Finite Group Databasebeta

← #10698448

Subgroups — #10698448

Order 768.

526 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 75 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#73845C211yesmaximal, characteristic
2C3×#1819C211yesmaximal, characteristic
3#71542C211yesmaximal, characteristic
4#71542C211yesmaximal, characteristic
5C3⋊#1304C211yesmaximal, characteristic
6C3⋊#1304C211yesmaximal, characteristic
7C3⋊#1304C211yesmaximal, characteristic
8#10393—31nomaximal, Sylow
9C22×C3×C22⋊C4C2211yescharacteristic
10C3×#379C2211yescharacteristic
11C3×#379C2211yescharacteristic
12#5247C2211yescharacteristic
13#5247C2211yescharacteristic
14#5247C2211yescharacteristic
15#5247C2211yescharacteristic
16#4949—22no—
17#4948—22no—
18#4948—22no—
19C3×#342—22no—
20#4526—22no—
21#4526—22no—
22#2842—31no—
23#1819—31no—
24#1819—31no—
25#1819D311yescharacteristic
26#1304—31no—
27#1304—31no—
28#1304—31no—
29C25×C3C2311yescharacteristic
30C23×C3×C4Q811yescharacteristic
31C23×C3×C4Q811yescharacteristic
32C2×C3×C22⋊C4—21no—
33C2×C3×C22⋊C4—21no—
34C2×C3×C22⋊C4—21no—
35C2×C3×C22⋊C4—21no—
36C2×C3×C22⋊C4—21no—
37C2×C3×C22⋊C4—21no—
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4D412yes—
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217D412yes—
55C2×#217D412yes—
56C2×C4⋊Dic3—22no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4—22no—
59C3⋊#95—22no—
60C22×C22⋊C4C2×D311yescharacteristic
61#399—62no—
62#396—62no—
63#396—62no—
64#379—31no—
65#379—31no—
66#379—31no—
67#379—31no—
68#379C2×D311yescharacteristic
69#379C2×D311yescharacteristic
70#342—22no—
71#342—62no—
72#342—62no—
73C24×C3—42no—
74C24×C3—81no—
75C24×C3—81no—
76C24×C3C2×D411yescharacteristic
77C24×C3C2×D411yescharacteristic
78C24×C3C2×D411yescharacteristic
79C24×C3C2×Q811yescharacteristic
80C24×C3#4111yescharacteristic
81C24×C3#4111yescharacteristic
82C24×C3#4111yescharacteristic
83C22×C3×C4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—41no—
91C22×C3×C4—81no—
92C22×C3×C4—81no—
93C22×Dic3—22no—
94C22×Dic3—22no—
95C22×Dic3—41no—
96C22×Dic3—41no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—42no—
102C3×C22⋊C4—42no—
103C3×C22⋊C4—81no—
104C3×C22⋊C4—81no—
105C3×C22⋊C4—81no—
106C3×C22⋊C4—81no—
107#217—42no—
108#217—42no—
109#217—42no—
110#217—42no—
111#217—81no—
112#217—81no—
113#217—81no—
114#217—81no—
115#217—81no—
116#217—81no—
117#217—81no—
118#217—81no—
119C4⋊Dic3—82no—
120Dic3⋊C4—82no—
121Dic3⋊C4—82no—
122C25C22×D311yescharacteristic
123C23×C4Dic611yescharacteristic
124C23×C4Dic611yescharacteristic
125C2×C4⋊C4—62no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—62no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—32no—
140C2×C22⋊C4—32no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4#6712yes—
152#95—62no—
153C23×C3—22no—
154C23×C3—22no—
155C23×C3—22no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—88no—
175C23×C3#12011yescharacteristic
176C23×C3C22⋊Q811yescharacteristic
177C23×C3C22⋊Q811yescharacteristic
178C23×C3C22⋊Q811yescharacteristic
179C23×C3#12411yescharacteristic
180C23×C3#12411yescharacteristic
181C23×C3#12411yescharacteristic
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C24—42no—
209C24—81no—
210C24—81no—
211C24C2×Dic611yescharacteristic
212C24#23511yescharacteristic
213C24D3×D411yescharacteristic
214C24D3×D411yescharacteristic
215C24#23711yescharacteristic
216C24#23711yescharacteristic
217C24C2×#6711yescharacteristic
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—121no—
231C22×C4—121no—
232C4⋊C4—242no—
233C4⋊C4—242no—
234C4⋊C4—242no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C22⋊C4—122no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—122no—
249C22⋊C4—241no—
250C22⋊C4—241no—
251C22⋊C4—241no—
252C22⋊C4—241no—
253C22⋊C4—241no—
254C22⋊C4—241no—
255C22⋊C4—241no—
256C22⋊C4—241no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—44no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3—82no—
281C22×C3—82no—
282C22×C3—82no—
283C22×C3—82no—
284C22×C3#39311yescharacteristic
285C22×C3#45812yes—
286C3×C4—82no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—161no—
290Dic3—82no—
291Dic3—82no—
292Dic3—161no—
293Dic3—161no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—88no—
316C23#87811yescharacteristic
317C23#87811yescharacteristic
318C23#88511yescharacteristic
319C23#88511yescharacteristic
320C23#92411yescharacteristic
321C23#93611yescharacteristic
322C23#93711yescharacteristic
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—241no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4—241no—
347C2×C4—241no—
348C2×C4—241no—
349C2×C3—22no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3#203011yescharacteristic
357C2×C3#203211yescharacteristic
358C2×C3#203211yescharacteristic
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—44no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22#494711yescharacteristic
387C22#505312yes—
388C4—82no—
389C4—81no—
390C4—81no—
391C4—161no—
392C4—242no—
393C4—242no—
394C4—481no—
395C4—481no—
396C3#1039311yescharacteristic, complemented, Sylow
397C2—22no—
398C2—41no—
399C2—41no—
400C2—41no—
401C2—41no—
402C2—41no—
403C2—41no—
404C2#6953611yescharacteristic
405C2#7002511yescharacteristic
406C2#7002511yescharacteristic
subgroupG/H
centerC22#4947
commutator, prime radical, cosocleC25×C3C23
FrattiniC25C22×D3
FittingC3×#1819C2
socleC22×C3#393
hypercenter#1819D3
hypocenterC3#10393
LCSG ▷ C25×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
396. C3#10393semidirect

Maximal subgroups

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

subgroupindex
#738452
C3×#18192
#715422
C3⋊#13042
#103933