Finite Group Databasebeta

← #11779499

Subgroups — #11779499

Order 1152.

490 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5563C211yesmaximal, characteristic
2#10592707C211yesmaximal, characteristic, complemented
3#10590386C211yesmaximal, characteristic, complemented
4#10590268C211yesmaximal, characteristic, complemented
5C3×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10589095C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71928—31nomaximal
9#68203—31nomaximal
10C32×C4×D4C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×#938—21no—
13C3×#930—21no—
14C3×#926—21no—
15C3×#892—21no—
16C3×#885—22no—
17C3×C4×D12C2211yescharacteristic
18C3×#869C2211yescharacteristic
19C2×#3719—22no—
20#63659—21no—
21#63635—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25#63573—21no—
26#63563—21no—
27C3×#555—31no—
28#5563D311yescharacteristic
29#5092—31no—
30#5008—31no—
31#5008—31no—
32#5008—31no—
33#5008—61no—
34#4842—31no—
35#4699—31no—
36C3×D4⋊C8—31no—
37#4546—31no—
38D4⋊(C3⋊C8)—31no—
39D4⋊(C3⋊C8)D311yescharacteristic, complemented
40#4481—31no—
41#4459—31no—
42C2×C32×D4D411yescharacteristic
43C22×C32×C4—21no—
44C2×#634—21no—
45C2×C3×#67—22no—
46C2×C3×D12D411yescharacteristic
47C2×C3×C4×D3—21no—
48C2×C3×Dic6D412yes—
49C32×C4⋊C4D411yescharacteristic
50C32×C22⋊C4—21no—
51C32×C42C2311yescharacteristic
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)—21no—
54C2×C3⋊(C3⋊C8)—21no—
55C3×#217—22no—
56C3×(C2×D3)⋊C4—22no—
57C3×(C2×D3)⋊C4—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3D411yescharacteristic
61C3×C4×Dic3—22no—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8—21no—
64#3719—44no—
65#1634—91noSylow
66C3×C4⋊Q8—31no—
67C3×#124—62no—
68C3×#121—61no—
69C3×#121—61no—
70C3×C4×D4—21no—
71C3×C4×D4—31no—
72C3×C4×D4C2×D311yescharacteristic
73C3×C4×D4C2×D311yescharacteristic
74C3×#117—61no—
75C3×#117—61no—
76C2×#216—62no—
77#938—21no—
78#930—21no—
79#926—21no—
80C2×Dic12—62no—
81#892—21no—
82#885—22no—
83C4×D12C2×D311yescharacteristic
84#869—31no—
85#869—31no—
86#869—61no—
87#869C2×D311yescharacteristic
88C3×C4⋊C8—31no—
89C3×C4×C8—31no—
90#832—61no—
91#832—62no—
92#832—61no—
93#832—61no—
94#821—61no—
95#817—61no—
96#808—61no—
97#808—62no—
98#808—61no—
99#808—61no—
100C4⋊(C3⋊C8)—31no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—61no—
103C4⋊(C3⋊C8)C2×D311yescharacteristic
104C4×C3⋊C8C2×D311yescharacteristic
105C23×C32—21no—
106C22×C3×D3—21no—
107C32×D4—21no—
108C32×D4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—41no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C3⋊Dic3—41no—
116#634—42no—
117C3×#67—82no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—22no—
120C3×D12—21no—
121C3×D12—21no—
122C3×C4×D3—41no—
123C3×C4×D3—41no—
124C3×Dic6—42no—
125C3⋊(C3⋊C8)—41no—
126C3×C3⋊C8—22no—
127C3×C3⋊C8—41no—
128#555—31no—
129#494—91no—
130#476—91no—
131#476—91no—
132#333—91no—
133D4⋊C8—31no—
134D4⋊C8—91no—
135C2×C3×Q8—32no—
136C2×C3×D4—21no—
137C2×C3×D4—31no—
138C2×C3×D4—62no—
139C2×C3×D4D1211yescharacteristic
140C2×C3×D4#6711yescharacteristic
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—41no—
144C22×C3×C4—61no—
145C2×#67—22no—
146C2×D12#6711yescharacteristic
147C2×C4×D3—21no—
148C2×Dic6—61no—
149C2×Dic6—61no—
150C2×Dic6—121no—
151C2×Dic6#6712yes—
152C2×C3×C8—61no—
153C2×C3×C8—61no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—61no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4D1211yescharacteristic
159C3×C4⋊C4#6711yescharacteristic
160C3×C22⋊C4—21no—
161C3×C22⋊C4—21no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—62no—
164C3×C22⋊C4—62no—
165C3×C22⋊C4—61no—
166C3×C42—21no—
167C3×C42—62no—
168C3×C42C22×D311yescharacteristic
169C3×C42C22×D311yescharacteristic
170#217—22no—
171#216—124no—
172(C2×D3)⋊C4—22no—
173(C2×D3)⋊C4—21no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4⋊Dic3—62no—
177C4⋊Dic3—61no—
178C4⋊Dic3—61no—
179C4⋊Dic3—61no—
180C4⋊Dic3—61no—
181C4⋊Dic3—121no—
182C4⋊Dic3#6711yescharacteristic
183C4×Dic3—22no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—62no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—61no—
189Dic12—124no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32#12011yescharacteristic
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—41no—
197C32×C4—21no—
198C32×C4—41no—
199C32×C4#13511yescharacteristic
200C32×C4#13511yescharacteristic
201C32×C4#13711yescharacteristic
202C32×C4#13711yescharacteristic
203C3⋊Dic3—81no—
204C3×Dic3—41no—
205C3×Dic3—42no—
206C2×Q16—182no—
207C4⋊Q8—31no—
208C4⋊Q8—91no—
209#124—62no—
210#121—61no—
211#121—61no—
212C4×D4—31no—
213C4×D4D3211yescharacteristic
214#117—61no—
215#117—61no—
216#106—181no—
217#106—181no—
218C4⋊C8—31no—
219C4⋊C8—91no—
220#102—181no—
221#102—181no—
222C4×C8—31no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—41no—
226C23×C3—61no—
227C22×D3—21no—
228C3×Q8—122no—
229C3×D4—22no—
230C3×D4—21no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—21no—
234C3×D4—41no—
235C3×D4—61no—
236C3×D4—61no—
237C3×D4—242no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—61no—
249C2×C3×C4—62no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4—121no—
253C2×C3×C4—121no—
254C2×C3×C4C2×D1211yescharacteristic
255C2×C3×C4D3×D411yescharacteristic
256C2×C3×C4D3×D411yescharacteristic
257C2×C3×C4C2×#6711yescharacteristic
258C2×C3×C4C2×#6711yescharacteristic
259C2×C3×C4C2×#6711yescharacteristic
260#67—82no—
261C2×Dic3—21no—
262C2×Dic3—22no—
263C2×Dic3—122no—
264C2×Dic3—121no—
265C2×Dic3—121no—
266D12—21no—
267D12—21no—
268C4×D3—41no—
269C4×D3—41no—
270Dic6—42no—
271Dic6—122no—
272Dic6—122no—
273Dic6—242no—
274C3×C8—62no—
275C3×C8—121no—
276C3⋊C8—22no—
277C3⋊C8—41no—
278C3⋊C8—122no—
279C3⋊C8—121no—
280C3⋊C8—121no—
281C2×C32—81no—
282C2×C32#45211yescharacteristic
283C2×C32#45211yescharacteristic
284C2×C32#45311yescharacteristic
285C3×D3—81no—
286C2×Q8—32no—
287C2×Q8—181no—
288C2×D4—31no—
289C2×D4—62no—
290C2×D4#62611yescharacteristic
291C22×C4—21no—
292C22×C4—61no—
293Q16—364no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—181no—
297C4⋊C4—31no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—181no—
301C4⋊C4—181no—
302C4⋊C4#62611yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—62no—
305C22⋊C4—62no—
306C22⋊C4—61no—
307C42—62no—
308C42C2×D3211yescharacteristic
309C22×C3—21no—
310C22×C3—42no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3—121no—
322C22×C3#88211yescharacteristic
323C22×C3#95311yescharacteristic
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—81no—
337C3×C4—121no—
338C3×C4—122no—
339C3×C4#90411yescharacteristic
340C3×C4#90911yescharacteristic
341C3×C4#91211yescharacteristic
342C3×C4#91511yescharacteristic
343C3×C4#95011yescharacteristic
344C3×C4#95011yescharacteristic
345C3×C4#95111yescharacteristic
346C3×C4#95111yescharacteristic
347Dic3—41no—
348Dic3—42no—
349Dic3—242no—
350Dic3—241no—
351Dic3—241no—
352C32#163411yescharacteristic, complemented, Sylow
353C23—21no—
354C23—61no—
355Q8—122no—
356Q8—362no—
357D4—21no—
358D4—21no—
359D4—61no—
360D4—61no—
361D4—242no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—61no—
366C2×C4—62no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—361no—
370C2×C4C2×#62611yescharacteristic
371C2×C4D3×#6711yescharacteristic
372C2×C4D3×#6711yescharacteristic
373C8—62no—
374C8—121no—
375C8—361no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—82no—
380C2×C3—81no—
381C2×C3—81no—
382C2×C3—241no—
383C2×C3#473311yescharacteristic
384C2×C3#478311yescharacteristic
385C2×C3#482211yescharacteristic
386C2×C3#523911yescharacteristic
387C2×C3#523911yescharacteristic
388C2×C3#524111yescharacteristic
389D3—81no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—121no—
394C22—121no—
395C22—121no—
396C22#6397711yescharacteristic
397C4—21no—
398C4—41no—
399C4—122no—
400C4—121no—
401C4—721no—
402C4#6382911yescharacteristic
403C4#6383111yescharacteristic
404C4#6392911yescharacteristic
405C4#6393311yescharacteristic
406C3—21no—
407C3#6820311yescharacteristic, complemented
408C3#7192811yescharacteristic, complemented
409C2—81no—
410C2—241no—
411C2#1059117311yescharacteristic
412C2#1059146311yescharacteristic
413C2#1059166611yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×D4C22
socleC22×C32#120
prime radicalC3×#5563C2
hypercenterC4×D4D32
hypocenterC32#1634
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #10592707C2semidirect
3. #10590386C2semidirect
4. #10590268C2semidirect
5. C3×D4⋊(C3⋊C8)C2semidirect
6. #10589095C2semidirect
7. #10589081C2semidirect
39. D4⋊(C3⋊C8)D3semidirect
352. C32#1634semidirect
407. C3#68203semidirect
408. C3#71928semidirect

Maximal subgroups

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

subgroupindex
C3×#55632
#105927072
#105903862
#105902682
C3×D4⋊(C3⋊C8)2
#105890952
#105890812
#719283
#682033