Finite Group Databasebeta

← #11781782

Subgroups — #11781782

Order 1152.

386 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 86 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#10592545C211yesmaximal, characteristic
2#10590386C211yesmaximal, characteristic, complemented
3#10590338C211yesmaximal, characteristic
4#10590313C211yesmaximal, characteristic
5C3×#4486C211yesmaximal, characteristic, complemented
6C3×#4480C211yesmaximal, characteristic, complemented
7#10589094C211yesmaximal, characteristic, complemented
8#68541—31nomaximal
9#68449—31nomaximal
10C4×C4⋊(C3⋊D3)C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3719—21no—
14C2×#3719—21no—
15C2×#3712—21no—
16C32×C4⋊C8C2211yescharacteristic
17#63648—21no—
18#63645—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63575—21no—
23#63564—21no—
24#63557—21no—
25#5092—31no—
26#5087—31no—
27#4846—31no—
28#4846—31no—
29#4846—31no—
30#4846—61no—
31#4838—31no—
32#4699—31no—
33#4688—31no—
34C3×#334—31no—
35C3×#332—31no—
36#4486—31no—
37#4486D311yescharacteristic, complemented
38#4482—31no—
39#4480D311yescharacteristic, complemented
40C2×C4⋊(C3⋊D3)D411yescharacteristic
41C2×C4×C3⋊D3—21no—
42C2×#634—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C32×C42C2311yescharacteristic
46#3752—21no—
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)D411yescharacteristic
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C2×C3×C3⋊C8—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3719—42no—
57#3719—42no—
58#3712—42no—
59#1689—91noSylow
60C3×C4⋊Q8—31no—
61C2×#216—61no—
62C2×#216—61no—
63C2×Dic12—61no—
64C2×Dic12—61no—
65C2×ΓL(1,𝔽25)—61no—
66C2×ΓL(1,𝔽25)—61no—
67C4×D12—31no—
68C4×D12—31no—
69C4×D12—61no—
70#869—31no—
71#869—31no—
72#869—61no—
73#869C2×D311yescharacteristic
74C3×C4⋊C8—21no—
75C3×C4⋊C8—31no—
76C3×C4⋊C8—31no—
77C3×C4⋊C8C2×D311yescharacteristic
78C3×C4⋊C8C2×D311yescharacteristic
79C3×C4×C8—31no—
80#821—61no—
81#821—61no—
82#821—61no—
83#821—62no—
84(C2×D3)⋊C8—61no—
85(C2×D3)⋊C8—61no—
86#818—61no—
87#818—61no—
88#818—61no—
89#818—62no—
90#809—61no—
91#807—61no—
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4⋊(C3⋊C8)C2×D311yescharacteristic
94C4×C3⋊C8C2×D311yescharacteristic
95C22×C3⋊D3—21no—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—41no—
101C4⋊(C3⋊D3)—21no—
102C4⋊(C3⋊D3)—21no—
103C4×C3⋊D3—41no—
104C4×C3⋊D3—41no—
105#634—41no—
106#634—41no—
107C2×C3×Dic3—41no—
108C3×Dic6—41no—
109C3×Dic6—41no—
110C32×C8—41no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8D812yes—
115#494—91no—
116#474—91no—
117#474—91no—
118#436—91no—
119#334—31no—
120#332—31no—
121#332—91no—
122C2×C3×Q8—61no—
123C2×D12—31no—
124C2×D12—31no—
125C2×D12—61no—
126C2×C4×D3—61no—
127C2×C4×D3—61no—
128C2×C4×D3—121no—
129C2×Dic6—21no—
130C2×Dic6—61no—
131C2×Dic6—61no—
132C2×Dic6—121no—
133C2×C3×C8—22no—
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—31no—
137C2×C3×C8—31no—
138C2×C3×C8—61no—
139C2×C3×C8—61no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—61no—
142C3×C42—21no—
143C3×C42C22×D311yescharacteristic
144C3×C42C22×D311yescharacteristic
145#216—122no—
146#216—122no—
147(C2×D3)⋊C4—61no—
148(C2×D3)⋊C4—61no—
149(C2×D3)⋊C4—121no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—61no—
155C4⋊Dic3—62no—
156C4⋊Dic3—61no—
157C4⋊Dic3—61no—
158C4⋊Dic3—61no—
159C4⋊Dic3—61no—
160C4⋊Dic3—121no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8D1211yescharacteristic
164C2×C3⋊C8D1211yescharacteristic
165Dic12—122no—
166Dic12—122no—
167ΓL(1,𝔽25)—122no—
168ΓL(1,𝔽25)—122no—
169C22×C32#12111yescharacteristic
170C2×C3⋊D3—41no—
171C2×C3⋊D3—41no—
172C2×C3⋊D3—41no—
173C32×C4—21no—
174C32×C4C2×D811yescharacteristic
175C32×C4AGL(1,ℤ/8)11yescharacteristic
176C32×C4#13711yescharacteristic
177C32×C4#13711yescharacteristic
178C3⋊Dic3—41no—
179C3⋊Dic3—81no—
180C3×Dic3—81no—
181C2×Q16—181no—
182C2×Q16—181no—
183C2×M16—181no—
184C4⋊Q8—31no—
185C4⋊Q8—91no—
186C4×D4—91no—
187#107—181no—
188#107—181no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8D3211yescharacteristic
192#102—181no—
193#102—181no—
194C22⋊C8—181no—
195C4×C8—31no—
196C22×D3—61no—
197C22×D3—61no—
198C22×D3—121no—
199C3×Q8—121no—
200C3×Q8—121no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—121no—
205C2×C3×C4C2×D1211yescharacteristic
206C2×C3×C4#23511yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4D3×D411yescharacteristic
209C2×C3×C4D3×D411yescharacteristic
210C2×C3×C4#23911yescharacteristic
211C2×Dic3—41no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—121no—
215C2×Dic3—122no—
216C2×Dic3—121no—
217C2×Dic3—121no—
218D12—62no—
219D12—61no—
220D12—61no—
221D12—61no—
222D12—61no—
223D12—121no—
224C4×D3—121no—
225C4×D3—121no—
226C4×D3—121no—
227C4×D3—121no—
228C4×D3—241no—
229C4×D3—241no—
230Dic6—41no—
231Dic6—41no—
232Dic6—121no—
233Dic6—121no—
234Dic6—121no—
235Dic6—121no—
236Dic6—241no—
237Dic6—241no—
238C3×C8—32no—
239C3×C8—42no—
240C3×C8—41no—
241C3×C8—41no—
242C3×C8—61no—
243C3×C8—121no—
244C3×C8—121no—
245C3⋊C8—21no—
246C3⋊C8—41no—
247C3⋊C8—41no—
248C3⋊C8D2412yes—
249C2×C32#45911yescharacteristic
250C2×C32#47011yescharacteristic
251C2×C32#47311yescharacteristic
252C3⋊D3—81no—
253C2×Q8—61no—
254C2×Q8—181no—
255C2×D4—91no—
256C22×C4—181no—
257Q16—362no—
258Q16—362no—
259M16—362no—
260C2×C8—21no—
261C2×C8—31no—
262C2×C8—31no—
263C2×C8—61no—
264C2×C8—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—91no—
268C4⋊C4—181no—
269C4⋊C4—181no—
270C22⋊C4—181no—
271C42C2×D3211yescharacteristic
272C22×C3—21no—
273C22×C3#88411yescharacteristic
274C22×C3#89511yescharacteristic
275C2×D3—122no—
276C2×D3—121no—
277C2×D3—121no—
278C2×D3—121no—
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3—241no—
283C2×D3—241no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—41no—
291C3×C4—241no—
292C3×C4C2×D2411yescharacteristic
293C3×C4#90811yescharacteristic
294C3×C4D3×D811yescharacteristic
295C3×C4#91411yescharacteristic
296C3×C4#91511yescharacteristic
297C3×C4#91511yescharacteristic
298C3×C4#91511yescharacteristic
299C3×C4#91811yescharacteristic
300Dic3—81no—
301Dic3—121no—
302Dic3—121no—
303Dic3—241no—
304Dic3—242no—
305Dic3—241no—
306Dic3—241no—
307C32#168911yescharacteristic, complemented, Sylow
308C23—181no—
309Q8—121no—
310Q8—121no—
311Q8—361no—
312Q8—361no—
313D4—181no—
314D4—181no—
315C2×C4—121no—
316C2×C4—181no—
317C2×C4—361no—
318C2×C4—361no—
319C2×C4—361no—
320C2×C4#379911yescharacteristic
321C2×C4D3×D1211yescharacteristic
322C2×C4#381111yescharacteristic
323C8—32no—
324C8—41no—
325C8—61no—
326C8—121no—
327C8—121no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3#475511yescharacteristic
332C2×C3#478611yescharacteristic
333C2×C3#481611yescharacteristic
334C2×C3#484311yescharacteristic
335C2×C3#486711yescharacteristic
336C2×C3#489611yescharacteristic
337D3—242no—
338D3—241no—
339D3—241no—
340C22—361no—
341C22—361no—
342C22—361no—
343C22#6391011yescharacteristic
344C4—21no—
345C4—241no—
346C4—361no—
347C4—721no—
348C4D3×D2411yescharacteristic
349C4#6379411yescharacteristic
350C4#6393111yescharacteristic
351C4#6395111yescharacteristic
352C3—21no—
353C3#6844911yescharacteristic, complemented
354C3#6854111yescharacteristic, complemented
355C2—721no—
356C2#1059052411yescharacteristic
357C2#1059076511yescharacteristic
358C2#1059099211yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊C8D32
hypocenterC32#1689
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. #10590386C2semidirect
5. C3×#4486C2semidirect
6. C3×#4480C2semidirect
7. #10589094C2semidirect
37. #4486D3semidirect
39. #4480D3semidirect
307. C32#1689semidirect
353. C3#68449semidirect
354. C3#68541semidirect

Maximal subgroups

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

subgroupindex
#105925452
#105903862
#105903382
#105903132
C3×#44862
C3×#44802
#105890942
#685413
#684493