Finite Group Databasebeta

← #11779968

Subgroups — #11779968

Order 1152.

465 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#10593960C211yesmaximal, characteristic
2#10592380C211yesmaximal, characteristic, complemented
3#10590385C211yesmaximal, characteristic, complemented
4#10590272C211yesmaximal, characteristic
5C3×#4459C211yesmaximal, characteristic
6C3×#4456C211yesmaximal, characteristic, complemented
7#10589096C211yesmaximal, characteristic, complemented
8#68270—31nomaximal
9#68191—31nomaximal
10#64078C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×C4×D12C2211yescharacteristic
13C3×C4×Dic6C2211yescharacteristic
14#63896—22no—
15Dic3×D12—21no—
16D3×C4⋊Dic3—21no—
17#63869—21no—
18#63856—21no—
19C2×#3719—21no—
20C32×C4×C8C2211yescharacteristic
21#63643—22no—
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24#63575—21no—
25#63567—21no—
26#63563—21no—
27#5579—31no—
28#5543—31no—
29#5031—31no—
30#5007—31no—
31#4849—31no—
32#4846—31no—
33#4681—31no—
34#4681—31no—
35#4681—61no—
36C3×Q8⋊C8—31no—
37C3×D4⋊C8—31no—
38#4481—31no—
39#4481—31no—
40#4459D311yescharacteristic
41#4456D311yescharacteristic, complemented
42C2×#634—21no—
43C2×#634D412yes—
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×Dic6D411yescharacteristic
47C2×D3×Dic3—22no—
48C2×C32×C8—21no—
49C32×C42C2311yescharacteristic
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)—21no—
54C3×(C2×D3)⋊C4—21no—
55C3×C4⋊Dic3D411yescharacteristic
56C3×C4⋊Dic3D411yescharacteristic
57C3×Dic3⋊C4—21no—
58C3×C4×Dic3—21no—
59C2×C3×C3⋊C8—21no—
60C2×C3×C3⋊C8—21no—
61#3724—22no—
62Dic3⋊Dic3—22no—
63(C2×D3)⋊Dic3—22no—
64#3719—42no—
65#1644—91noSylow
66C3×C4×Q8—31no—
67C3×C4×D4—31no—
68#946—61no—
69C2×#216—61no—
70D4×Dic3—61no—
71C2×C4⋊Dic3—61no—
72#924—61no—
73C2×Dic12—61no—
74#896—62no—
75D3×C4⋊C4—61no—
76#890—61no—
77#878—62no—
78C4×D12—61no—
79C4×D12C2×D311yescharacteristic
80#870—61no—
81#869—31no—
82#869—31no—
83#869—31no—
84#869—31no—
85#869—61no—
86#869—61no—
87C4×Dic6C2×D311yescharacteristic
88C3×C4⋊C8—31no—
89C3×C4⋊C8—31no—
90C3×C4×C8—21no—
91C3×C4×C8C2×D311yescharacteristic
92C3×C4×C8C2×D311yescharacteristic
93#832—61no—
94#821—61no—
95#818—61no—
96#816—62no—
97#816—62no—
98#816—64no—
99#816—61no—
100#810—61no—
101#807—61no—
102C4⋊(C3⋊C8)C2×D311yescharacteristic
103C4⋊(C3⋊C8)C2×D311yescharacteristic
104C22×C3×D3—21no—
105C2×C32×C4C2×D411yescharacteristic
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—41no—
110#634—42no—
111#634—41no—
112#634—41no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C3×D12—21no—
117C3×D12Q1612yescomplemented
118C3×C4×D3—41no—
119C3×C4×D3—41no—
120C3×Dic6—21no—
121C3×Dic6—21no—
122D3×Dic3—82no—
123C32×C8—41no—
124C3×C3⋊C8—41no—
125C3×C3⋊C8—41no—
126#554—91no—
127#487—91no—
128#474—91no—
129#462—91no—
130#333—91no—
131Q8⋊C8—31no—
132D4⋊C8—31no—
133C2×C3×Q8—31no—
134C2×C3×D4—31no—
135C22×C3×C4—61no—
136C22×Dic3—62no—
137C2×D12D1211yescharacteristic
138C2×C4×D3—21no—
139C2×C4×D3—62no—
140C2×Dic6—32no—
141C2×Dic6—32no—
142C2×Dic6—62no—
143C2×Dic6—61no—
144C2×Dic6—61no—
145C2×Dic6—121no—
146C2×Dic6D1211yescharacteristic
147C2×C3×C8—22no—
148C2×C3×C8—21no—
149C2×C3×C8—21no—
150C2×C3×C8—61no—
151C2×C3×C8—61no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—61no—
155C3×C22⋊C4—61no—
156C3×C42—21no—
157C3×C42—61no—
158C3×C42C22×D311yescharacteristic
159C3×C42C22×D311yescharacteristic
160#217—62no—
161#216—122no—
162(C2×D3)⋊C4—21no—
163(C2×D3)⋊C4—62no—
164C4⋊Dic3—62no—
165C4⋊Dic3—61no—
166C4⋊Dic3—61no—
167C4⋊Dic3—61no—
168C4⋊Dic3—61no—
169C4⋊Dic3—62no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172C4⋊Dic3—61no—
173C4⋊Dic3—61no—
174C4⋊Dic3—121no—
175C4⋊Dic3—121no—
176C4⋊Dic3—121no—
177C4⋊Dic3D1211yescharacteristic
178C4⋊Dic3D1211yescharacteristic
179Dic3⋊C4—21no—
180Dic3⋊C4—62no—
181Dic3⋊C4—62no—
182Dic3⋊C4—62no—
183C4×Dic3—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186Dic12—122no—
187C22×C32#12011yescharacteristic
188C2×C3×D3—22no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C32×C4—21no—
192C32×C4C2×Q1611yescharacteristic
193C32×C4#13511yescharacteristic
194C32×C4AGL(1,ℤ/8)11yescharacteristic
195C32×C4#13711yescharacteristic
196C3⋊Dic3—42no—
197C3⋊Dic3—81no—
198C3×Dic3—41no—
199C3×Dic3—41no—
200C3×Dic3—41no—
201C2×Q16—181no—
202C4⋊Q8—91no—
203C4⋊Q8—91no—
204#125—181no—
205C22⋊Q8—182no—
206C22⋊Q8—181no—
207C4×Q8—31no—
208C4×D4—31no—
209C4×D4—181no—
210C2×C4⋊C4—181no—
211#107—181no—
212C4⋊C8—31no—
213C4⋊C8—31no—
214#103—181no—
215#103—182no—
216#102—181no—
217C4×C8D3211yescharacteristic
218C23×C3—61no—
219C22×D3—21no—
220C3×Q8—61no—
221C3×Q8—61no—
222C3×D4—32no—
223C3×D4—61no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—121no—
231C2×C3×C4—121no—
232C2×C3×C4C2×D1211yescharacteristic
233C2×C3×C4C2×D1211yescharacteristic
234C2×C3×C4D3×D411yescharacteristic
235C2×C3×C4D3×D411yescharacteristic
236C2×C3×C4D3×D411yescharacteristic
237C2×C3×C4D3×D411yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3—122no—
244C2×Dic3—122no—
245C2×Dic3—121no—
246C2×Dic3—121no—
247C2×Dic3—242no—
248D12—21no—
249D12Dic1212yescomplemented
250C4×D3—41no—
251C4×D3—41no—
252C4×D3—242no—
253Dic6—21no—
254Dic6—21no—
255Dic6—122no—
256Dic6—122no—
257Dic6—124no—
258Dic6—121no—
259Dic6—121no—
260Dic6—121no—
261Dic6—121no—
262Dic6—241no—
263Dic6—241no—
264C3×C8—42no—
265C3×C8—41no—
266C3×C8—41no—
267C3×C8—121no—
268C3×C8—121no—
269C3⋊C8—41no—
270C3⋊C8—41no—
271C2×C32#44911yescharacteristic
272C2×C32#45111yescharacteristic
273C2×C32#45511yescharacteristic
274C3×D3—42no—
275C2×Q8—31no—
276C2×Q8—92no—
277C2×Q8—181no—
278C2×D4—31no—
279C22×C4—61no—
280C22×C4—182no—
281Q16—362no—
282C2×C8—21no—
283C2×C8—61no—
284C2×C8—61no—
285C4⋊C4—31no—
286C4⋊C4—31no—
287C4⋊C4—61no—
288C4⋊C4—182no—
289C4⋊C4—182no—
290C4⋊C4—181no—
291C4⋊C4—181no—
292C4⋊C4—181no—
293C4⋊C4—181no—
294C22⋊C4—61no—
295C22⋊C4—182no—
296C42—61no—
297C42C2×D3211yescharacteristic
298C22×C3—21no—
299C22×C3—62no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3#88211yescharacteristic
303C22×C3#88211yescharacteristic
304C2×D3—22no—
305C2×D3—41no—
306C2×D3—41no—
307C3×C4—22no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—121no—
315C3×C4—121no—
316C3×C4—121no—
317C3×C4#90411yescharacteristic
318C3×C4C2×Dic1211yescharacteristic
319C3×C4#90811yescharacteristic
320C3×C4#90911yescharacteristic
321C3×C4#91111yescharacteristic
322C3×C4#91511yescharacteristic
323C3×C4#91611yescharacteristic
324C3×C4D3×Q1611yescharacteristic
325Dic3—41no—
326Dic3—41no—
327Dic3—41no—
328Dic3—122no—
329Dic3—122no—
330Dic3—242no—
331Dic3—242no—
332Dic3—241no—
333Dic3—241no—
334C32#164411yescharacteristic, complemented, Sylow
335C23—61no—
336Q8—61no—
337Q8—61no—
338Q8—362no—
339Q8—361no—
340Q8—361no—
341D4—32no—
342D4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—182no—
349C2×C4—361no—
350C2×C4—722no—
351C2×C4D3×D1211yescharacteristic
352C2×C4D3×D1211yescharacteristic
353C2×C4#381111yescharacteristic
354C8—41no—
355C8—121no—
356C8—121no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—122no—
361C2×C3#473411yescharacteristic
362C2×C3#473911yescharacteristic
363C2×C3#478111yescharacteristic
364C2×C3#480911yescharacteristic
365C2×C3#482511yescharacteristic
366C2×C3#482511yescharacteristic
367D3—42no—
368C22—62no—
369C22—121no—
370C22—121no—
371C22#6392311yescharacteristic
372C4—21no—
373C4—121no—
374C4—121no—
375C4—121no—
376C4—362no—
377C4—721no—
378C4#6379511yescharacteristic
379C4D3×Dic1211yescharacteristic
380C4#6380011yescharacteristic
381C4#6380611yescharacteristic
382C3—21no—
383C3#6819111yescharacteristic, complemented
384C3#6827011yescharacteristic, complemented
385C2—122no—
386C2#1059039811yescharacteristic
387C2#1059095211yescharacteristic
388C2#1059096811yescharacteristic
subgroupG/H
centerC22#63923
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×C8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4×C8D32
hypocenterC32#1644
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. #10592380C2semidirect
3. #10590385C2semidirect
6. C3×#4456C2semidirect
7. #10589096C2semidirect
41. #4456D3semidirect
117. C3×D12Q16semidirect
249. D12Dic12semidirect
334. C32#1644semidirect
383. C3#68191semidirect
384. C3#68270semidirect

Maximal subgroups

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

subgroupindex
#105939602
#105923802
#105903852
#105902722
C3×#44592
C3×#44562
#105890962
#682703
#681913