D4×Q8C2semidirect
2. Q8×#67C2semidirect
3. C2×#944C2semidirect
4. #5729C2semidirect
5. D3×C4⋊Q8C2semidirect
6. #5722C2semidirect
7. #5721C2semidirect
8. #5636C2semidirect
9. #5632C2semidirect
10. #5623C2semidirect
11. #5622C2semidirect
12. #5578C2semidirect
13. #5567C2semidirect
15. #5539C2semidirect
16. D4×Dic6C2semidirect
92. D4×Q8D3semidirect
113. #237Q8semidirect
240. C3×D4C2×Q8semidirect
250. #67C2×Q8semidirect
253. C2×Dic3C2×Q8semidirect
265. C4×D3C2×Q8semidirect
271. Dic6C2×Q8semidirect
375. D4D3×Q8semidirect
427. C3#3503semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Q82
Q8×#672
C2×#9442
#57292
D3×C4⋊Q82
#57222
#57212
#56362
#56322
#56232
#56222
#55782
#55672
#55462
#55392
D4×Dic62
C4×#2372
#35033