Finite Group Databasebeta

← #10696124

Subgroups — #10696124

Order 768.

542 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3109C211yesmaximal, characteristic
2#83320C211yesmaximal, characteristic, complemented
3#82315C211yesmaximal, characteristic, complemented
4#71798C211yesmaximal, characteristic, complemented
5#71615C211yesmaximal, characteristic, complemented
6#70813C211yesmaximal, characteristic, complemented
7#67755C211yesmaximal, characteristic, complemented
8#10027—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#481—21no—
13C3×#476—21no—
14C3×#459—21no—
15C3×#447—21no—
16C3×#442—22no—
17C3×#432C2211yescharacteristic
18#5218—21no—
19C2×#830—22no—
20#5094—21no—
21#5075—21no—
22#5041—22no—
23#5019—21no—
24#5019—21no—
25#5018—22no—
26#5008—21no—
27#5007—21no—
28#4970—21no—
29#4926C2211yescharacteristic
30C2×C4⋊(C3⋊C8)C2211yescharacteristic
31C2×#803C2211yescharacteristic
32#4540—22no—
33#4465—22no—
34#3246—31no—
35#3110—31no—
36#3109D311yescharacteristic
37#1955—31no—
38#1531—31no—
39#1530—31no—
40#1290—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×C3×D8—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#214—22no—
60C2×#214—22no—
61#926—21no—
62#924—21no—
63#924—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D412yes—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#106—22no—
72C3×C4⋊C8—21no—
73C3×#102—22no—
74C3×#102—22no—
75C3×C22⋊C8—21no—
76C3×#97D412yes—
77#832—22no—
78#832—22no—
79#830—21no—
80#830—44no—
81#810—22no—
82#810—22no—
83#808—22no—
84#807—22no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90#803D412yes—
91#533—31no—
92#532C2×D311yescharacteristic
93C2×C4×D4C2×D311yescharacteristic
94#484—61no—
95#481—21no—
96#476—21no—
97#476—61no—
98#475—61no—
99#474—61no—
100#465—62no—
101#461—61no—
102#461—61no—
103#460—61no—
104#459—21no—
105#450—61no—
106#447—21no—
107#442—22no—
108#440—62no—
109#432—31no—
110#432C2×D311yescharacteristic
111C2×C4⋊C8—31no—
112C2×C22⋊C8—62no—
113C2×#97—31no—
114#330—62no—
115#329—62no—
116C24×C3—21no—
117C3×#41—82no—
118C2×C3×Q8—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—42no—
125C2×C3×D4—41no—
126C22×C3×C4—21no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C2×Dic6—21no—
136C2×Dic6—21no—
137C3×D8—44no—
138C2×C3×C8—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151#217—41no—
152#214—44no—
153#214—82no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—42no—
167C2×#41—21no—
168C22×D4#6711yescharacteristic
169C23×C4—21no—
170C2×SD16—62no—
171C2×SD16—62no—
172C2×D8—22no—
173C22×C8—32no—
174C22×C8—61no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177C4⋊Q8#6711yescharacteristic
178#127#6711yescharacteristic
179#125—61no—
180#124—21no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183#121—21no—
184#121—21no—
185C4×Q8—61no—
186C4×Q8—61no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193#117—61no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C42C22×D311yescharacteristic
197#107—62no—
198#106—22no—
199#106—62no—
200C4⋊C8—21no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#103—62no—
205#103—62no—
206#102—22no—
207#102—22no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C22⋊C8—124no—
213#97—32no—
214#97—61no—
215#97—61no—
216#97#6712yes—
217C23×C3—22no—
218C23×C3—42no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3#12111yescharacteristic
224C3×Q8—81no—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—41no—
228C3×D4—41no—
229C3×D4—42no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—81no—
233C3×D4—82no—
234C2×C3×C4—22no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—82no—
242C2×C3×C4—82no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12711yescharacteristic
251C2×Dic3—41no—
252C2×Dic3—41no—
253Dic6—42no—
254Dic6—41no—
255Dic6—41no—
256C3×C8—42no—
257C3⋊C8—42no—
258C3⋊C8—42no—
259C3⋊C8—42no—
260C24—21no—
261#41—82no—
262C2×Q8—21no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—42no—
271C2×D4—41no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4D3×D411yescharacteristic
279C22×C4#23711yescharacteristic
280C22×C4C2×#6711yescharacteristic
281SD16—124no—
282SD16—242no—
283D8—44no—
284C2×C8—22no—
285C2×C8—62no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—121no—
289C2×C8—121no—
290C2×C8—122no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—121no—
300C4⋊C4—121no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—121no—
306C42—121no—
307C42D3×D411yescharacteristic
308C42D3×D411yescharacteristic
309C42C2×#6711yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—44no—
314C22×C3—42no—
315C22×C3—82no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3#39211yescharacteristic
322C22×C3#47112yes—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—81no—
329C3×C4—81no—
330C3×C4#45311yescharacteristic
331C3×C4#45411yescharacteristic
332Dic3—81no—
333Dic3—81no—
334C23—22no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23#93911yescharacteristic
341Q8—81no—
342Q8—122no—
343Q8—121no—
344Q8—121no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—42no—
350D4—41no—
351D4—41no—
352D4—81no—
353D4—82no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4#93711yescharacteristic
368C2×C4#93711yescharacteristic
369C2×C4#93811yescharacteristic
370C2×C4#93911yescharacteristic
371C2×C4#94011yescharacteristic
372C2×C4#95311yescharacteristic
373C8—42no—
374C8—122no—
375C8—122no—
376C8—122no—
377C2×C3—22no—
378C2×C3—82no—
379C2×C3—161no—
380C2×C3#200611yescharacteristic
381C2×C3#201011yescharacteristic
382C2×C3#201911yescharacteristic
383C22—21no—
384C22—21no—
385C22—44no—
386C22—42no—
387C22—82no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22#520012yes—
394C22#522311yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—81no—
401C4—81no—
402C4—241no—
403C4—241no—
404C4#519811yescharacteristic
405C4#520111yescharacteristic
406C3#1002711yescharacteristic, complemented, Sylow
407C2—22no—
408C2—82no—
409C2—161no—
410C2#7171511yescharacteristic
411C2#7209711yescharacteristic
412C2#7209811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3109C2
socleC22×C3#392
hypercenter#3109D3
hypocenterC3#10027
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
2. #83320C2semidirect
3. #82315C2semidirect
4. #71798C2semidirect
5. #71615C2semidirect
6. #70813C2semidirect
7. #67755C2semidirect
406. C3#10027semidirect

Maximal subgroups

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

subgroupindex
C3×#31092
#833202
#823152
#717982
#716152
#708132
#677552
#100273