Finite Group Databasebeta

← #11779979

Subgroups — #11779979

Order 1152.

470 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 88 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#10593958C211yesmaximal, characteristic
2#10592545C211yesmaximal, characteristic
3#10590385C211yesmaximal, characteristic, complemented
4#10590287C211yesmaximal, characteristic, complemented
5C3×#4484C211yesmaximal, characteristic, complemented
6C3×#4481C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic
8#68506—31nomaximal
9#68270—31nomaximal
10C4×C4⋊(C3⋊D3)C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C3×C4×Dic6C2211yescharacteristic
14#63900—21no—
15#63894—21no—
16#63888—22no—
17#63871—21no—
18#63869—21no—
19C2×#3719—21no—
20C32×C4⋊C8C2211yescharacteristic
21#63648—21no—
22#63645—21no—
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25#63567—21no—
26#63566—22no—
27#5730—31no—
28#5579—31no—
29#5079—31no—
30#5031—31no—
31#4849—31no—
32#4846—31no—
33#4846—31no—
34#4846—61no—
35#4681—31no—
36C3×#333—31no—
37C3×Q8⋊C8—31no—
38#4484D311yescharacteristic, complemented
39#4483—31no—
40#4481D311yescharacteristic, complemented
41#4459—31no—
42C2×C4⋊(C3⋊D3)D411yescharacteristic
43C2×C4×C3⋊D3—21no—
44C2×#634—21no—
45C2×C3×Dic6D411yescharacteristic
46C2×C3×Dic6D412yes—
47C2×#624—22no—
48C2×C32×C8—21no—
49C32×C42C2311yescharacteristic
50#3752—21no—
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)D411yescharacteristic
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3—21no—
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#3722—22no—
64#3719—42no—
65#1644—91noSylow
66C3×C4⋊Q8—31no—
67C3×C4×Q8—31no—
68#946—62no—
69#946—61no—
70C2×#216—61no—
71C2×Dic12—61no—
72#897—61no—
73#896—62no—
74#893—61no—
75D3×C4⋊C4—61no—
76D3×C4⋊C4—61no—
77#890—61no—
78C4×D12—31no—
79C4×D12—31no—
80C4×D12—61no—
81C4×D12—61no—
82#870—61no—
83#869—31no—
84#869—31no—
85#869—61no—
86#869C2×D311yescharacteristic
87C4×Dic6C2×D311yescharacteristic
88C3×C4⋊C8—21no—
89C3×C4⋊C8—31no—
90C3×C4⋊C8C2×D311yescharacteristic
91C3×C4⋊C8C2×D311yescharacteristic
92C3×C4×C8—31no—
93#835—62no—
94#821—61no—
95#821—61no—
96#821—62no—
97#818—61no—
98#818—61no—
99#818—62no—
100#816—62no—
101#816—61no—
102#810—61no—
103C4⋊(C3⋊C8)C2×D311yescharacteristic
104C4×C3⋊C8C2×D311yescharacteristic
105C22×C3⋊D3—21no—
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—41no—
111C4⋊(C3⋊D3)—21no—
112C4⋊(C3⋊D3)Q1612yescomplemented
113C4×C3⋊D3—41no—
114C4×C3⋊D3—41no—
115#634—41no—
116#634—41no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—21no—
120C3×Dic6—21no—
121C3×Dic6—21no—
122C3×Dic6—42no—
123#624—82no—
124C32×C8—41no—
125C3×C3⋊C8—41no—
126C3×C3⋊C8—41no—
127#554—91no—
128#487—91no—
129#474—91no—
130#462—91no—
131#333—31no—
132Q8⋊C8—31no—
133D4⋊C8—91no—
134C2×C3×Q8—31no—
135C2×C3×Q8—32no—
136C2×D12—31no—
137C2×D12—31no—
138C2×D12—61no—
139C2×C4×D3—61no—
140C2×C4×D3—61no—
141C2×C4×D3—62no—
142C2×C4×D3—62no—
143C2×C4×D3—121no—
144C2×Dic6—61no—
145C2×Dic6—61no—
146C2×Dic6—121no—
147C2×Dic6D1211yescharacteristic
148C2×Dic6D1212yes—
149C2×C3×C8—22no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C2×C3×C8—61no—
153C2×C3×C8—61no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—61no—
157C3×C4⋊C4—61no—
158C3×C42—21no—
159C3×C42—61no—
160C3×C42C22×D311yescharacteristic
161C3×C42C22×D311yescharacteristic
162#216—122no—
163(C2×D3)⋊C4—62no—
164(C2×D3)⋊C4—62no—
165(C2×D3)⋊C4—61no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—121no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173C4⋊Dic3—62no—
174C4⋊Dic3—62no—
175C4⋊Dic3—61no—
176C4⋊Dic3—61no—
177C4⋊Dic3—61no—
178C4⋊Dic3—61no—
179C4⋊Dic3—121no—
180C4⋊Dic3D1211yescharacteristic
181Dic3⋊C4—21no—
182Dic3⋊C4—62no—
183Dic3⋊C4—62no—
184Dic3⋊C4—62no—
185C4×Dic3—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188Dic12—122no—
189C22×C32#12011yescharacteristic
190C2×C3⋊D3—22no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C32×C4—21no—
194C32×C4C2×Q1611yescharacteristic
195C32×C4#13511yescharacteristic
196C32×C4AGL(1,ℤ/8)11yescharacteristic
197C32×C4#13711yescharacteristic
198C3⋊Dic3—41no—
199C3⋊Dic3—81no—
200C3×Dic3—41no—
201C3×Dic3—41no—
202C3×Dic3—42no—
203C2×Q16—181no—
204C4⋊Q8—31no—
205C4⋊Q8—91no—
206#125—181no—
207C22⋊Q8—182no—
208C22⋊Q8—181no—
209C4×Q8—31no—
210C4×D4—91no—
211C4×D4—181no—
212C2×C4⋊C4—181no—
213#107—181no—
214C4⋊C8—31no—
215C4⋊C8D3211yescharacteristic
216#103—181no—
217#103—182no—
218#102—181no—
219C4×C8—31no—
220C22×D3—61no—
221C22×D3—61no—
222C22×D3—121no—
223C3×Q8—61no—
224C3×Q8—61no—
225C3×Q8—122no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—62no—
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—22no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—121no—
244C2×Dic3—122no—
245C2×Dic3—121no—
246C2×Dic3—121no—
247D12—32no—
248D12—32no—
249D12—62no—
250D12—61no—
251D12—61no—
252D12—121no—
253C4×D3—121no—
254C4×D3—121no—
255C4×D3—121no—
256C4×D3—121no—
257C4×D3—241no—
258C4×D3—241no—
259C4×D3—242no—
260C4×D3—242no—
261Dic6—21no—
262Dic6—21no—
263Dic6—42no—
264Dic6—121no—
265Dic6—121no—
266Dic6—121no—
267Dic6—121no—
268Dic6—241no—
269Dic6—241no—
270C3×C8—42no—
271C3×C8—41no—
272C3×C8—41no—
273C3×C8—121no—
274C3×C8—121no—
275C3⋊C8—41no—
276C3⋊C8—41no—
277C2×C32#44911yescharacteristic
278C2×C32#45111yescharacteristic
279C2×C32#45511yescharacteristic
280C3⋊D3—42no—
281C2×Q8—31no—
282C2×Q8—32no—
283C2×Q8—181no—
284C2×D4—91no—
285C22×C4—182no—
286C22×C4—181no—
287Q16—362no—
288C2×C8—21no—
289C2×C8—61no—
290C2×C8—61no—
291C4⋊C4—31no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—91no—
296C4⋊C4—182no—
297C4⋊C4—182no—
298C4⋊C4—181no—
299C4⋊C4—181no—
300C22⋊C4—182no—
301C22⋊C4—181no—
302C42—61no—
303C42C2×D3211yescharacteristic
304C22×C3—21no—
305C22×C3#88211yescharacteristic
306C22×C3#88211yescharacteristic
307C2×D3—62no—
308C2×D3—62no—
309C2×D3—122no—
310C2×D3—121no—
311C2×D3—121no—
312C2×D3—121no—
313C2×D3—121no—
314C2×D3—241no—
315C2×D3—241no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—41no—
323C3×C4—121no—
324C3×C4—121no—
325C3×C4—122no—
326C3×C4#90411yescharacteristic
327C3×C4#90811yescharacteristic
328C3×C4#91411yescharacteristic
329C3×C4#91511yescharacteristic
330C3×C4#91511yescharacteristic
331C3×C4D3×Q1611yescharacteristic
332C3×C4D3×Q1611yescharacteristic
333C3×C4#91911yescharacteristic
334Dic3—41no—
335Dic3—42no—
336Dic3—41no—
337Dic3—121no—
338Dic3—121no—
339Dic3—241no—
340Dic3—242no—
341Dic3—241no—
342Dic3—241no—
343C32#164411yescharacteristic, complemented, Sylow
344C23—181no—
345Q8—61no—
346Q8—61no—
347Q8—122no—
348Q8—361no—
349Q8—361no—
350D4—92no—
351D4—181no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—62no—
355C2×C4—181no—
356C2×C4—361no—
357C2×C4—361no—
358C2×C4—361no—
359C2×C4—722no—
360C2×C4D3×D1211yescharacteristic
361C2×C4D3×D1211yescharacteristic
362C2×C4#381111yescharacteristic
363C8—41no—
364C8—121no—
365C8—121no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3#473411yescharacteristic
370C2×C3#474611yescharacteristic
371C2×C3#480911yescharacteristic
372C2×C3#480911yescharacteristic
373C2×C3#481011yescharacteristic
374C2×C3#482511yescharacteristic
375D3—122no—
376D3—122no—
377D3—242no—
378C22—182no—
379C22—361no—
380C22—361no—
381C22#6392311yescharacteristic
382C4—21no—
383C4—121no—
384C4—122no—
385C4—121no—
386C4—361no—
387C4—721no—
388C4#6379411yescharacteristic
389C4#6380811yescharacteristic
390C4#6393111yescharacteristic
391C4#6394411yescharacteristic
392C3—21no—
393C3#6827011yescharacteristic, complemented
394C3#6850611yescharacteristic, complemented
395C2—362no—
396C2#1059060511yescharacteristic
397C2#1059096711yescharacteristic
398C2#1059101611yescharacteristic
subgroupG/H
centerC22#63923
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#120
prime radicalC2×C4⋊(C3⋊D3)D4
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
3. #10590385C2semidirect
4. #10590287C2semidirect
5. C3×#4484C2semidirect
6. C3×#4481C2semidirect
38. #4484D3semidirect
40. #4481D3semidirect
112. C4⋊(C3⋊D3)Q16semidirect
343. C32#1644semidirect
393. C3#68270semidirect
394. C3#68506semidirect

Maximal subgroups

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

subgroupindex
#105939582
#105925452
#105903852
#105902872
C3×#44842
C3×#44812
#105890822
#685063
#682703