Finite Group Databasebeta

← #10698925

Subgroups — #10698925

Order 768.

452 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3036C211yesmaximal, characteristic
2#71814C211yesmaximal, characteristic, complemented
3#71804C211yesmaximal, characteristic, complemented
4#71605C211yesmaximal, characteristic, complemented
5#71601C211yesmaximal, characteristic, complemented
6#70493C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#10465—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#463—22no—
13C3×#461—21no—
14C3×#459—21no—
15C3×#423C2211yescharacteristic
16C3×#417—21no—
17C2×#832—21no—
18#4936C2211yescharacteristic
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4547—21no—
23#4544—21no—
24#4537—22no—
25#4529—21no—
26#4525—22no—
27#4480—22no—
28#3036D311yescharacteristic
29#1804—31no—
30#1570—31no—
31#1540—31no—
32#1517—31no—
33#1460—31no—
34#1286—31no—
35C2×C3×#41—21no—
36C22×C3×D4D411yescharacteristic
37C23×C3×C4—21no—
38C3×AGL(1,ℤ/8)—42no—
39C3×AGL(1,ℤ/8)—42no—
40C2×C3×SD16—22no—
41C2×C3×D8—22no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×C4⋊Dic3—21no—
58C22×C3⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8D412yes—
61C3×#103—22no—
62C3×#102—22no—
63#832—42no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830—21no—
67#830D412yes—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8—21no—
74C2×AGL(1,ℤ/8)—21no—
75#532C2×D311yescharacteristic
76C2×C4×D4C2×D311yescharacteristic
77#463—22no—
78#461—21no—
79#459—21no—
80#431—31no—
81#431—31no—
82#423C2×D311yescharacteristic
83C2×C4⋊C8—31no—
84#417—21no—
85C2×#102—61no—
86#382—31no—
87#340—61no—
88#334—62no—
89#332—61no—
90#331—61no—
91#327—62no—
92#323—62no—
93C24×C3—21no—
94C3×#41—41no—
95C3×#41—41no—
96C3×#41—41no—
97C3×#41—41no—
98C2×C3×Q8—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—41no—
108C2×C3×D4SD1612yes—
109C22×C3×C4—21no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—82no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×Dic3—41no—
118C3×SD16—44no—
119C3×D8—44no—
120C3×M16—41no—
121C3×M16—41no—
122C2×C3×C8—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C2×#41—21no—
143C22×D4#6711yescharacteristic
144C23×C4—21no—
145AGL(1,ℤ/8)—42no—
146AGL(1,ℤ/8)—42no—
147C2×SD16—22no—
148C2×D8—22no—
149C2×M16—21no—
150C22×C8—61no—
151C4⋊Q8#6711yescharacteristic
152#127#6711yescharacteristic
153#124—21no—
154#121—21no—
155#121—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4#6711yescharacteristic
164C2×C22⋊C4—21no—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8#6712yes—
172#103—22no—
173#102—22no—
174#102—122no—
175C22⋊C8—32no—
176C22⋊C8—61no—
177C4×C8—61no—
178C4×C8—61no—
179#95—61no—
180#95—61no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3—41no—
185C23×C3—81no—
186C23×C3—81no—
187C23×C3#12111yescharacteristic
188C3×Q8—41no—
189C3×Q8—41no—
190C3×D4—22no—
191C3×D4—22no—
192C3×D4—41no—
193C3×D4—41no—
194C3×D4—41no—
195C3×D4—41no—
196C3×D4—81no—
197C3×D4—81no—
198C3×D4—81no—
199C3×D4—81no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—82no—
205C2×C3×C4—82no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4#12411yescharacteristic
215C2×C3×C4C2×SD1611yescharacteristic
216C2×C3×C4#13511yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222C3×C8—42no—
223C3⋊C8—42no—
224C3⋊C8—81no—
225C3⋊C8—81no—
226C24—21no—
227#41—41no—
228#41—41no—
229#41—41no—
230#41—41no—
231C2×Q8—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4#21412yes—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—82no—
247C22×C4—121no—
248C22×C4D3×D411yescharacteristic
249C22×C4#23711yescharacteristic
250C22×C4C2×#6711yescharacteristic
251SD16—44no—
252D8—44no—
253M16—41no—
254M16—41no—
255C2×C8—22no—
256C2×C8—62no—
257C2×C8—121no—
258C2×C8—121no—
259C2×C8—121no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C42#23711yescharacteristic
272C42#23711yescharacteristic
273C42C2×#6711yescharacteristic
274C42C2×#6711yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—42no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3#39411yescharacteristic
286C22×C3#46511yescharacteristic
287C22×C3#46911yescharacteristic
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4#44911yescharacteristic
295C3×C4#45011yescharacteristic
296C3×C4#48111yescharacteristic
297C3×C4#48211yescharacteristic
298Dic3—161no—
299C23—22no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—81no—
304C23—81no—
305C23#93911yescharacteristic
306Q8—41no—
307Q8—41no—
308D4—22no—
309D4—22no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316D4—81no—
317D4—81no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4#93211yescharacteristic
331C2×C4C2×#21411yescharacteristic
332C2×C4#93511yescharacteristic
333C2×C4#93511yescharacteristic
334C2×C4#93611yescharacteristic
335C2×C4#93811yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4#94911yescharacteristic
338C2×C4#95011yescharacteristic
339C2×C4#95311yescharacteristic
340C8—42no—
341C8—122no—
342C8—241no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—82no—
347C2×C3—161no—
348C2×C3#203411yescharacteristic
349C2×C3#204811yescharacteristic
350C2×C3#205311yescharacteristic
351C22—21no—
352C22—21no—
353C22—42no—
354C22—82no—
355C22—82no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#503911yescharacteristic
362C22#504111yescharacteristic
363C22#522211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—481no—
371C4#503411yescharacteristic
372C4#503511yescharacteristic
373C4#521811yescharacteristic
374C4#523811yescharacteristic
375C3#1046511yescharacteristic, complemented, Sylow
376C2—21no—
377C2—21no—
378C2—82no—
379C2—161no—
380C2#7168011yescharacteristic
381C2#7209211yescharacteristic
382C2#7211511yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3#394
hypercenter#3036D3
hypocenterC3#10465
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #71814C2semidirect
3. #71804C2semidirect
4. #71605C2semidirect
5. #71601C2semidirect
6. #70493C2semidirect
7. #67713C2semidirect
375. C3#10465semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#718142
#718042
#716052
#716012
#704932
#677132
#104653