Finite Group Databasebeta

← #10697624

Subgroups — #10697624

Order 768.

536 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2563C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic, complemented
3#82296C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10249—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#393—22no—
13C3×#390—21no—
14C3×#382—21no—
15C3×#380—22no—
16C3×C4×C4⋊C4C2211yescharacteristic
17#5215—21no—
18#5093—21no—
19#5091—22no—
20#5078—21no—
21#5009—22no—
22#5008—21no—
23#5007—21no—
24#4970—22no—
25#4964—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30D4⋊(C3⋊C8)—22no—
31#4452—22no—
32#3230—31no—
33#3077—31no—
34#2563D311yescharacteristic
35#1549—31no—
36#1503—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C4⋊C4D412yes—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58#926—21no—
59#924—21no—
60#924—21no—
61C2×#208—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#95—24no—
68C3×#95D412yes—
69#832—22no—
70#832—22no—
71#830—21no—
72#830—21no—
73#810—24no—
74#808—22no—
75#807—22no—
76#807—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×C4×D4C2×D311yescharacteristic
86#500—61no—
87#487—61no—
88#484—62no—
89#478—62no—
90#477—62no—
91#476—61no—
92#474—61no—
93#431—31no—
94#431—31no—
95#428—61no—
96#423—31no—
97#418—61no—
98#393—22no—
99#390—21no—
100#382—21no—
101#380—22no—
102C4×C4⋊C4C2×D311yescharacteristic
103#332—62no—
104#328—62no—
105D4⋊C8—62no—
106C24×C3—21no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—82no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4Q1612yes—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—44no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C2×Dic6—21no—
126C2×Dic6—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—84no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151#208—41no—
152#208—41no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C22×Q8—21no—
157C22×D4#6711yescharacteristic
158C23×C4—21no—
159C2×M16—61no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166#125—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175#117—61no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4#6711yescharacteristic
179C2×C4⋊C4#6712yes—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—24no—
182C2×C42—21no—
183C2×C42C22×D311yescharacteristic
184#107—62no—
185#107—64no—
186#106—62no—
187C4⋊C8—32no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#103—64no—
192#102—62no—
193#102—62no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196C4×C8—61no—
197C4×C8—61no—
198#95—24no—
199#95#6712yes—
200C23×C3—22no—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3#12011yescharacteristic
206C3×Q8—82no—
207C3×Q8—82no—
208C3×D4—22no—
209C3×D4—22no—
210C3×D4—41no—
211C3×D4—41no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—44no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—82no—
222C2×C3×C4—82no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229C2×C3×C4C2×Q1611yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4#13711yescharacteristic
232C2×C3×C4#13711yescharacteristic
233C2×Dic3—41no—
234C2×Dic3—41no—
235Dic6—42no—
236Dic6—81no—
237C3⋊C8—42no—
238C3⋊C8—42no—
239C3⋊C8—81no—
240C24—21no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—82no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4#21612yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—44no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261M16—121no—
262M16—121no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—121no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—84no—
282C22⋊C4—121no—
283C42—41no—
284C42—41no—
285C42—121no—
286C42D3×D411yescharacteristic
287C42D3×D411yescharacteristic
288C42D3×D411yescharacteristic
289C42#23711yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—44no—
293C22×C3—42no—
294C22×C3—82no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#39211yescharacteristic
298C22×C3#45111yescharacteristic
299C22×C3#45211yescharacteristic
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—82no—
304C3×C4—81no—
305C3×C4—82no—
306C3×C4#45111yescharacteristic
307C3×C4#45211yescharacteristic
308C3×C4#45412yes—
309C3×C4#46711yescharacteristic
310C3×C4#47011yescharacteristic
311Dic3—81no—
312Dic3—81no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23#95311yescharacteristic
319Q8—82no—
320Q8—82no—
321Q8—122no—
322Q8—241no—
323D4—22no—
324D4—22no—
325D4—41no—
326D4—41no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4#93711yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93811yescharacteristic
343C2×C4#93911yescharacteristic
344C2×C4#93911yescharacteristic
345C2×C4#94011yescharacteristic
346C2×C4#94211yescharacteristic
347C2×C4C2×#21611yescharacteristic
348C2×C4#95011yescharacteristic
349C2×C4#95111yescharacteristic
350C8—122no—
351C8—122no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—82no—
356C2×C3#200111yescharacteristic
357C2×C3#200311yescharacteristic
358C2×C3#200811yescharacteristic
359C22—21no—
360C22—21no—
361C22—44no—
362C22—42no—
363C22—82no—
364C22—81no—
365C22—81no—
366C22#522311yescharacteristic
367C22#522811yescharacteristic
368C22#523911yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—81no—
374C4—82no—
375C4—241no—
376C4—241no—
377C4#504711yescharacteristic
378C4#505011yescharacteristic
379C4#505111yescharacteristic
380C4#505211yescharacteristic
381C4#506212yes—
382C3#1024911yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—82no—
386C2#7170011yescharacteristic
387C2#7173011yescharacteristic
388C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2563C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#2563D3
hypocenterC3#10249
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
2. #83224C2semidirect
4. #71775C2semidirect
6. #69471C2semidirect
7. #67620C2semidirect
382. C3#10249semidirect

Maximal subgroups

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

subgroupindex
C3×#25632
#832242
#822962
#717752
#715972
#694712
#676202
#102493