Finite Group Databasebeta

← #10704760

Subgroups — #10704760

Order 768.

440 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 71 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#76336C211yesmaximal, characteristic
2#68144C211yesmaximal, characteristic, complemented
3D3×#324C211yesmaximal, characteristic, complemented
4C3×#1318C211yesmaximal, characteristic, complemented
5#67640C211yesmaximal, characteristic, complemented
6#67452C211yesmaximal, characteristic, complemented
7#67452C211yesmaximal, characteristic, complemented
8#11553—31nomaximal, Sylow
9C2×#938C411yescharacteristic
10C2×D3×C4⋊C4C2211yescharacteristic
11C3×#390C2211yescharacteristic
12#5221C411yescharacteristic
13#4977C2211yescharacteristic
14#4956—21no—
15#4956—21no—
16#4952—21no—
17#4952—21no—
18C4×(C2×D3)⋊C4C411yescharacteristic
19#4936C411yescharacteristic
20#4726—21no—
21D3×C22⋊C8—21no—
22C3×#324C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4528C2211yescharacteristic
25#4528C2211yescharacteristic
26#4469—21no—
27#4469—21no—
28#2504—31no—
29#1463—31no—
30C2×#324—31no—
31#1324—31no—
32#1324—31no—
33#1318—31no—
34#1318D311yescharacteristic, complemented
35C22×C3×D4C2×C411yescharacteristic
36C22×#67—21no—
37C22×C4×D3C2×C411yescharacteristic
38C22×C4×D3D411yescharacteristic
39C22×C4×D3D411yescharacteristic
40C2×C3×C4⋊C4C2311yescharacteristic
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42C2×C411yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47#938—41no—
48#938—41no—
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×Dic3⋊C4D411yescharacteristic
53C2×Dic3⋊C4D411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55C2×ΓL(1,𝔽25)—41no—
56C2×D3×C8—41no—
57D3×C4⋊C4—41no—
58D3×C4⋊C4—41no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830—21no—
67#830—21no—
68(C2×D3)⋊C8—41no—
69(C2×D3)⋊C8—41no—
70C2×#121—31no—
71C22×C4⋊C4—31no—
72#407—61no—
73C2×C22⋊C8—61no—
74#397—61no—
75#397—61no—
76#390—31no—
77#390C2×D311yescharacteristic
78#386—61no—
79#386—61no—
80#382—31no—
81#382—31no—
82C4×C22⋊C4—31no—
83#324—31no—
84#324—31no—
85#324—61no—
86#324—61no—
87#324C2×D311yescharacteristic
88#324C2×D311yescharacteristic
89C24×C3—21no—
90C23×D3C22⋊C411yescharacteristic
91C2×C3×D4—21no—
92C2×C3×D4—21no—
93C2×C3×D4—41no—
94C2×C3×D4—41no—
95C22×C3×C4—21no—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C2×#67—42no—
100C2×#67—42no—
101C2×#67—41no—
102C2×#67—41no—
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3C22⋊C411yescharacteristic
106C22×Dic3C22⋊C411yescharacteristic
107C22×Dic3C22⋊C411yescharacteristic
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3—21no—
116C2×C4×D3—41no—
117C2×C4×D3—81no—
118C2×C4×D3—81no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C3×C4⋊C4—41no—
122C3×C22⋊C4—81no—
123C3×C22⋊C4—81no—
124C3×C42—41no—
125#217—42no—
126#217—81no—
127#217—81no—
128(C2×D3)⋊C4—42no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133C4×Dic3—21no—
134C4×Dic3—21no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137ΓL(1,𝔽25)—82no—
138D3×C8—82no—
139C22×D4—61no—
140C22×D4C4×D311yescharacteristic
141C23×C4—31no—
142C23×C4—31no—
143C23×C4—31no—
144C2×M16—121no—
145C22×C8—121no—
146#121—121no—
147#121—121no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—121no—
153C2×C4⋊C4—121no—
154C2×C4⋊C4C22×D311yescharacteristic
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C42—31no—
162C2×C42C4×D311yescharacteristic
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167C22⋊C8—121no—
168C22⋊C8—121no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—61no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3C2×C22⋊C411yescharacteristic
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—22no—
185C22×D3—41no—
186C22×D3#9911yescharacteristic
187C22×D3#9911yescharacteristic
188C3×D4—81no—
189C3×D4—81no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4#13111yescharacteristic
200C2×C3×C4#13111yescharacteristic
201#67—82no—
202#67—82no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C4×D3—41no—
217C4×D3—41no—
218C4×D3—41no—
219C4×D3—41no—
220C4×D3—42no—
221C4×D3—42no—
222C4×D3—81no—
223C4×D3—81no—
224C3×C8—81no—
225C3×C8—81no—
226C3⋊C8—81no—
227C3⋊C8—81no—
228C24—21no—
229C24—31no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4—122no—
237C2×D4—122no—
238C22×C4—21no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—121no—
253C22×C4—241no—
254C22×C4—241no—
255C22×C4C2×C4×D311yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258M16—242no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—121no—
262C2×C8—121no—
263C2×C8—242no—
264C4⋊C4—41no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—122no—
272C22⋊C4—122no—
273C22⋊C4—241no—
274C22⋊C4—241no—
275C42—41no—
276C42—61no—
277C42—61no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3#40811yescharacteristic
288C22×C3C2×#9911yescharacteristic
289C22×C3#41311yescharacteristic
290C2×D3—21no—
291C2×D3—21no—
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—21no—
295C2×D3—42no—
296C2×D3—42no—
297C2×D3#35512yes—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4—81no—
303Dic3—41no—
304Dic3—41no—
305Dic3—41no—
306Dic3—81no—
307Dic3—81no—
308C23—31no—
309C23—31no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—62no—
318C23—61no—
319C23—61no—
320C23—121no—
321C23D3×C22⋊C411yescharacteristic
322D4—81no—
323D4—81no—
324D4—242no—
325D4—242no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—241no—
355C2×C4—241no—
356C2×C4#90111yescharacteristic
357C2×C4#90711yescharacteristic
358C8—81no—
359C8—81no—
360C8—241no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—161no—
365C2×C3#146811yescharacteristic
366C2×C3#212711yescharacteristic
367C2×C3C2×#35511yescharacteristic
368D3—22no—
369D3—41no—
370C22—21no—
371C22—21no—
372C22—32no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—61no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—122no—
386C22—122no—
387C22#472711yescharacteristic
388C22D3×#9911yescharacteristic
389C22#474511yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—121no—
396C4—121no—
397C4—121no—
398C4—241no—
399C4—241no—
400C3#1155311yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—62no—
404C2—121no—
405C2—161no—
406C2#6812211yescharacteristic
407C2#6915011yescharacteristic
408C2D3×#35511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1318C2
socleC22×C3C2×#99
prime radicalC2×#938C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1318D3
hypocenterC3#11553
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #68144C2semidirect
3. D3×#324C2semidirect
4. C3×#1318C2semidirect
5. #67640C2semidirect
6. #67452C2semidirect
7. #67452C2semidirect
34. #1318D3semidirect
400. C3#11553semidirect

Maximal subgroups

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

subgroupindex
#763362
#681442
D3×#3242
C3×#13182
#676402
#674522
#115533